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# List of the 230 crystallographic 3D space groups

## Triclinic system

Class # Space groups
$1,!$ 1 $mathrm\left\{P\right\}1,!$
$bar\left\{1\right\}$ 2 $mathrm\left\{P\right\}bar\left\{1\right\}$

## Monoclinic system

Class # Space groups
$2,!$ 3-5 $mathrm\left\{P\right\}2,!$ $mathrm\left\{P\right\}2_\left\{1\right\},!$ $mathrm\left\{C\right\}2,!$
$mathrm\left\{m\right\},!$ 6-9 $mathrm\left\{Pm\right\},!$ $mathrm\left\{Pc\right\},!$ $mathrm\left\{Cm\right\},!$ $mathrm\left\{Cc\right\},!$
$2/m,!$ 10-15 $mathrm\left\{P\right\}2/mathrm\left\{m\right\},!$ $mathrm\left\{P\right\}2_\left\{1\right\}/mathrm\left\{m\right\},!$ $mathrm\left\{C\right\}2/mathrm\left\{m\right\},!$ $mathrm\left\{P\right\}2/mathrm\left\{c\right\},!$
$mathrm\left\{P\right\}2_\left\{1\right\}/mathrm\left\{c\right\},!$ $mathrm\left\{C\right\}2/mathrm\left\{c\right\},!$

## Orthorhombic system

Class # Space groups
$222,!$ 16-24 $mathrm\left\{P\right\}222,!$ $mathrm\left\{P\right\}222_\left\{1\right\},!$ $mathrm\left\{P\right\}2_\left\{1\right\}2_\left\{1\right\}2,!$ $mathrm\left\{P\right\}2_\left\{1\right\}2_\left\{1\right\}2_\left\{1\right\},!$ $mathrm\left\{C\right\}222_\left\{1\right\},!$ $mathrm\left\{C\right\}222,!$ $mathrm\left\{F\right\}222,!$ $mathrm\left\{I\right\}222,!$
$mathrm\left\{I\right\}2_\left\{1\right\}2_\left\{1\right\}2_\left\{1\right\},!$
$mathrm\left\{mm\right\}2,!$ 25-46 $mathrm\left\{Pmm\right\}2,!$ $mathrm\left\{Pmc\right\}2_\left\{1\right\},!$ $mathrm\left\{Pcc\right\}2,!$ $mathrm\left\{Pma\right\}2,!$ $mathrm\left\{Pca\right\}2_\left\{1\right\},!$ $mathrm\left\{Pnc\right\}2,!$ $mathrm\left\{Pmn\right\}2_\left\{1\right\},!$ $mathrm\left\{Pba\right\}2,!$
$mathrm\left\{Pna\right\}2_\left\{1\right\},!$ $mathrm\left\{Pnn\right\}2,!$ $mathrm\left\{Cmm\right\}2,!$ $mathrm\left\{Cmc\right\}2_\left\{1\right\},!$ $mathrm\left\{Ccc\right\}2,!$ $mathrm\left\{Amm\right\}2,!$ $mathrm\left\{Aem\right\}2,!$ $mathrm\left\{Ama\right\}2,!$
$mathrm\left\{Aea\right\}2,!$ $mathrm\left\{Fmm\right\}2,!$ $mathrm\left\{Fdd\right\}2,!$ $mathrm\left\{Imm\right\}2,!$ $mathrm\left\{Iba\right\}2,!$ $mathrm\left\{Ima\right\}2,!$
$mathrm\left\{mmm\right\},!$ 47-74 $mathrm\left\{Pmmm\right\},!$ $mathrm\left\{Pnnn\right\},!$ $mathrm\left\{Pccm\right\},!$ $mathrm\left\{Pban\right\},!$ $mathrm\left\{Pmma\right\},!$ $mathrm\left\{Pnna\right\},!$ $mathrm\left\{Pmna\right\},!$ $mathrm\left\{Pcca\right\},!$
$mathrm\left\{Pbam\right\},!$ $mathrm\left\{Pccn\right\},!$ $mathrm\left\{Pbcm\right\},!$ $mathrm\left\{Pnnm\right\},!$ $mathrm\left\{Pmmn\right\},!$ $mathrm\left\{Pbcn\right\},!$ $mathrm\left\{Pbca\right\},!$ $mathrm\left\{Pnma\right\},!$
$mathrm\left\{Cmcm\right\},!$ $mathrm\left\{Cmce\right\},!$ $mathrm\left\{Cmmm\right\},!$ $mathrm\left\{Cccm\right\},!$ $mathrm\left\{Cmme\right\},!$ $mathrm\left\{Ccce\right\},!$ $mathrm\left\{Fmmm\right\},!$ $mathrm\left\{Fddd\right\},!$
$mathrm\left\{Immm\right\},!$ $mathrm\left\{Ibam\right\},!$ $mathrm\left\{Ibca\right\},!$ $mathrm\left\{Imma\right\},!$

Note. An e plane is a double glide plane, one having glides in two different directions. They are found in five space groups, all in the orthorhombic system and with a centered lattice. The use of the symbol e became official with the 5th edition of the International tables for crystallography, volume A (2002).

## Tetragonal system

Class # Space groups
$4,!$ 75-80 $mathrm\left\{P\right\}4,!$ $mathrm\left\{P\right\}4_\left\{1\right\},!$ $mathrm\left\{P\right\}4_\left\{2\right\},!$ $mathrm\left\{P\right\}4_\left\{3\right\},!$ $mathrm\left\{I\right\}4,!$ $mathrm\left\{I\right\}4_\left\{1\right\},!$
$bar\left\{4\right\},!$ 81-82 $mathrm\left\{P\right\}bar\left\{4\right\},!$ $mathrm\left\{I\right\}bar\left\{4\right\},!$
$bar\left\{4\right\}/mathrm\left\{m\right\},!$ 83-88 $mathrm\left\{P\right\}4/mathrm\left\{m\right\},!$ $mathrm\left\{P\right\}4_\left\{2\right\}/mathrm\left\{m\right\},!$ $mathrm\left\{P\right\}4/mathrm\left\{n\right\},!$ $mathrm\left\{P\right\}4_\left\{2\right\}/mathrm\left\{n\right\},!$ $mathrm\left\{I\right\}4/mathrm\left\{m\right\},!$ $mathrm\left\{I\right\}4_\left\{1\right\}/mathrm\left\{a\right\},!$
$422,!$ 89-98 $mathrm\left\{P\right\}422,!$ $mathrm\left\{P\right\}42_\left\{1\right\}2,!$ $mathrm\left\{P\right\}4_\left\{1\right\}22,!$ $mathrm\left\{P\right\}4_\left\{1\right\}2_\left\{1\right\}2,!$ $mathrm\left\{P\right\}4_\left\{2\right\}22,!$ $mathrm\left\{P\right\}4_\left\{2\right\}2_\left\{1\right\}2,!$
$mathrm\left\{P\right\}4_\left\{3\right\}22,!$ $mathrm\left\{P\right\}4_\left\{3\right\}2_\left\{1\right\}2,!$ $mathrm\left\{I\right\}422,!$ $mathrm\left\{I\right\}4_\left\{1\right\}22,!$
$4mathrm\left\{mm\right\},!$ 99-110 $mathrm\left\{P\right\}4mathrm\left\{mm\right\},!$ $mathrm\left\{P\right\}4mathrm\left\{bm\right\},!$ $mathrm\left\{P\right\}4_\left\{2\right\}mathrm\left\{cm\right\},!$ $mathrm\left\{P\right\}4_\left\{2\right\}mathrm\left\{nm\right\},!$ $mathrm\left\{P\right\}4mathrm\left\{cc\right\},!$ $mathrm\left\{P\right\}4mathrm\left\{nc\right\},!$
$mathrm\left\{P\right\}4_\left\{2\right\}mathrm\left\{mc\right\},!$ $mathrm\left\{P\right\}4_\left\{2\right\}mathrm\left\{bc\right\},!$ $mathrm\left\{I\right\}4mathrm\left\{mm\right\},!$ $mathrm\left\{I\right\}4mathrm\left\{cm\right\},!$ $mathrm\left\{I\right\}4_\left\{1\right\}mathrm\left\{md\right\},!$ $mathrm\left\{I\right\}4_\left\{1\right\}mathrm\left\{cd\right\},!$
$bar\left\{4\right\}2mathrm\left\{m\right\},!$ 111-122 $mathrm\left\{P\right\}bar\left\{4\right\}2mathrm\left\{m\right\},!$ $mathrm\left\{P\right\}bar\left\{4\right\}2mathrm\left\{c\right\},!$ $mathrm\left\{P\right\}bar\left\{4\right\}2_\left\{1\right\}mathrm\left\{m\right\},!$ $mathrm\left\{P\right\}bar\left\{4\right\}2_\left\{1\right\}mathrm\left\{c\right\},!$ $mathrm\left\{P\right\}bar\left\{4\right\}mathrm\left\{m\right\}2,!$ $mathrm\left\{P\right\}bar\left\{4\right\}mathrm\left\{c\right\}2,!$
$mathrm\left\{P\right\}bar\left\{4\right\}mathrm\left\{b\right\}2,!$ $mathrm\left\{P\right\}bar\left\{4\right\}mathrm\left\{n\right\}2,!$ $mathrm\left\{I\right\}bar\left\{4\right\}mathrm\left\{m\right\}2,!$ $mathrm\left\{I\right\}bar\left\{4\right\}mathrm\left\{c\right\}2,!$ $mathrm\left\{I\right\}bar\left\{4\right\}2mathrm\left\{m\right\},!$ $mathrm\left\{I\right\}bar\left\{4\right\}2mathrm\left\{d\right\},!$
$4/mathrm\left\{mmm\right\},!$ 123-142 $mathrm\left\{P\right\}4/mathrm\left\{mmm\right\},!$ $mathrm\left\{P\right\}4/mathrm\left\{mcc\right\},!$ $mathrm\left\{P\right\}4/mathrm\left\{nbm\right\},!$ $mathrm\left\{P\right\}4/mathrm\left\{nnc\right\},!$ $mathrm\left\{P\right\}4/mathrm\left\{mbm\right\},!$ $mathrm\left\{P\right\}4/mathrm\left\{mnc\right\},!$
$mathrm\left\{P\right\}4/mathrm\left\{nmm\right\},!$ $mathrm\left\{P\right\}4/mathrm\left\{ncc\right\},!$ $mathrm\left\{P\right\}4_\left\{2\right\}/mathrm\left\{mmc\right\},!$ $mathrm\left\{P\right\}4_\left\{2\right\}/mathrm\left\{mcm\right\},!$ $mathrm\left\{P\right\}4_\left\{2\right\}/mathrm\left\{nbc\right\},!$ $mathrm\left\{P\right\}4_\left\{2\right\}/mathrm\left\{nnm\right\},!$
$mathrm\left\{P\right\}4_\left\{2\right\}/mathrm\left\{mbc\right\},!$ $mathrm\left\{P\right\}4_\left\{2\right\}/mathrm\left\{mnm\right\},!$ $mathrm\left\{P\right\}4_\left\{2\right\}/mathrm\left\{nmc\right\},!$ $mathrm\left\{P\right\}4_\left\{2\right\}/mathrm\left\{ncm\right\},!$ $mathrm\left\{I\right\}4/mathrm\left\{mmm\right\},!$ $mathrm\left\{I\right\}4/mathrm\left\{mcm\right\},!$
$mathrm\left\{I\right\}4_\left\{1\right\}/mathrm\left\{amd\right\},!$ $mathrm\left\{I\right\}4_\left\{1\right\}/mathrm\left\{acd\right\},!$

## Rhombohedral (trigonal) system

Class # Space groups
$3,!$ 143-146 $mathrm\left\{P\right\}3,!$ $mathrm\left\{P\right\}3_\left\{1\right\},!$ $mathrm\left\{P\right\}3_\left\{2\right\},!$ $mathrm\left\{R\right\}3,!$
$bar\left\{3\right\},!$ 147-148 $mathrm\left\{P\right\}bar\left\{3\right\},!$ $mathrm\left\{R\right\}bar\left\{3\right\},!$
$32,!$ 149-155 $mathrm\left\{P\right\}312,!$ $mathrm\left\{P\right\}321,!$ $mathrm\left\{P\right\}3_\left\{1\right\}12,!$ $mathrm\left\{P\right\}3_\left\{1\right\}21,!$ $mathrm\left\{P\right\}3_\left\{2\right\}12,!$ $mathrm\left\{P\right\}3_\left\{2\right\}21,!$ $mathrm\left\{R\right\}32,!$
$3mathrm\left\{m\right\},!$ 156-161 $mathrm\left\{P\right\}3mathrm\left\{m\right\}1,!$ $mathrm\left\{P\right\}31mathrm\left\{m\right\},!$ $mathrm\left\{P\right\}3mathrm\left\{c\right\}1,!$ $mathrm\left\{P\right\}31mathrm\left\{c\right\},!$ $mathrm\left\{R\right\}3mathrm\left\{m\right\},!$ $mathrm\left\{R\right\}3mathrm\left\{c\right\},!$
$bar\left\{3\right\}mathrm\left\{m\right\}$ 162-167 $mathrm\left\{P\right\}bar\left\{3\right\}1mathrm\left\{m\right\},!$ $mathrm\left\{P\right\}bar\left\{3\right\}1mathrm\left\{c\right\},!$ $mathrm\left\{P\right\}bar\left\{3\right\}mathrm\left\{m\right\}1,!$ $mathrm\left\{P\right\}bar\left\{3\right\}mathrm\left\{c\right\}1,!$ $mathrm\left\{R\right\}bar\left\{3\right\}mathrm\left\{m\right\},!$ $mathrm\left\{R\right\}bar\left\{3\right\}mathrm\left\{c\right\},!$

## Hexagonal system

Class # Space groups
$6,!$ 168-173 $mathrm\left\{P\right\}6,!$ $mathrm\left\{P\right\}6_\left\{1\right\},!$ $mathrm\left\{P\right\}6_\left\{5\right\},!$ $mathrm\left\{P\right\}6_\left\{2\right\},!$ $mathrm\left\{P\right\}6_\left\{4\right\},!$ $mathrm\left\{P\right\}6_\left\{3\right\},!$
$bar\left\{6\right\},!$ 174 $mathrm\left\{P\right\}bar\left\{6\right\},!$
$6/mathrm\left\{m\right\},!$ 175-176 $mathrm\left\{P\right\}6/mathrm\left\{m\right\},!$ $mathrm\left\{P\right\}6_\left\{3\right\}/mathrm\left\{m\right\},!$
$622,!$ 177-182 $mathrm\left\{P\right\}622,!$ $mathrm\left\{P\right\}6_\left\{1\right\}22,!$ $mathrm\left\{P\right\}6_\left\{5\right\}22,!$ $mathrm\left\{P\right\}6_\left\{2\right\}22,!$ $mathrm\left\{P\right\}6_\left\{4\right\}22,!$ $mathrm\left\{P\right\}6_\left\{3\right\}22,!$
$6mathrm\left\{mm\right\},!$ 183-186 $mathrm\left\{P\right\}6mathrm\left\{mm\right\},!$ $mathrm\left\{P\right\}6mathrm\left\{cc\right\},!$ $mathrm\left\{P\right\}6_\left\{3\right\}mathrm\left\{cm\right\},!$ $mathrm\left\{P\right\}6_\left\{3\right\}mathrm\left\{mc\right\},!$
$bar\left\{6\right\}mathrm\left\{m\right\}2,!$ 187-190 $mathrm\left\{P\right\}bar\left\{6\right\}mathrm\left\{m\right\}2,!$ $mathrm\left\{P\right\}bar\left\{6\right\}mathrm\left\{c\right\}2,!$ $mathrm\left\{P\right\}bar\left\{6\right\}2mathrm\left\{m\right\},!$ $mathrm\left\{P\right\}bar\left\{6\right\}2mathrm\left\{c\right\},!$
$6/mathrm\left\{mmm\right\},!$ 191-194 $mathrm\left\{P\right\}6/mathrm\left\{mmm\right\},!$ $mathrm\left\{P\right\}6/mathrm\left\{mcc\right\},!$ $mathrm\left\{P\right\}6_\left\{3\right\}/mathrm\left\{mcm\right\},!$ $mathrm\left\{P\right\}6_\left\{3\right\}/mathrm\left\{mmc\right\},!$

## Cubic system

Class # Space groups
$23,!$ 195-199 $mathrm\left\{P\right\}23,!$ $mathrm\left\{F\right\}23,!$ $mathrm\left\{I\right\}23,!$ $mathrm\left\{P\right\}2_\left\{1\right\}3,!$ $mathrm\left\{I\right\}2_\left\{1\right\}3,!$
$mathrm\left\{m\right\}bar\left\{3\right\},!$ 200-206 $mathrm\left\{Pm\right\}bar\left\{3\right\},!$ $mathrm\left\{Pn\right\}bar\left\{3\right\},!$ $mathrm\left\{Fm\right\}bar\left\{3\right\},!$ $mathrm\left\{Fd\right\}bar\left\{3\right\},!$ $mathrm\left\{Im\right\}bar\left\{3\right\},!$ $mathrm\left\{Pa\right\}bar\left\{3\right\},!$ $mathrm\left\{Ia\right\}bar\left\{3\right\},!$
$432,!$ 207-214 $mathrm\left\{P\right\}432,!$ $mathrm\left\{P\right\}4_\left\{2\right\}32,!$ $mathrm\left\{F\right\}432,!$ $mathrm\left\{F\right\}4_\left\{1\right\}32,!$ $mathrm\left\{I\right\}432,!$ $mathrm\left\{P\right\}4_\left\{3\right\}32,!$ $mathrm\left\{P\right\}4_\left\{1\right\}32,!$ $mathrm\left\{I\right\}4_\left\{1\right\}32,!$
$bar\left\{4\right\}3mathrm\left\{m\right\},!$ 215-220 $mathrm\left\{P\right\}bar\left\{4\right\}3mathrm\left\{m\right\},!$ $mathrm\left\{F\right\}bar\left\{4\right\}3mathrm\left\{m\right\},!$ $mathrm\left\{I\right\}bar\left\{4\right\}3mathrm\left\{m\right\},!$ $mathrm\left\{P\right\}bar\left\{4\right\}3mathrm\left\{n\right\},!$ $mathrm\left\{F\right\}bar\left\{4\right\}3mathrm\left\{c\right\},!$ $mathrm\left\{I\right\}bar\left\{4\right\}3mathrm\left\{d\right\},!$
$mathrm\left\{m\right\}bar\left\{3\right\}mathrm\left\{m\right\},!$ 221-230 $mathrm\left\{Pm\right\}bar\left\{3\right\}mathrm\left\{m\right\},!$ $mathrm\left\{Pn\right\}bar\left\{3\right\}mathrm\left\{n\right\},!$ $mathrm\left\{Pm\right\}bar\left\{3\right\}mathrm\left\{n\right\},!$ $mathrm\left\{Pn\right\}bar\left\{3\right\}mathrm\left\{m\right\},!$ $mathrm\left\{Fm\right\}bar\left\{3\right\}mathrm\left\{m\right\},!$ $mathrm\left\{Fm\right\}bar\left\{3\right\}mathrm\left\{c\right\},!$ $mathrm\left\{Fd\right\}bar\left\{3\right\}mathrm\left\{m\right\},!$ $mathrm\left\{Fd\right\}bar\left\{3\right\}mathrm\left\{c\right\},!$
$mathrm\left\{Im\right\}bar\left\{3\right\}mathrm\left\{m\right\},!$ $mathrm\left\{Ia\right\}bar\left\{3\right\}mathrm\left\{d\right\},!$