space groups

List of the 230 crystallographic 3D space groups

Triclinic system

Class # Space groups
1,! 1 mathrm{P}1,!
bar{1} 2 mathrm{P}bar{1}

Monoclinic system

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

Orthorhombic system

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

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

Rhombohedral (trigonal) system

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

Hexagonal system

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

Cubic system

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

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