What Is the Conjugate Base of H2SO4?
The University of Waterloo science page lists HSO4 as the conjugate base of H2SO4. H2SO4 is the chemical name for sulfuric acid, and the conjugate base is hydrogen sulfate. Even though HSO4 is also an acid, it can either be a conjugate acid or a conjugate base depending on context.
A conjugate base is not necessarily a basic molecule. Conjugate bases are defined as molecules that are formed when an acid loses a hydrogen ion. Although these molecules may still be acidic in most situations, they can still regain that lost hydrogen ion to form the original acid. Therefore, it makes the most sense to talk about conjugate acids and bases in pairs.
The concept of conjugate acids and bases comes from the Bronsted-Lowry definition of acids. According to this definition, an acid is any molecule that can "donate" or give up a hydrogen ion (a proton). Bases are molecules that can "accept" or take a proton.
In the case of sulfuric acid and hydrogen sulfate, sulfuric acid can never be a conjugate base because it is impossible for sulfuric acid to accept another proton. Hydrogen sulfate, the conjugate base of sulfuric acid, can also lose its hydrogen ion to produce sulfate (SO4). In that case, hydrogen sulfate is the conjugate acid of sulfate, and sulfate is the conjugate base of hydrogen sulfate.