If you don't like Andrew Stacey's choice of free groups, consider the group of (finite or infinite, as you like) permutations on a set G. As in Andrew's example, a group isomorphism is a set isomorphism, so again there are at least X-many group isomorphism classes, where X "counts" the number of set-isomorphism classes, e.g. cardinals. I prefer using Choice to simplify arguments, but I suspect the above example can be done in ZF without choice, especially if you restrict to torsion permutation groups.

Gerhard "Ask Me About System Design" Paseman, 2010.01.25