The number of abelian groups of order $n$ (call it $a(n)$ is a studied subject (see <http://oeis.org/A000688>), but I can't seem to find any asymptotic results. Obviously, there is no asymptotic for $a(n)$ itself (it is much too irregular), but there should an asymptotic for $\sum\_{n\leq x} a(n),$ but I can't seem to find a reference.

Roberto answered the above, but another question is whether one has any distributional results (how high are the maxima, is there a limiting distribution, etc).