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).



Asymptotics for the number of abelian groups of order at most $x.$
The problem was studied quite a bit; a complete summary will be complicated to give (in any case I cannot). A standard reference for classical results on this is **A. Ivić "The Riemann Zeta-function: Theory and Applications"** (1985); it seems there is a recent Dover edition. Ivić has various papers on this problem, too. The book has a chapter on this subject (14.5 Non-isomorphic abelian groups of a given order). At the moment I cannot recall what exactly is in there.


**Some information related to this type of problem:**


The correct asymptotics for $\sum\_{n\in \mathbb{N}} a(n)$ were obtained by 
Erdős and Szekeres (1934), they proved: 
$$\sum\_{n\le x} a(n) \sim A x + O(\sqrt{x})$$
with $A= \prod\_{n \ge 2}\zeta(n)$. 


By now there are more precise results known. For example the paper by Kendall and Rankin (1947), mentioned in another answer, gives $Ax + B x^{1/2} + O(x^{1/3} \log x )$, and developments continued, like the result mentioned by Roberto Pignatelli.


The to my knowledge latest improvement there is by Sargos and Wu (2000)
$$Ax + Bx^{1/2} + Cx^{1/3} + O(x^{55/219} (\log x)^7 )$$
The first to get the third term of the main term was Richert (1952). 
For the historical development of the error see the introduction of a paper by Calderón (2003).


Related questions were also studied. For example, Kendall and Rankin (1947) showed that the "local densities" of $a(n)$ exist, that is $\sum\_{n \le x, \, a(n)=k}1 \sim d\_k x$ (good error terms are known). 
There are also estimates "in short intervals" so on $\sum\_{ x \le n \le y, \, a(n)=k}1$.


For two recent papers on this see: 


*Emre Alkan*, [**On the enumeration of finite abelian and solvable groups**](http://dx.doi.org/10.1016/S0022-314X(03)00055-6), *J. Number Theory* **101** (2003), no. 2, 404--423.


*Ekkehard Krätzel*, [**The distribution of values of the enumerating function of finite, non-isomorphic abelian groups in short intervals**](http://dx.doi.org/10.1007/s00013-008-2732-7), *Arch. Math. (Basel)* **91** (2008), no. 6, 518--525.


The introductions and references there will lead to various additional paper. (The first is freely accessible.) 


*References*


C. Calderón: Asymptotic estimates on finite abelian groups. Publications De L’institut Mathematique, Nouvelle série, 74, 57-70 (2003).


P. Erdős, G. Szekeres: Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem (in German), Acta Litt. Sci. Szeged 7 (1934), 95--102; Zentralblatt 10,294. [Free PDF](https://www.renyi.hu/~p_erdos/1934-05.pdf)


*D. G. Kendall and R. A. Rankin*, [**On the number of Abelian groups of a given order**](http://www.ams.org/mathscinet-getitem?mr=22569), *Quart. J. Math., Oxford Ser.* **18** (1947), 197--208.


*Hans-Egon Richert*, [**Über die Anzahl Abelscher Gruppen gegebener Ordnung. I**](http://www.ams.org/mathscinet-getitem?mr=50577), *Math. Z.* **56** (1952), 21--32.


*P. Sargos and J. Wu*, [**Multiple exponential sums with monomials and their applications in number theory**](http://dx.doi.org/10.1023/A:1006777803163), *Acta Math. Hungar.* **87** (2000), no. 4, 333--354. 


