number of maximal subgroups of the symmetric group What is the asymptotics of the number of the maximal subgroups of $S_n$ (as a function of $n$)? This must be written down somewhere...
EDIT I am actually more interested in the number of conjugacy classes of maximal subgroups (the difference is graphically illustrated by Derek's and Gerry's comments.)
 A: With regard to the conjugacy class question, you should refer to this:

Liebeck, Martin W.; Shalev, Aner Maximal subgroups of symmetric groups.
  J. Comb. Theory, Ser. A 75, No.2, 341-352 (1996).

The following is a quote from the ZBMath review by W. Knapp:

The purpose of this paper is to give estimates on the number of conjugacy classes of maximal subgroups of the finite symmetric groups $S_n$, on $n$ letters in terms of $n$. It is shown that this number is of the form $(\frac12+o(1))n$. The main work has to be done in establishing that $S_n$ has at most $n^{6/11+o(1)}$ conjugacy classes of primitive maximal subgroups. Of course, the O’Nan-Scott Theorem and the classification of finite simple groups are used.

The same paper is also relevant to the original question of the OP (about the number of maximal subgroups). A further quote from the review:

In the course of the proof, it is shown that any finite almost simple group has at most $n^{17/11+o(1)}$ maximal subgroups of index $n$.

