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.)



number of maximal subgroups of the symmetric group
With regard to the conjugacy class question, you should refer to this:



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> Liebeck, Martin W.; Shalev, Aner *Maximal subgroups of symmetric groups*.
>  J. Comb. Theory, Ser. A 75, No.2, 341-352 (1996).
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The following is a quote from the ZBMath review by W. Knapp:



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> 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.
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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:



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> 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 $n$.
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