Maximal Subgroups of Symmetric Groups

We show thatSnhas at mostn6/11+o(1)conjugacy classes of primitive maximal subgroups. This improves annclog3nbound given by Babai. We conclude that the number of conjugacy classes of maximal subgroups ofSnis of the form (12+o(1))n. It also follows that, for largen,Snhas less thann ! maximal subgroups. This confirms a special case of a conjecture of Wall. Improving a recent result from [MSh], we also show that any finite almost simple group has at mostn17/11+o(1)maximal subgroups of indexn.