If you look at my answer to [this question,][1] you will see that the number of proper subgroups grows *at least* as fast as $2^{n^2/6},$ so **much** faster than $n!.$ This is a result of Laci Pyber.


  [1]: http://mathoverflow.net/questions/211260/which-subgroup-order-of-the-symmetric-group-is-the-most-frequent/211263#211263