As Francesco Polizzi mentions, the answer is no alredy for ${\frak S}_4$: there are $30$ subgroups, but $4!=24$.

Here are some more (small) calculations:

For ${\frak S}_5$, there are $156$ subgroups ($5!=120$).

For ${\frak S}_6$, there are $1455$ subgroups ($6!=720$).

For ${\frak S}_7$, there are $11300$ subgroups ($7!=5040$).

You can find the number of subgroups up to ${\frak S}_{18}$ in here:

* Derek Holt, [Enumerating subgroups of the symmetric group](https://homepages.warwick.ac.uk/~mareg/download/papers/symsubs/symsubs.pdf)

Also, according to [**this**](http://math.stackexchange.com/q/79139) math.stackexchange, the asymptotics for the number of subgroups is $\log(\#\text{sub}) = \Theta(n^2)$.

As for the number of subgroups for arbitrary ${\frak S}_n$, the problem seems wide open.