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
Also, according to this 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.