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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