Number of Sylow $3$ subgroups of $S_4$,not knowing the structure of $S_4$

Question

I am trying to find out the number of Sylow $3$ subgroups of $S_4$, Order of $S_4$ is $24$,we write $24$ as $24=2*2*2*3$, so we both have sylow $2$ subgroups and sylow $3$ subgroups. Let $n_2$ and $n_3$ are their respective numbers. By using Sylow's 1st Theorem The possible values of $n_3$ are $1,4$ and for $n_2=1,3$.We know $S_4$ has order $3$ for the elements $ (1,2,3),(2,3,4),(3,4,1),(1,2,4)$,Each of them generates distinct cyclic subgroups of $S_4$,Thus we get $n_3=4$.Now we can think each of Sylow $2$ subgroups as the rotation,reflection of a square having vertices $1,2,3,4$ about the axes through the center which is $D_4$.We get such three Sylow $3$ subgroups ie.$n_2=3$.I have solved my problem using the structure of $S_4$,But suppose I have no such idea on the algebraic structure of $S_4$ and trying to do it using only Sylow's Theorems.can I be successful?Is it necessary to study the algebraic structure of any Symmetric group to find it's Sylow $p$ subgroups?