Let $G$ be a group of order 24. How many subgroups of $G$ are there, given that there are exactly 8 elements of order 3 in $G$?
Tried: since $24=2^3\times 3$, there must be a subgroup of order 1,2,4,8,3, and 24 respectively. Also, it's easy to show that $\{a,b,c,d,a^2,b^2,c^2,d^2\}$ is the set of elements of order 3. So essentially we have 4 subgroups of order 3. But how about those of order 2,4,8?
There are three groups that satisfy your hypotheses $S_4$, $A_4 \times C_2$, and ${\rm SL}(2,3)$, and they each have different numbers of subgroups (30, 26, and 15), so the question does not have a unique answer.