In a paper that I am reading, the author said that there exists a simple group of order 168 and it is known to have two distinct conjugacy classe of subgroups of order 24.
My question: Are they probably referring to $PSL(3,2)$? If so, what are the the subgroups of order 24 explicitly.
As there is just one simple group of order 168, they are, and it is more easily seen to be $\text{GL}(3,2)$.
You will get one class of subgroups isomorphic to $S_4$ by taking the block matrices $$ \left[ \begin{matrix} A & 0\\ v&1\\ \end{matrix} \right] $$ where the $2\times 2$ matrix $A$ is of course non-singular.
These stabilise a [whatever] in the projective plane, and as I recall it the other class of $S_4$'s stabilise a [dual whatever].