Is there a way to directly determine to which subgroup a symmetric group is the symmetry group of a polyhedron isomorphic to? In example, I know that the symmetry group of a tetrahedron is isomorphic to the alternating group $A_4 \leq S_4$ or that the symmetric group of a square is isomorphic to another subgroup of $S_4$ with order $8$.
Is there a way to think about symmetry groups of platonic solids isomorphics without having to examine each one individually?