Subgroup structure of symmetric group:S4

The symmetric group of degree four has many subgroups.

Note that since $$S_4$$ is a complete group, every automorphism is inner, so the classification of subgroups upto conjugacy is equivalent to the classification of subgroups upto automorphism. In other words, every subgroup is an automorph-conjugate subgroup.

Table listing number of subgroups by order
These numbers satisfy the congruence condition on number of subgroups of given prime power order: the number of subgroups of order $$p^r$$ for a fixed nonnegative integer $$r$$ is congruent to 1 mod $$p$$. For $$p = 2$$, this means the number is odd, and for $$p = 3$$, this means the number is congruent to 1 mod 3 (so it is among 1,4,7,...)

Classification based on partition given by orbit sizes
For any subgroup of $$S_4$$, the natural action on $$\{ 1,2,3,4 \}$$ induces a partition of the set $$\{ 1,2,3 \}$$ into orbits, which in turn induces an unordered integer partition of the number 4. Below, we classify this information for the subgroups.