TITLE: Subgroups of $S_6$
QUESTION [8 upvotes]: What are all the subgroups of $S_6$ up to isomorphism? I have been able to find a few listings but not one I can be sure is complete.

REPLY [7 votes]: The GroupProps Subwiki is a decent resource for information about small groups (up to isomorphism) and their subgroups.
Drilling down a level we learn that $S_6$ is unique among the symmetric groups in having a different number of conjugacy classes (56) from automorphism classes (37) of subgroups.  That is, there are subgroups of $S_6$ which are isomorphic under an outer automorphism of $S_6$ but not under an inner automorphism (conjugacy).
Unfortunately the latter Wiki page has a very incomplete "Table classifying subgroups up to conjugacy", listing only 4 of the 56 conjugacy classes.  Of course equivalence up to isomorphism is even coarser than equivalence up to automorphism (of the whole group $S_6$).  A paper at arxiv.org lists 29 such subgroups up to isomorphism (in an applied context):
$$ D_6, D_3, D_2, Z_6, Z_3, Z_2, D_3\times D_3, Z_3×Z_3, (Z_3\times Z_3)⋊ Z_2,
D_3 \times Z_3, Z_2 \times Z_2 \times Z_2, S_3 ≀ Z_2, Z_4, Z_5, D_4, Z_2 \times Z_4, D_4, D_5, A_4, D_4 \times Z_2, Z_5 ⋊ Z_4,
A_4 \times Z_2, S_4, (Z_3 \times Z_3) ⋊ Z_4, S_4 \times Z_2, A_5, S_5, A_6, S_6 $$
and references a PDF that lists by order/index the 56 conjugacy classes of $S_6$ subgroups from this Web site.  There is an error in this list, as $D_4$ appears twice and the trivial group $Z_1$ is omitted.
For comparison with Jack Schmidt's Answer, the $Z_n$ here are the cyclic groups denoted $C_n$ by GAP. As Derek Holt points out, the notation here for dihedral group of order $2n$ is $D_n$, where GAP uses $D_{2n}$.  There's a redundancy in that $D_3 \cong S_3$, yet here both are used.