How many subgroups are there in S4?

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SUMMARY

The discussion centers on identifying all subgroups of the symmetric group S4. The user has enumerated 30 subgroups, including the trivial group, S4 itself, and various combinations of permutations such as the alternating group and specific transpositions. A reference from PlanetMath suggests there are 29 subgroups, excluding S4, indicating a potential discrepancy in subgroup counting. The consensus is that the user’s count is accurate when including S4.

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TaylorWatts
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Not a true homework question, but I'm trying to find all subgroups of S4.

Including the identity and the group itself, I've found 30. Is that correct?

I've got groups such as:

trivial

s4

alternating group

{identity, (12)}, {identity, (13)} etc - 6 of these

{identity, (123), (132)}, {identity, (124), (142)} etc - 4 of these

{identity, (12)(34)}, {identity, (13)(24)} etc - 3 of these

{identity, (1243), (14)(23), (1342)} etc - 3 of these

{identity, (13), (24), (12)(34), (13)(24), (14)(23), (1234), (1432)} - 3 of these

{identity, (12), (34), (12)(34)} etc - 3 of these

{identity, (123), (132), (12), (23), (13)}, {identity, (124), (142), (12), (24), (14)} etc - 4 of these.

{identity, (12)(34), (13)(24), (14)(23)}

Am I missing any?
 
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Somebody else added them up too. http://planetmath.org/encyclopedia/SubgroupsOfS_4.html They got 29, but they didn't count S4 itself, I don't think. So you are at least pretty close to right. Check it out. Compare notes.
 
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