How many subgroups are there in S4?

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The discussion focuses on identifying all subgroups of the symmetric group S4, with one participant claiming to have found 30 subgroups, including the identity and S4 itself. Various subgroup types are mentioned, such as the trivial group, the alternating group, and several combinations of transpositions and cycles. There is a reference to another source that lists 29 subgroups, which may not include S4 itself. Participants are encouraged to compare their findings and verify the counts. The conversation highlights the complexity of subgroup classification within S4.
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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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