TITLE: How to enumerate subgroups of each order of $S_4$ by hand
QUESTION [24 upvotes]: I would like to count subgroups of each order
(2, 3, 4, 6, 8, 12) of $S_4$,
and, hopefully, convince others that I counted them correctly.  In
order to do this by hand in the term exam, I need a clever way to do
this because there can be as many subgroups of a group of order 24 as
$2^{23}$.
Do you know how to do this? 
(I would be most grateful if you could tell me what part of the answer to the old question answers my question before voting to close.)

REPLY [2 votes]: I know I am late but this might be helpful for people who are coming here looking for an answer.
In order to find all the subgroups, you must go through each element and see what it generates. To begin with, list the group
S4={(1),(12),(13),(14),(23),(24),(34),(123),(132),(142),(124),(134),(143),(234),(243),(1234),(1243),(1324),(1342),(1423),(1432),(12)(34),(13)(24),(14)(23)}
Iterate through each element and see what it generates.
In order to figure out what an element generates, you multiply it by itself until you get the identity. All the elements you get when multiply are part of the subgroup. The amount of times you need to do this will be the order, or the number of elements in the subgroup. Here are some examples of how to generate subgroup.
<(1)> = {(1)}

<(1 2)> = {(1), (1 2)}
(1 2)(1 2) = (1)

<(1 3)> = {(1), (1 3)}
(1 3)(1 3) = (1)

<(1 2 3)> = {(1), (1 2 3), (1 3 2)}
(1 2 3)(1 2 3) = (1 3 2)
(1 2 3)(1 2 3)(1 2 3) = (1 3 2)(1 2 3) = (1)

<(1 2 3 4)> = {(1), (1 2 3 4), (2 4)(1 3), (1 4 3 2)}
(1 2 3 4)(1 2 3 4) = (2 4)(1 3)
(1 2 3 4)(1 2 3 4)(1 2 3 4) = (1 4 3 2)
(1 2 3 4)(1 2 3 4)(1 2 3 4)(1 2 3 4) = (1)

<(1 2)(3 4)> = {(1),(1 2)(3 4)}
(1 2)(3 4)(1 2)(3 4) = (1)

Some of the generated subgroups will be duplicated. However, there will not be more than 4!=24 because that is how many elements there are to generate subgroups.
I hope this helped and let me know if I made any mistakes.