TITLE: Subgroups of $S_{4}$ that contain the kernel of a homomorphism $\varphi:S_{4}\to S_{3}$
QUESTION [5 upvotes]: This is an exercise from Artin's Algebra 2nd Ed. The exercise refers to the following example from the text:

The three partitions of the set of four indices 
  $\{\mathbf{1},\mathbf{2},\mathbf{3},\mathbf{4}\}$ into pairs of subsets of order two are :$$\Pi_{1}\colon\{\mathbf{1},\mathbf{2}\}\cup\{\mathbf{3},\mathbf{4}\},\quad\Pi_{2}\colon\{\mathbf{1},\mathbf{3}\}\cup\{\mathbf{2},\mathbf{4}\},\quad\Pi_{3}\colon\{\mathbf{1},\mathbf{4}\}\cup\{\mathbf{2},\mathbf{3}\}.$$ An element of the symmetric group $S_{4}$ permutes the four indices, and by doing so it also permutes these three partions. This defines a map $\varphi$ from $S_{4}$ to the group of permutations of the set $\{\Pi_{1},\Pi_{2},\Pi_{3}\}$, which is just the symmetric group $S_{3}$.

The example goes on to show that $\varphi$ is a homomorphism and that

Its kernel can be computed. It is the subgroup of $S_{4}$ consisting of the identity and the three products of disjoint transpositions:
  $$K=\left\{1,~(\mathbf{1}~\mathbf{2})(\mathbf{3}~\mathbf{4}),~(\mathbf{1}~\mathbf{3})(\mathbf{2}~\mathbf{4}),~(\mathbf{1}~\mathbf{4})(\mathbf{2}~\mathbf{3})\right\}.$$

The exercise then asks:


Determine the six subgroups of $S_{4}$ that contain $K$.


I have two difficulties with this exercise. How was $K$ computed? And how can I determine the subgroups of $S_{4}$ that contain $K$ without having to just use brute force to compute all the possible cycle permutations?
So far, my naive approach has been to compose elements from $S_{4}$ in an effort to find a composition that gives an element of $K$. While I think this would work eventually, I know there must be an easier way. 
I should mention that I have always had a difficult time working in $S_{n}$. I'm not familiar with the subgroups of $S_{n}$ and anytime I've had to work with a symmetric group I inevitably have to sit down and write out a table for each permutation and/or the composition of permutations.
I haven't tagged this as homework because I'm self studying. (Trying to improve my grasp of permutations and permutation groups.)

REPLY [2 votes]: Here is one way you can do it directly by hand: Write down explicitly (I know this may be a painful process but I've done it before) where each element of $S_4$ goes into $S_3$ under the homomorphism. Your six subgroups of $S_4$ that contain the kernel $K$ by the correspondence theorem are precisely the preimages of subgroups of $S_3$. To compute the preimages of subgroups of $S_3$, take for example the subgroup
$$\{e,(123),(132)\}.$$
If you already computed where each element in $S_4$ went in $S_3$, then the preimage of this subgroup is precisely the union of the preimages of each cycle in the list above.