Find all the subgroups of a subgroup of $S_4$

Question

I have $H=\langle \ (1\ 2\ 3 4), \ (1 \ 2)(3 \ 4) \ \rangle $, and I need to find all the subgroups of $H$.

I have found the elements of $H$ and their order.

Also I have found (by using Lagrange's Theorem) that these subgroups must have order 1,2,4 or 8.

And lastly I have found the possible order 1,2,8 subgroups.

My question is are there any other theorems (like Lagrange's) or clever tricks that I can use to narrow down my work in finding the order 4 subgroups?

Because the best idea I've got now is to try $\{I,x,y,z\}$ where $I$ is the identity element and $x,y,z \in H$, and this looks messy and unordered.

Thanks.

Answer 1

If you are interested in solve it using any software you could use GAP, you can check How to find all subgroups of a group in GAP in order to solve your question.

Answer 2

Observe that

$$(12)(34)^2=(1234)^4=1\;,\;\;(12)(34)(1234)(12)(34)=(1432)=(1234)^{-1}$$

so that in fact $\;H\cong D_4=\,$ the dihedral group of order $\;8\;$ , and from here you can deduce all its subgroups (five subgroups of order two, three of order four and both trivial subgroups)