Lattice of subgroups of $S_{n}$

Question

There is a method of calculating all the subgroups of dihedral group $D_{2n}$.I want to know that is there any method to find all the subgroups of $S_{4}, A_{4} $. I have found all the subgroups of both $S_{4}$ and $A_{4}$ somewher on google but want to know how to find them. Secondly I want to know that how can we say that our lattice diagram is true? And why it is called lattice of subgroups? Thanks in advance

Answer 1

For $S_4$ and $A_4$, the groups are small enough that one can find all their subgroups by hand. For a general symmetric group $S_n$, as far as I know, there is no method to find all its subgroups (although the O'Nan-Scott theorem gives information on the maximal subgroups of symmetric groups).

As for why we call it the lattice of subgroups, it is because subgroups, together with inclusion, form a lattice.