How do I find all the subgroups of the symmetry group of an equilateral triangle?

Question

Generally speaking, finding all the subgroups of a finite group is a very difficult problem. But because $S_3$ is finite and of small order, it is possible to use brute force to find all of the subgroups. I need to know both a general method for calculating the subgroup structure by brute force and see how this is applied to the group $S_3$ step-by-step. If I don't see it done step-by-step, I cannot understand it.

Answer 1

Here's a brute force method (for finite groups $G$).

First, find all the cyclic subgroups, that is, for each $g$ in $G$, find the subgroup generated by $g$.

Then, find all the two-generator subgroups. For each cyclic subgroup $H$, and each element $g$ in $G$ but not in $H$, find the subgroup generated by $H$ and $g$, that is, the smallest subgroup containing both $H$ and $g$.

Then, find all the three-generator subgroups, then all the four-generator subgroups, etc.

There are shortcuts. Keep in mind that the order of a proper subgroup can't exceed half the order of the group, so as soon as you see that some set of generators gives you more than half the elements of the group, you can stop.