How to find non-cyclic subgroups of a group? I am trying to find all of the subgroups of a given group. To do this, I follow the following steps:


*

*Look at the order of the group. For example, if it is $15$, the subgroups can only be of order $1,3,5,15$.

*Then find the cyclic groups.

*Then find the non cyclic groups.


But i do not know how to find the non cyclic groups. For example, let us consider the dihedral group $D_4$, then the subgroups are of the orders $1,2,4$ or $8$. I find all cyclic groups. Then, I saw that there are non-cyclic groups of order $4$. How can I find them? I appreciate any help. Thanks.

In general, finding the subgroups (of a biggish group) won't be easy.
In the case of $D_4$, the only non-cyclic groups besides $D_4$ itself can only be of order $4$. So you are looking at subgroups of $G$ that consist of the identity, and three involutions (elements of order $2$) $a, b, c = ab$.
Now try out the various possibilities, avoiding repetitions.
