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.

One thing you can try is find the groups of each order. A group of order $2$ must be isomorphic to $\mathbb{Z}_2$, which contains identity and another element of order $2$. How many elements of order $2$ are there?
For groups of order $4$, they are isomorphic to either $\mathbb{Z}_4$ or $\mathbb{Z}_2\times\mathbb{Z}_2$. In $\mathbb{Z}_4$, it contains an element of order $4$, so what is it? The other case is similar.
You can also find them using Group Explorer.
