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 the $n=15=3\cdot 5$ case, recall that every group of order $p$ prime is cyclic. This leaves you with the subgroups of order $15$. How many are there?
Of course, this is not as easy in general. For general finite groups, the classification is a piece of work. Finite Abelian groups are easier, as they fall in the classification of finitely-generated Abelian groups.
Now, $D_4$ is not that bad. The only nontrivial thing is to find all the subgroups of order $4$. Cyclic ones correspond to order $4$ elements in $D_4$. Noncyclic ones are of the form $\{\pm 1,\pm z\}$ where $z$ is an order $2$ element in $D_4$. Since $D_4$ has eight elements, it is fairly easy to determine all these order $2$ and $4$ elements.
