Classify groups of order $36$

Question

Question is to classify all groups of order $36$.

I do not even know if it is of my level. Let me try this.

The Sylow theorems say that there are Sylow $2$ subgroups of order $4$ and Sylow $3$ subgroups of order $9$.

The possible numbers of Sylow $2$ subgroups are $1,3,9$

The possible numbers of Sylow $3$ subgroups are $1,4$

Suppose $G$ has $4$ Sylow $3$ subgroups, each subgroup has $8$ non identity elements, total $32$ non identity elements from $4$ Sylow subgroups. Remaining $4$ elements would become one Sylow $2$ subgroups.

This says that if $G$ has $4$ Sylow $3$ subgroups then $G$ has $1$ Sylow $2$ subgroup and thus, this Sylow $2$ subgroup is normal and so $G$ is not simple.

Suppose $G$ has $1$ Sylow $3$ subgroup then it is a normal subgroup and so $G$ is simple. In this case we can not decide on the number of Sylow $2$ subgroups.. It can be any one of $1,3,9$..

I can classify abelian groups

1. $\mathbb{Z}_4\times \mathbb{Z}_9$
2. $\mathbb{Z}_4\times \mathbb{Z}_3\times \mathbb{Z}_3$
3. $\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_9$
4. $\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_3\times \mathbb{Z}_3\cong \mathbb{Z}_6\times\mathbb{Z}_6$

One more piece of information is that if there are $4$ Sylow $3$-subgroups then there is exactly one Sylow $2$ subgroup so normal.. If there is only one Sylow $3$ subgroups it is normal... So, any such group has either a normal Sylow $2$ subgroup or a Sylow $3$ subgroup..

I could not go beyond this.

Answer 1

There are $4$ abelian groups of order $36$ and $10$ non-abelian solvable groups of order $36$. We can do the classification according to the following case distinction. Let $G$ be a group of order $36$.

1.) There are only $2$ groups of order $36$ having no normal subgroup of order $9$, namely $C_3\times A_4$ and $C_9\ltimes (C_2\times C_2)$.

2.) There are $12$ groups of order $36$ having a normal subgroup of order $9$, namely the $4$ abelian groups, and $8$ others, e.g., a nontrivial extension of the dihedral group $D_9$ by $C_2$, $C_9\ltimes (C_2\times C_2)$, $(C_3\times C_3)\ltimes C_4$, $(C_3\times C_3)\ltimes (C_2\times C_2)$, and $D_{18}$.

Reference: These Notes.