The lamplighter group can be defined by the semidirect product: $$ L_2=(\mathbb{Z} _2) \wr \mathbb{Z} \cong \bigoplus_{i=-\infty}^{\infty}\mathbb{Z}_{2} \rtimes_\phi\mathbb{Z},$$
where $\phi(1)$ "shifts" every element in $\bigoplus_{i=-\infty}^{\infty}\mathbb{Z}_{2} $ to the right by $1$.
I am trying to classify all the non-abelian subgroups of the Lamplighter group. I know the obvious non-abelian subgroups of the form: $K \rtimes_\phi H$, where $K$ is a non-trivial subgroup of $\bigoplus_{i=-\infty}^{\infty}\mathbb{Z}_{2}$ and $H$ is a non-trivial subgroup of $\mathbb{Z}$. I was wondering if all the non-abelian subgroups are of this form.
Any idea or reference would be really appreciated.