Examples of abelian subgroups of non-abelian groups.

Let $G$ be any group, Abelian or not, and let $g\in G$. Then $\langle g\rangle=\{g^n:n\in\Bbb Z\}$, the subgroup of $G$ generated by $g$, is Abelian.

The smallest non-abelian group is the symmetric group $S_3$ of order $6$. So all its proper subgroups are abelian (the trivial subgroup, three subgroups of order $2$ and one subgroup of order $3$).

The more simple example is the trivial subgroup $\{e\}$ of any non-abelian group where $e$ is the identity element.

Another example less simple is the quaternion group and for example $\{\pm1\}$ is an abelian subgroup.