Non-abelian group, where torsion elements form subgroup

Question

We proved in the lectures, that for an abelian group, the torsion elements (elements of finite order) form a subgroup. I also found an example for a non abelian group, where the torsion elements do not form a subgroup. Now I am wondering: Is there a non abelian group G, for which the torsion elements form a non trivial subgroup? So where they are not only the neutral element and not the whole group.