Clearly, every abelian group has this property, because all subgroups of an abelian group are normal subgroups, but there are non-abelian examples as well. The most familiar (and smallest) is the quaternion group of order 8, denoted by Q8.
It can be shown that every Hamiltonian group is a direct sum of the form G = Q8 + B + D, where B is the direct sum of some number of copies of the cyclic groupC2, and D is a periodic abelian group with all elements of odd order.
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