Subgroup Isomorphism Problem for units of integral group rings

The Subgroup Isomorphism Problem for Integral Group Rings asks for which finite groups U it is true that if U is isomorphic to a subgroup of V(ZG), the group of normalized units of the integral group ring of the finite group G, it must be isomorphic to a subgroup of G. The smallest groups known not to satisfy this property are the counter-examples to the Isomorphism Problem constructed by M. Hertweck. However, the only groups known to satisfy it are cyclic groups of prime power order and elementary-abelian p-groups of rank 2. We give a positive solution to the Subgroup Isomorphism Problem for C-4 x C-2. Moreover, we prove that if the Sylow 2-subgroup of G is a dihedral group, any 2-subgroup of V(ZG) is isomorphic to a subgroup of G.