A group $G$ is Schur-nontrivial if the Schur multipler $H^2(G,U(1))$ is not the trivial group.

I am trying to find an example of a Schur-nontrivial group which does not contain a subgroup of the form $H\times H$ where $H$ is a finite abelian group (thus $H\times H$ is a group of central type). Every example I have seen (browsing Groupprops, for example) does contain such a subgroup.

I have not yet looked through all of the finite simple groups, as I am not very familiar with them. I am particularly interested in nice groups which could perhaps represent a physical symmetry.