A non-abelian group $G$ has property $X$ if there exist non-trivial groups $A$, $B$ such that $A$ is abelian, $B$ is non-abelian, with
$$G \cong A \times B.$$
In essence, property $X$ is saying that the non-abelian group can be "broken down" into a simpler part that governs the non-abelianness, and another part that is abelian. For example, $S_3$ does not have property $X$, whereas $S_3 \times C_2$ clearly does.
Is there a name for the types of non-abelian groups with property $X$? Or without it?