Groups with a nontrivial nonideal kernel

We classify finite groups $G$, such that the group algebra, $\mathbb{Q}G$ (over the field of rational numbers $\mathbb{Q}$), is the direct product of the group algebra $\mathbb{Q}[G\mathrm{/}N]$ of a proper factor group $G\mathrm{/}N$, and some division rings.