Let $G$ and $H$ be two finite groups. Let $r(G)$ be the order of the set of conjugacy classes of $G$. We know $$r(G\times H)=r(G)\times r(H).$$ My problem is: if there is a semi-direct product $G\rtimes H$ such that $G\rtimes H$ cannot be decomposed in the form $G\times H$, then do we have $$r(G\rtimes H)<r(G\times H)?$$