Let $\displaystyle G=H \rtimes K$ (or even just $\displaystyle G=H\times K$).

Is it true that if $\displaystyle \bar{G}\leq G$ then either $\displaystyle \bar{G}\leq H$, $\displaystyle \bar{G}\leq K$, or $\displaystyle \bar{G}=\bar{H}\rtimes \bar{K}$ where $\displaystyle \bar{H}\leq H$ and $\displaystyle \bar{K}\leq K$?

I know the equivalent isn't true for homomorphic images, but I'm not sure about semidirect products...