Subgroups of a finite group whose algebra of invariants is a complete intersection

Let $G$ be a finite subgroup of $GL(V)$, where $V$ is a finite-dimensional vector space over the field $K$ and $\operatorname{char}K\nmid|G|$. We show that if the algebra of invariants $K(V)^G$ of the symmetric algebra of $V$ is a complete intersection then $K(V)^H$ is also a complete intersection for all subgroups $H$ of $G$ such that $H=\{\sigma\in G|\sigma(v)=v\text{\rm{ for all }}v\in V^H\}$.