Restriction of a reflection group to a reflection hyperplane

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Let $G$ be a finite reflection group acting on a euclidean vector space $V$ of dimension $n$. Choose any parabolic subgroup $G'$ and let $H_{G'}$ be the subspace of points fixed under $G'$ and $\overline{G}$ be the group generated by all reflections which do not contain ${H}_{G'}$. Is there a way to understand which groups $\overline{G}$ one gets by studying the Dynkin diagram?