Let $(W,S)$ be a finite irreducible Coxeter-System of rank $n$ and $E$ be a real reflection representation of $W$. Let $x\in E$ and suppose that the isotropy group of $x$ is generated by one element in $S$. Now which are the subgroups of rank $n-1$ that do not stabilize $x$? 



Subgroups of finite reflection groups that do not fix a point
You have to consider the Coxeter Graph: The subgroups you are looking for are those that you get by removing one edge...


