Let $G\subseteq GL(V)$ be a complex finite reflection group. I would like to understand the stabilizer subgroups of $G$, and their normalizers. By this I mean (the conjugacy classes of) subgroups $H<G$ such that there exists $v\in V$ such that $Stab_G(v)=H$, together with $N_G(H)$.
Ideally there would be some table somewhere, with the poset of stabilizer subgroups. I was able to compute everything (rather painfully) when $dim(V)\leq 3$ or when $G=G(n,m,p)$, so I am only missing the exceptional groups of rank at least $4$, that is 28-37 in this list (Shephard-Todd classification).
Alternatively, since I am also trying to learn to use Magma/Sage/GAP, I was wondering if there is a way to extract the information I need from these programs.