# Conjugacy classes of stabilizer subgroups

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 $$H such that there exists $$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.