I'm interested in which finite groups can arise as $$ N(H)/H $$ for $ H $ a connected subgroup of a compact connected simple Lie group $ G $.
One obvious family of examples is take $ H $ to be the maximal torus then $ N(H)/H $ is the Weyl group of $ G $. For some other examples I looked at $ N(H)/H $ is just cyclic 2. Also all the $ N(H)/H $ given in the second column of tables 5,6,7,8 of [https://arxiv.org/abs/math/0605784] seem to be Coxeter groups or at least complex reflection groups.
Is it possible that $ N(H)/H $ is always a Coxeter group? (again I'm assuming $ H $ a connected subgroup of a connected Lie group $ G $)
Seems like a bit of a crazy conjecture, but mostly I'm just interested in understanding the structure of the finite group $ N(H)/H $.