Say we have a group $G$ and a set of Higgs fields in a representation $R$ of $G$. One of the Higgs fields in $R$ gets a VEV, how can I determine the remaining subgroup after this symmetry breaking?
In the standard model things are easy, because the Higgs field that gets a VEV has nonzero Isospin and Hypercharge, but zero Electric Charge. Therefore $$SU(2)\otimes U(1)_Y \rightarrow U(1)_Q$$
There is no room for ambiguities.
Nevertheless, for some bigger group it's not sufficient to know that the corresponding Higgs field has zero charge under the corresponding subgroup $G'$ because there can be some bigger subgroup $G' \subset G''$ under which the Higgs field has zero charge, too.
This task is commonly called "finding the maximal little group" and I'm looking for a concrete way of determine this "maximal little group" if some Higgs field gets a VEV.
Any ideas would be much appreciated!