Here is a pattern of group embedding, $$ SU(4) \supset SU(3) \times U(1) $$ such that the irrep of SU(4) can be decomposed as the sum of tensor product of irrep of SU(3) and irrep of U(1).
Two questions:
I wonder what will be the embedding for SO(6), $$ SO(6) \supset SU(2) \times G, $$ where, such that the irrep of SO(6) can be decomposed as the sum of tensor product of irrep of SU(2) and irrep of $G$?
alternatively what will be the embedding for $Spin(6)\simeq SU(4)$, $$ Spin(6)\simeq SU(4) \supset SU(2) \times G', $$ where, such that the irrep of SU(4) can be decomposed as the sum of tensor product of irrep of SU(2) and irrep of $G'$?
I only ask the proper ways to write the group $G$ and $G'$ here.
Here $G$ and $G'$ may be a product of several groups.