A subgroup name is not enough
Branching restricts a representation along a specified homomorphism. The abstract groups alone do not identify that map. Kernel, image, and charge conventions are part of the calculation.
The regular map
The regular block inclusion sends U to diag(U,1). It is injective. Restricting the fundamental and adjoint gives:
The principal map
The spin-one representation defines the principal homomorphism from SU(2) to SU(3). Its kernel is {±I} and its image is SO(3). The same parent representations now restrict as follows:
Verify the result
Start from the fundamental times its conjugate, which is the sum of the singlet and adjoint. Restrict both factors, then remove the singlet. The regular and principal results both have dimension eight, but different spin content. The principal restriction contains only integer spins, as required by the kernel.
Work with the actual map
Embeddings and branching contrasts the regular and principal maps. Continue to the SU(5) worked restriction and retain its abelian charges and finite kernel.