When thinking in terms of Dynkin diagrams, I am naively used to see that the diagram for a subgroup can be extracted from the diagram of the group by removing some roots. Now, I noticed that for SO(10) and its subgroup SU(4)xSU(2)xSU(2), no such removal happens, the Dynkin diagrams have the same number of nodes.
How usual is this? For which cases we can find subgroups with the same number of roots (the same number of nodes in the Dynkin diagram, I mean) that the main group?
For the aforementionated case, if one tries to remove a further root then the inclusion relationship does not hold anymore: from SO(10) we got to SO(8), but from 4-2-2 we go either to SU(3)xSU(2)xSU(2) or to SU(4)xSU(2), and neither of them are subgroups of SO(8).