Active directory question

does it not make sense that a person that has the authority to manage high level administrative group membership be an administrator, domain admin etc? I have been tasked with cleaning up admin by reducing and delegating administrative tasks, but am experiencing administrative overlap that seems to require domain admin membership for certain people. I have created 2 groups and have not been able to create a group with non domain admin members, that can add members to the domain admin group….even with full control at the highes level…..Am I missing something?