In the case of the former, the CMO may be a supermassive black hole, a nuclear star cluster, or even both together.

[1] The most massive galaxies are thought to always contain a supermassive black hole (SBH); these galaxies do not contain nuclear star clusters, and the CMO is identified with the SBH.

Fainter galaxies usually contain a nuclear star cluster (NSC).

In most of these galaxies, it is not known whether a supermassive black hole is present, and the CMO is identified with the NSC.

[2] A few galaxies, for instance the Milky Way and NGC 4395, are known to contain both a SBH and a NSC.