Can a Succubus use its Charm action to make my undead minion attack me?

I animated a corpse (into a zombie) using the animate dead spell. During a fight with a succubus, the succubus tried to charm the zombie using its Charm ability:

One humanoid the fiend can see within 30 feet of it must succeed on a DC 15 Wisdom saving throw or be magically charmed for 1 day. The charmed target obeys the fiend’s verbal or telepathic commands. If the target suffers any harm or receives a suicidal command, it can repeat the saving throw, ending the effect on a success. If the target successfully saves against the effect, or if the effect on it ends, the target is immune to this fiend’s Charm for the next 24 hours.

The zombie is not immune to being charmed according to its stat block – but the animate dead spell says that I command the undead how to act.

Can the zombie be charmed into fighting me? In general, can creatures you command (familiars, summoned, animated) be charmed into fighting their master?