$\phi_n (z) = z^n$ find all subgroups between $A = \ker(\phi_3)$ and $B= \ker(\phi_{12})$

Question

$\phi_n :\mathbb C^* \to \mathbb C^*$

$\phi_n (z) = z^n$

I need to find all subgroups between $A = \ker(\phi_3)$ and $B = \ker(\phi_{12})$.

what I tried -

I know that if $x \in A$ then $x \in \ker(\phi_6)$ and $x \in \ker(\phi_9)$

EDITED: so I think that $A \subseteq \ker(\phi_6) \subseteq B$

is that true? also how do I prove they are subgroups ? and how do I prove there are no more subgroups.

any help will be appreciated

thanks