I am looking for examples of CM fields whose Galois group is not abelian. By a CM field K I mean an imaginary quadratic extension of a totally real field $K_0$. If the extension is not Galois I take the Galois closure $L/K$.

Obviously the degree of such a field is even.

When $[K:Q]=2,$ K is an imaginary quadratic field, so its Galois group is $\mathbb{Z}/2\mathbb{Z}$.

My question is: what are the possible Galois groups for $[K:Q]=4$ and 6?

For instance, if I consider $K=K_0(\mu_3)$, with $K_0$ totally real of degree 4 and $\mu_3$ the roots of unity of order 3, what are the possible Galois groups?

Thanks for your help!