Let $K_0$ be totally real of degree $2,3$ and $K/K_0$ a totally imaginary quadratic extension. Let $L_0$ and $L$ be the Galois closures of $K_0$ and $K$ over ${\mathbb Q}$. Then $Gal(L)$ maps onto $Gal (K)$ with kernel an abelian $2$ group (a vector space over ${\mathbb F}_2$) .

So the issue is: given a totally real number field $K_0$ over ${\mathbb Q}$ (of degree $2,3$), what can its Galois group be? It is easy to see that it can be cyclic of order $2$, cyclic of order $3$ or $S_3$.

Let $K_0$ be totally real of degree $2,3$ and $K/K_0$ a totally imaginary quadratic extension. Let $L_0$ and $L$ be the Galois closures of $K_0$ and $K$ over ${\mathbb Q}$. Then $Gal(L)$ maps onto $Gal (L_0)$ with kernel an abelian $2$ group (a vector space over ${\mathbb F}_2$) .

So the issue is: given a totally real number field $K_0$ over ${\mathbb Q}$ (of degree $2,3$), what can its Galois group be? It is easy to see that it can be cyclic of order $2$, cyclic of order $3$ or $S_3$.