I know the following theorem.
Let $V, W$ be two valuation domains with the same fraction field $K$. Suppose $V$ and $W$ share the same maximal ideal. Then $V=W$.
In other words, fixed a field $K$, there exist an injective function $$M_- : \{ \mbox{valuation domains $V \subset K$ with $K= \operatorname{Frac}(V)$} \} \longrightarrow \{ \mbox{subgroups of $K$} \}$$ mapping every valuation domain $V$ to its maximal ideal $M_V$.
My question is: is it possible to characterize all subgroups $M \subset K$ of the form $M_V$?
What I know is that they must satisfy the following property: $$\mbox{both $M$ and $K \setminus M$ are multiplicatively closed}$$ Moreover, given a valuation domain $V$ with maximal ideal $M$, I know that $V=(M:M) = \{ x \in K : xM \subset M \}$.
Obviously, this last condition shows that they can be characterized by means of $(M:M)$, requiring that it is a valuation domain. But I don't find this satisfactory, so I am wondering if more can be done.