The possible degrees of $\mathbb{Q}(a,b)$ in terms of the degrees of $a$ and $b$

Question

Let $n,m \geq 1$ be natural numbers. Is there a characterization of those natural numbers $d$ for which there are algebraic numbers $a,b$ of degrees $n,m$ such that $\mathbb{Q}(a,b)$ has degree $d$ over $\mathbb{Q}$? Two necessary conditions are $\mathrm{lcm}(n,m) \mid d$ and $d \leq nm$. (In particular, if $n,m$ are coprime, only $ d=nm$ is possible.) Are they sufficient? Or do we actually have $d \mid nm$? I have chosen $\mathbb{Q}$ just to fix ideas, maybe the same analysis works for any field (of characteristic zero). So an answer treating this more general case is appreciated as well.