Given a tower of field extensions, does this equality involving Galois group orders hold in general?

Suppose we have a tower of field extensions:

$\overline{F} \subset K \subset E \subset F$

Is it true in general that $|G(K/F)| = |G(K/E)| \cdot |G(E/F)|$?

I was able to verify some specific examples, like $\mathbb{Q}(\sqrt[3]{2}, \omega)$ for $x^3-2$ and another extension, but how could I show that this holds in general for all such towers of extensions?