Comparative size of lattice of subgroups and lattice of the intermediate fields

Question

If an extension of fields $E/F$ is normal and separable, Galois theory tells us that the intermediate fields $E \supset M \supset F$ are in 1-1 correspondence with the proper of the Galois group $\operatorname{Gal}(E/F)$. But if the extension is for instance not normal, then which lattice is in general bigger? The one of the subgroups or the one of the intermediate fields? Are there easy example for the case "the latice of the intermediate fields is bigger"/"the lattice of the subgroups is bigger"?

Answer 1

For a purely inseparable normal extension (for instance, $k(X,Y)$ over $k(X^p,Y^p)$ for $k$ of characteristic $p$), the Galois group is trivial but the lattice of intermediate fields isn't.

In general distinct subgroups of the Galois group will determine subfields as fixed fields, but one need not get all subfields.