Let $K/F$ be a finite field extension.
If $K/F$ is Galois then it is well known that there is a bijection between subgroups of $Gal(K/F)$ and subfields of $K/F$.
Since finding subgroups of a finite group is always easy (at least in the meaning that we can find every subgroup by brute-force or otherwise) this gives a nice way of finding subfields and proving they are the only ones.
What can we do in the case that $K/F$ is not a Galois extension ? that is: How can I find all subfields of a non-Galois field extension ?