  In Hopf-Galois theory, every $H$-Hopf-Galois structure on a field extension
$K/k$ gives rise to an injective map $\mathcal{F}$ from the set of $k$-sub-Hopf
algebras of $H$ into the intermediate fields of $K/k$. Recent papers on the
failure of the surjectivity of $\mathcal{F}$ reveal that there exist many
Hopf-Galois structures for which there are many more subfields than sub-Hopf
algebras. This paper surveys and illustrates group-theoretical methods to
determine $H$-Hopf-Galois structures on finite separable extensions in the
extreme situation when $H$ has only two sub-Hopf algebras.
