  A Hopf Galois structure on a finite field extension $L/K$ is a pair
$(H,\mu)$, where $H$ is a finite cocommutative $K$-Hopf algebra and $\mu$ a
Hopf action. In this paper we present a program written in the computational
algebra system Magma which gives all Hopf Galois structures on separable field
extensions of degree up to eleven and several properties of those. Besides, we
exhibit several results on Hopf Galois structures inspired by the program
output. We prove that if $(H,\mu)$ is an almost classically Hopf Galois
structure, then it is the unique Hopf Galois structure with underlying Hopf
algebra $H$, up to isomorphism. For $p$ an odd prime, we prove that a separable
extension of degree $p^2$ may have only one type of Hopf Galois structure and
determine those of cyclic type; we determine as well the Hopf Galois structures
on separable extensions of degree $2p$. We highlight the richness of the
results obtained for extensions of degree 8 by computing an explicit example
and presenting some tables which summarizes these results.
