I'm looking that in the Fusion System categories, the p-subgroups that are essential, are centric (by definition) and radical (by implication of the definition), but I want to know if there is an example about a p-subgroup that is radical and centric, but not essential. Anyone knows an example? 



Radical and Centric not Essential P-group
Yes, take any prime $p$, and consider the fusion system for ${\rm GL}(4,p)$ ( ie objects the $p$-subgroups of ${\rm GL}(4,p)$ morphisms induced by conjugations).
Take a maximal parabolic $P$ such that $P/U \cong {\rm GL}(3,p)$, where $U$ is the unipotent radical of $P$. Then $U$ is centric and radical but is not essential (as ${\rm GL}(3,p)$ has no strongly $p$-embedded subgroup).


