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Let $X$ be a finite simple undirected graph with a subgroup $G$ of the full automorphism group $\mathrm{Aut}\left(X\right)$. Then $X$ is said to be $(G,s)$-transitive for a positive integer $s$, if $G$ is transitive on $s$-arcs but not on $(s+1)$-arcs, and $s$-transitive if it is $\left(\mathrm{Aut}\right(X),s)$-transitive. Let $G_v$ be a stabilizer of a vertex $v\in V\left(X\right)$ in $G$. Up to now, the structures of vertex stabilizers $G_v$ of cubic, tetravalent or pentavalent $(G,s)$-transitive graphs are known. Thus, in this paper, we give the structure of the vertex stabilizers $G_v$ of connected hexavalent $(G,s)$-transitive...