  When a 3-manifold admits an openbook decomposition, we get a Heegaard
splitting by thickening a page. This splitting surface has a special multi
curves coming from the binding. In this paper, we consider the subgroup of the
Goeritz group of this Heegaard splitting, which is the mapping class group of
the 3-manifold preserving the given Heegaard splitting, consisting of elements
preserving the binding. This subgroup turned out to be the quotient of the
subgroup of the orientation preserving mapping class group consisting of
elements commuting with the monodromy by the subgroup generated by the Dehn
twists along the boundary curves. We also get a criterion for the existence of
an element of the Goeritz group which fixes the binding as a set and reverses
the orientation. At last, we give some example of computation of a Goeritz
group.
