Hasse diagrams and orbit class spaces

Let X be a topological space and G be a group of homeomorphisms of X. Let be an equivalence relation on X defined by if the closure of the G-orbit of x is equal to the closure of the G-orbit of y. The quotient space is called the orbit class space and is endowed with the natural order inherited from the inclusion order of the closure of the classes, so that, if such a space is finite, one can associate with it a Hasse diagram. We show that the converse is also true: any finite Hasse diagram can be realized as the Hasse diagram of an orbit class space built from a dynamical system (X,G) where X is a compact space and G is a finitely generated group of homeomorphisms of X.