An investigation of point symmetry patterns on the regular hexagonal tessellation is presented. This tessellation has three point symmetry groups. However, the restriction to the hexagonal tessellation causes some symmetry subgroups to be repeated in ways that are geometrically unique and others that are geometrically equivalent, resulting in a total of 14 geometrically distinct symmetry groups. Each symmetry group requires a particular set of motif symmetries to allow its construction. Examples of symmetric patterns are shown for several simple motif families.