Every civil office in a political subdivision filled by appointment shall be vacant upon the happening of any one of the events listed in section 32-560 except as provided in section 32-561. The resignation of the incumbent of such a civil office may be made as provided in section 32-562. Vacancies in such a civil office shall be filled as provided in sections 32-567 and 32-574 and shall be subject to section 32-563.