Conn. Gen. Stat. § 9-373b

A candidate for an office appearing on a ballot for a state election, as defined in section 9-1 , may appear on the ballot as a candidate for more than one major or minor party, as defined in section 9-372 , for the same office, provided a candidate of such minor party for the office of Governor, Secretary of the State, State Treasurer, State Comptroller or Attorney General received at least fifteen thousand votes at the previous state election for any such office.