The optical properties of a novel nanostructure consisting of a hexagonal array of aligned vertically threelayered metal-dielectric-metal nanodisks on a silver film are theoretically studied through the finite-difference time-domain method. The novel nanostructure exhibits three obvious optical transmission bands due to the excitation of subradiant plasmon modes, superradiant plasmon modes, and Fano resonances. Surface plasmon polaritons of the underlying Ag film also play a significant role on these three optical transmission bands via coupling with localized surface plasmons of nanodisk pairs. Moreover, the nanostructure also exhibits a good tunability of optical response by modifying the sizes of cylinders, the thickness of underlying metal film, and the dielectric constant of middle layer. These results demonstrate the nanostructure with great advantages in optical sensors and filters.