The surface plasmon resonances of a nanostructure composed of a single-split ring and a double-split disk is studied theoretically by the finite element method. When the incident light is normal to the structure surface, a magnetic Fano resonance can be generated by the interference between a bright magnetic mode and a dark magnetic mode. High order magnetic modes and double magnetic Fano resonances can be generated when the double-split disk, the cavity, and the gap of the single-split ring offset together along the negative x-axis. On the basis of this structure, further modulating the gap width of the single-split ring, magnetic modes intensity is enhanced in the near-infrared region and triple magnetic Fano resonances is generated; similarly, by modulating the upper split angle of the double-split disk, high-order magnetic mode is obtained in the visible region and triple magnetic Fano resonances is generated. In addition, the maximum sensitivity and magnetic field enhancement of the structure can reach 1 400 nm/RIU and 69.7, respectively. These optical properties make the structure has potential application value in the field of ultra-sensitive biosensor and multi-control magnetic Fano switch.