The periodic arrays of edge-to-edge nanorod oligomer on glass substrate are designed to investigate the magnetic resonances and to realize the Fano resonances. With the transverse excitation, that is, the polarized electric field of the incident light is perpendicular to the long axis of the nanorods, the planar symmetric nanorod trimer can obtain a single Fano resonance, while the metal-insulator-metal (MIM) nanorod oligomer can obtain double Fano resonances. The near field electromagnetic distributions and far field extinction spectra of the nanorod arrays are analyzed in visible to the near-infrared regions using finite element method. And the characteristics of its resonance peaks and its realization mechanism are studied. The analysis shows that the near field coupling and superposition of the localized surface plasmon resonance mode of nanorods excite its magnetic surface plasmons (MSPs) to obtain the Fano resonances. In particular, the introduction of MIM nanorods provides more possibilities for the realization of double or even multiple Fano resonances. The Fano resonances of designed nanorod oligomer arrays have the advantages of low resonant loss and high quality with the bandwidth of 30 to 50 nm, which can be potentially used in multi-wavelength biosensor, optical switch and other devices.