In this work, we have carried out a systematic study on the optical properties of gold nano-octahedra via numerical simulations. We obtained broadening scattering spectra dominated by a hybridized bonding mode originated from the interaction between intrinsic plasmon modes when increasing the size of gold nano-octahedra in vacuum. Once placing the nano-octahedra on a high refractive index dielectric substrate, a Fano dip can be induced in the scattering spectra due to the mutual interference between a dipolar mode and a quadrupolar mode. Moreover, we found that the interference become stronger by increasing the refractive index in substrate, and the broadening scattering spectra split into two hybridized modes in a single gold nano-octahedron: the anti-bonding mode and the bonding mode. Our results for the first time give a clear description of the complex plasmonic properties of a gold nano-octahedron and provide insights in designing appropriate nano-octahedral structures for applications.