A magneto-mechanical lattice proposed in this paper is one of the nonlinear lattices in which intrinsic localized mode(ILM) exists. A bifurcation diagram concerning ILMs is investigated with respect to the magnitude of nonlinear coupling force. In addition, the possibility of the existence of moving ILM is discussed based on the phase structure around an unstable ILM which is numerically examined by computing unstable manifolds of the unstable ILM.