The fracture problem of the magnetic electro-elastic material with four cracks in a square hole is investigated under the action of anti-plane shear force. Based on the linear elastic fracture theory and the Riemann-Schwarz analytical continuation theorem, by constructing a suitable numerical conformal mapping function and the residue theorem, the boundary value problem of the analytical function is transformed into Cauchy integral equations, then the explicit expression of fracture mechanics parameters at the crack tip under magnetoelectric impermeable boundary conditions is obtained. The effectiveness of the method is verified by comparing with the existing results. The effects of hole size, four crack lengths and mechano-electro-magnetic load on crack propagation parameters are described by numerical examples. The results show that the effect of the horizontal right crack on the crack growth is more significant. Vertical crack affects the propagation trend of horizontal crack. Under magnetoelectric impermeable boundary conditions, the stress intensity factor of crack tip increases with the increase of mechanical load. However, the propagation of crack under electric and magnetic load is closely related to the magnitude and direction of mechanical load. The analytical method developed provides an effective way to solve the problem of intelligent composite materials in complex multi-connected domains, and the research results provide a scientific basis for the optimal design of composite materials or structures with cracks.