The invariant shapes for close formation flying with inter-craft electromagnetic force ensure several potential space applications. However, the 6-DOF relative equilibrium problem has not been systematically investigated. This paper mainly analyzes the invariant shapes of relative equilibrium for the three-spacecraft electromagnetic formation, and studies the families of invariant shape solutions with real and constant magnetic moments as well as their linear stability. The problem is examined based on the full nonlinear coupled dynamic models for collinear and general triangular configurations. The relative equilibrium conditions are analyzed to determine whether an invariant shape do exist, and the corresponding families of invariant shape solutions are identified for static and spinning configurations respectively. Finally, the linear stability of such invariant shapes is numerically discussed, which have shown that most invariant shapes are unstable and controllable.