The buckling of a three-ply sandwich composite plate is studied in the context of finite deformation incompressible nonlinear elasticity. As in the case of a noncomposite plate, buckled configurations involving either barrelling or flexure are possible. However, the ordering of the failure thrusts for certain composite configurations is more complicated than the ordering that occurs in the noncomposite case. In particular, under suitable conditions, the thrusts at which barrelling and flexural buckling modes occur can be interlaced. Furthermore, transitions in the ordering of the different failure modes may occur as the aspect ratio of the plate is varied.