Sandwich panels with aluminum alloy face sheets and a hierarchical composite square honeycomb core have been manufactured and tested in out-of-plane compression. The prismatic direction of the square honeycomb is aligned with the normal of the overall sandwich panel. The cell walls of the honeycomb comprise sandwich plates made from glass fiber/epoxy composite faces and a polymethacrylimide foam core. Analytical models are presented for the compressive strength based on three possible collapse mechanisms: elastic buckling of the sandwich walls of the honeycomb, elastic wrinkling, and plastic microbuckling of the faces of the honeycomb. Finite element calculations confirm the validity of the analytical expressions for the perfect structure, but in order for the finite element simulations to achieve close agreement with the measured strengths it is necessary to include geometric imperfections in the simulations. Comparison of the compressive strength of the hierarchical honeycombs with that of monolithic composite cores shows a substantial increase in performance by using the hierarchical topology.