A three-way three-mode extension of De Boeck and Rosenberg's (1988) two-way two-mode hierarchical classes model is presented for the analysis of individual differences in binary object x attribute arrays. In line with the two-way hierarchical classes model, the three-way extension represents both the association relation among the three modes and the set-theoretical relations among the elements of each mode. An algorithm for fitting the model is presented and evaluated in a simulation study. The model is illustrated with data on psychiatric diagnosis. Finally, the relation between the model and extant models for three-way data is discussed.