After faults arise in a hypercube, it is often desirable to reconfigure the faulty hypercube in such a way as to retain as many fault-free nodes as possible, because system performance tends to be in proportion to the computational power, and a reconfigured hypercube with more nodes is likely to retain performance better. This inspires us to identify maximum incomplete subcubes in a faulty hypercube, as the subcube so reconfigured is often much larger than that reconfigured according to earlier schemes. Here we propose an efficient algorithm for determining maximum incomplete subcubes in faulty hypercubes. The basic idea is to construct a maximum incomplete subcube from a number of healthy complete subcubes of distinct sizes. To this end, an efficient procedure for finding all maximum fault-free complete subcubes in a faulty hypercube is introduced, and then an efficient algorithm for determining maximum incomplete subcubes is presented.