We present, a classification of the finite irreducible 2-subgroups of
GL(4, C); that is, we give a parametrised list of representatives for
the conjugacy classes of such groups. Each group listed is defined by
a generating set of monomial matrices. There are essentially three pos
sibilities for the projection of an irreducible monomial 2-group into
the group of all permutation matrices. The classification problem acco
rdingly falls into three separate cases. Each case may be handled by a
general method consisting of three major steps. Techniques for applyi
ng the method to the most difficult case are developed in detail, so t
hat the other two cases may then be dealt with routinely. The techniqu
es used include elementary character theory, a method for drawing the
Hasse diagram of the submodule lattice of a direct sum, and cohomology
theory, particularly the calculation of 2-cohomology by means of the
Lyndon-Hochschild-Serre spectral sequence. Related questions concernin
g isomorphism between the listed groups, and Schur indices over Q, are
also considered.