  While the finite-dimensional modules of the dihedral 2-groups over fields of
characteristic 2 were classified over 30 years ago, very little is known about
the tensor products of such modules. In this article, we compute the Loewy
length of the tensor product of two modules of a dihedral two-group in
characteristic 2. As an immediate consequence, we determine when such a tensor
product has a projective direct summand.
