  We construct lattices on six dimensional not completely solvable almost
abelian Lie groups, for which the Mostow condition does not hold. For the
corresponding compact quotients, we compute the de Rham cohomology (which does
not agree in general with the Lie algebra one) and a minimal model. We show
that some of these solvmanifolds admit not invariant symplectic structures and
we study formality and Lefschetz properties.
