  We study the four infinite families KA(n), KB(n), KD(n), KQ(n) of finite
dimensional Hopf (in fact Kac) algebras constructed respectively by A. Masuoka
and L. Vainerman: isomorphisms, automorphism groups, self-duality, lattices of
coideal subalgebras. We reduce the study to KD(n) by proving that the others
are isomorphic to KD(n), its dual, or an index 2 subalgebra of KD(2n). We
derive many examples of lattices of intermediate subfactors of the inclusions
of depth 2 associated to those Kac algebras, as well as the corresponding
principal graphs, which is the original motivation.
  Along the way, we extend some general results on the Galois correspondence
for depth 2 inclusions, and develop some tools and algorithms for the study of
twisted group algebras and their lattices of coideal subalgebras. This research
was driven by heavy computer exploration, whose tools and methodology we
further describe.
