We define a nontrivial homomorphism from the Hecke algebra of type $B$ onto a reduced algebra of the Hecke algebra of type $A$ at roots of unity. We use this homomorphism to describe semisimple quotients of the Hecke algebra of type $B$ at roots of unity. Using these quotients we determine subfactors obtained from the inclusion of Hecke algebra of type $A$ into Hecke algebras of type $B$. We also study intermediate subfactors related to the Hecke algebra of type $D$.