  We show that any two Hadamard subfactors arising from a pair of distinct
complex Hadamard matrices of order 3 are either equal or inner conjugate via a
unitary in the relative commutant of their intersection. Moreover, when the
Hadamard subfactors are not equal, we prove the factoriality of their
intersection, and quite interestingly, it turns out to be a vertex model
subfactor. We compute the first relative commutant and characterize this
subfactor by identifying it with a particular type of Krishnan-Sunder
subfactor. A few key invariants, including the Pimsner-Popa probabilistic
number, the angle, and the Connes- St{\o}rmer relative entropy for the pair of
Hadamard subfactors are computed to understand their relative position.
