Finding four six-dimensional mutually unbiased bases (MUBs) is a long-standing open problem in quantum information. By assuming that they exist and contain the identity matrix, we investigate whether the remaining three MUBs have an $$H_2$$ H2 -reducible matrix, namely a $$6\times 6$$ 6×6 complex Hadamard matrix (CHM) containing a $$2\times 2$$ 2×2 subunitary matrix. We show that every $$6\times 6$$ 6×6 CHM containing at least 23 real entries is an $$H_2$$ H2 -reducible matrix. It relies on the fact that the CHM is complex equivalent to one of the two constant $$H_2$$ H2 -reducible matrices. They, respectively, have exactly 24 and 30 real entries, and both have more than eighteen $$2\times 2$$ 2×2 subunitary matrices. It turns out that such $$H_2$$ H2 - reducible matrices do not belong to the remaining three MUBs. This is the corollary of a stronger claim; namely, any $$H_2$$ H2 -reducible matrix belonging to the remaining three MUBs has exactly nine or eighteen $$2\times 2$$ 2×2 subunitary matrices.