  We study the intermediate quantum groups $H_N\subset G\subset U_N^+$. The
basic examples are $H_N,K_N,O_N,U_N,H_N^+,K_N^+,O_N^+,U_N^+$, which form a
cube. Any other example $G$ sits inside the cube, and by using standard
operations, namely intersection $\cap$ and generation $<\,,>$, can be projected
on the faces and edges. We prove that under the strongest possible axioms,
namely (1) easiness, (2) uniformity, and (3) geometric coherence of the various
projection operations, the 8 basic solutions are the only ones.
