Let $G$ and $H$ be two Hopf algebras, and $\pi: G \to H$ a Hopf algebra map. We will call an algebra of the form
$$
M:= \lbrace m \in G ~ | ~ m_{(1)} \otimes \pi(m_{(2)}) = m \otimes 1 \rbrace
$$
a *quantum homogeneous space*.

There are many well known examples of quantum homogeneous spaces where $G$ is a faithfully flat module over $M$: the quantum spheres, the quantum projective spaces, or more generally the quantum flag manifolds. What I would like are examples of quantum homogeneous spaces for which $G$ is **not** a faithfully flat module over $M$?

It is known that quantum-$SU(2)$ is not faithfully flat over some of the non-standard Podles spheres. However, these are not quantum homogenous spaces in the sense given above.