  We investigate the existence and the properties of normalized ground states
of a nonlinear Schr\"odinger equation on a quantum hybrid formed by two planes
connected at a point. The nonlinearities are of power type and
$L^2$-subcritical, while the matching condition between the two planes
generates two point interactions of different strengths on each plane, together
with a coupling condition between the two planes. We prove that ground states
exist for every value of the mass and two different qualitative situations are
possible depending on the matching condition: either ground states concentrate
on one of the plane only, or ground states distribute on both the planes and
are positive, radially symmetric, decreasing and present a logarithmic
singularity at the origin of each plane. Moreover, we discuss how the mass
distributes on the two planes and compare the strengths of the logarithmic
singularities on the two planes when the parameters of the matching condition
and the powers of the nonlinear terms vary.
