  We study the coisotropic subgroup structure of standard SL_q(2,R) and the
corresponding embeddable quantum homogeneous spaces. While the subgroups S^1
and R_+ survive undeformed in the quantization as coalgebras, we show that R is
deformed to a family of quantum coisotropic subgroups whose coalgebra can not
be extended to an Hopf algebra. We explicitly describe the quantum homogeneous
spaces and their double cosets.
