We study the discrete groups $\Lambda$ whose duals embed into a given compact quantum group, $\widehat{\Lambda }\subset G$. In the matrix case $G\subset U_n^{+}$ the embedding condition is equivalent to having a quotient map ${\Gamma }_U\to \Lambda$, where $F=\{{\Gamma }_U\mid U\in U_n\}$ is a certain family of groups associated to $G$. We develop here a number of techniques for computing $F$, partly inspired from Bichon’s classification of group dual subgroups $\widehat{\Lambda }\subset S_n^{+}$. These results are motivated by Goswami’s notion of quantum isometry group, because a compact connected Riemannian manifold cannot have non-abelian...