Given a compact connected Lie group $G$, we estimated the constant for the Sobolev inequality for an arbitrary Hörmander sub-Laplacian on $G$. The constant was estimated in terms of quantities depending on the intrinsic geometry of the group and on the Lebesgue parameters. We also provided applications of this result to the estimation of the operator norm for $L^p$- $L^q$-bounded subelliptic pseudo-differential operators in subelliptic Hörmander classes.