Let $ G $ be a simple Lie group and let $ \Gamma $ be a lattice in $ G $. If $ \Gamma $ is not contained in any connected subgroup of $ G $ does that imply that $ \Gamma $ is not contained in any positive dimensional subgroup of $ G $?
Context: I was thinking about the lattice $ SL(2,\mathbb{Z}) $ in $ SL(2,\mathbb{C}) $. $ SL(2,\mathbb{Z}) $ is not contained in any proper connected subgroup of $ SL(2,\mathbb{C}) $. And it is also true that $ SL(2,\mathbb{Z}) $ is not contained in any positive dimensional proper subgroup.