Let $G$ be a Lie group and $H$ a Lie subgroup. Then $H$ contains the connected component of the identity $G^0$ if and only if $G/H$ is a discrete manifold.
One direction is immediate since $G/G^0$ is discrete. Now if $G/H$ is discrete then $H$ is open and obviously contains the identity so it contains an open set $H\cap G^0$. Not sure how to show that it contains all of $G^0$ unless I have connectedness of $H$ though.
$H$ is in fact clopen. It is a more general fact that a clopen subset of any topological space is a union of connected components (of that topological space): Why are clopen sets a union of connected components?
In particular this implication holds for any topological group. But of course the other one does not.