Compact subgroup and quotient implies compactness?

Question

Let $G$ be a topological group with $K \le G$ a compact subgroup such that $G/K$ is compact. Is $G$ compact, too?

Answer 1

$q: G \rightarrow G/K$ is a perfect map: continuous, all fibres are homeomorphic to $K$, hence compact, $q$ is closed (I take this on faith from the OP). We then apply this standard fact on perfect maps (and my answer for a proof sketch) to conclude compactness of $G$.