  In this note we classify all homogeneous spaces $G/H$ admitting a
$G$-invariant $G_2$-structure, assuming that $G$ is a compact Lie group and $G$
acts effectively on $G/H$. They include a subclass of all homogeneous spaces
$G/H$ with a $G$-invariant $\tilde G_2$-structure, where $G$ is a compact Lie
group. There are many new examples with nontrivial fundamental group. We study
a subclass of homogeneous spaces of high rigidity and low rigidity and show
that they admit families of invariant coclosed $G_2$-structures (resp. $\tilde
G_2$-structures).
