  We investigate the existence of left-invariant closed G$_2$-structures on
seven-dimensional non-solvable Lie groups, providing the first examples of this
type. When the Lie algebra has trivial Levi decomposition, we show that such a
structure exists only when the semisimple part is isomorphic to
$\mathfrak{sl}(2,\mathbb{R})$ and the radical is unimodular and centerless.
Moreover, we classify unimodular Lie algebras with non-trivial Levi
decomposition admitting closed G$_2$-structures.
