# Overgroups of the stabilizer of an isotropic vector in orthogonal groups

Suppose $$V$$ is a $$2n+2$$-dimensional vector space over $$\mathbb{F}_q$$. Suppose also that there exists an isotropic vector $$v \in V$$. It is known that the stabilizer in $$GO_{2n+2}^{\varepsilon}(q)$$ of $$v$$ is a subgroup of type $$q^{2n}\rtimes GO_{2n}^{\varepsilon}(q)$$. Let $$G$$ be a subgroup of $$GO_{2n+2}^{\varepsilon}(q)$$ such that $$q^{2n}\rtimes GO_{2n}^{\varepsilon}(q) \leqslant G < GO_{2n+2}^{\varepsilon}(q)$$. Is it true that $$G$$ always stabilizes a one-dimensional subspace spanned by $$v$$?