Is there a vector bundle of rank $\geq2$ admitting no nonzero proper vector sub-bundles?

Question

Is there a real vector bundle on a topological space $X$, of rank $\geq2$, which admits no nonzero proper vector sub-bundles on $X$?

Certainly there are vector bundles admitting no nonvanishing global section, hence admitting no nonzero globally-trivial vector sub-bundle; however, I'm not sure how to find a rank$\geq2$ vector bundle with no nonzero proper sub-bundle, trivial or otherwise.

Answer 1

Since $H^1(S^2, \mathbb{Z}_2) = 0$, any real line bundle over $S^2$ is trivial and thus admits a nowhere-vanishing section. But $TS^2$ does not (because, e.g., $e(TS^2) = \chi(TS^2)$ is nonzero), so $TS^2$ has no line subbundle.