Homotopy equivalence of a subspace

Question

Suppose $X$ is a CW complex which is homotopy equivalent to a wedge sum of $k$ spheres $S^d$, for some integers $k,d \geq 1$. Is it true that $X$ contains a subcomplex $Y$ which is homotopy equivalent to a sphere $S^d$? If so, how can it be proved? Otherwise, is there an easy counterexample? (Sorry if this is a silly question, I'm just starting to learn about topology.) Thanks.