The subdifferential of a function is a generalization for nonsmooth functions of the concept of gradient. It is frequently used in variational analysis, particularly in the context of nonsmooth optimization. The present work proposes algorithms to reconstruct a polyhedral subdifferential of a function from the computation of finitely many directional derivatives. We provide upper bounds on the required number of directional derivatives when the space is $\mathbb{R}^1$ and $\mathbb{R}^2$, as well as in $\mathbb{R}^n$ where subdifferential is known to possess at most three vertices.