  This work provides calculus for the Fr\'echet and limiting subdifferential of
the pointwise supremum given by an arbitrary family of lower semicontinuous
functions. We start our study showing fuzzy results about the Fr\'echet
subdifferential of the supremum function. Posteriorly, we study in finite- and
infinite-dimensional settings the limiting subdifferential of the supremum
function. Finally, we apply our results to the study of the convex
subdifferential; here we recover general formulae for the subdifferential of an
arbitrary family of convex functions.
