  A classical extremal, or Tur\'an-type problem asks to determine ${\rm ex}(G,
H)$, the largest number of edges in a subgraph of a graph $G$ which does not
contain a subgraph isomorphic to $H$. Alon and Shikhelman introduced the
so-called generalized extremal number ${\rm ex}(G,T,H)$, defined to be the
maximum number of subgraphs isomorphic to $T$ in a subgraph of $G$ that
contains no subgraphs isomorphic to $H$. In this paper we investigate the case
when $G = Q_n$, the hypercube of dimension $n$, and $T$ and $H$ are smaller
hypercubes or cycles.
