  We prove that if $G$ is a finite flat group scheme of $p$ power rank over a
perfect field of characteristic $p$, then the second crystalline cohomology of
its classifying stack $H^2_{crys}(BG)$ recovers the Dieudonn\'e module of $G$.
We also provide a calculation of crystalline cohomology of classifying stack of
abelian varieties. We use this to prove that crystalline cohomology of the
classifying stack of a $p$-divisible group is a symmetric algebra (in degree
$2$) on its Dieudonn\'e module. We also prove mixed characteristic analogues of
some of these results using prismatic cohomology.
