  Let $G$ be a finite group and $\mathsf{k}$ a field of characteristic $p$. It
is conjectured in a paper of the first author and John Greenlees that the thick
subcategory of the stable module category StMod$(\mathsf{k}G)$ consisting of
modules whose cohomology is finitely generated over
$\mathsf{H}^*(G,\mathsf{k})$ is generated by finite dimensional modules and
modules with no cohomology. If the centraliser of every element of order $p$ in
$G$ is $p$-nilpotent, this statement follows from previous work. Our purpose
here is to prove this conjecture in two cases with non $p$-nilpotent
centralisers. The groups involved are ${\mathbb Z}/3^r\times\Sigma_3$ ($r> 0$)
in characteristic three and ${\mathbb Z}/2\times A_4$ in characteristic two.
  As a consequence, in these cases the bounded derived category of $C^*BG$
(cochains on $BG$ with coefficients in $\mathsf{k}$) is generated by $C^*BS$,
where $S$ is a Sylow $p$-subgroup of $G$.
