  Given a closed smooth manifold $M$ of even dimension $2n\ge6$ with finite
fundamental group, we show that the classifying space ${\rm BDiff}(M)$ of the
diffeomorphism group of $M$ is of finite type and has finitely generated
homotopy groups in every degree. We also prove a variant of this result for
manifolds with boundary and deduce that the space of smooth embeddings of a
compact submanifold $N\subset M$ of arbitrary codimension into $M$ has finitely
generated higher homotopy groups based at the inclusion, provided the
fundamental group of the complement is finite. As an intermediate result, we
show that the group of homotopy classes of simple homotopy self-equivalences of
a finite CW complex with finite fundamental group is up to finite kernel
commensurable to an arithmetic group.
