  On a stratified Lie group $G$ equipped with hypoelliptic heat kernel measure,
we study the behavior of the dilation semigroup on $L^p$ spaces of
log-subharmonic functions. We consider a notion of strong hypercontractivity
and a strong logarithmic Sobolev inequality, and show that these properties are
equivalent for any group $G$. Moreover, if $G$ satisfies a classical
logarithmic Sobolev inequality, then both properties hold. This extends similar
results obtained by Graczyk, Kemp and Loeb in the Euclidean setting.
