Is the log of closed connected subgroups a vector space?

Question

Let $G$ be a simply connected nilpotent Lie group (so that the exponential map $\exp:{\mathfrak g}\to G$ from the Lie agebra ${\mathfrak g}$ to $G$ is a diffeomorphism, and hence so is the inverse $\log:G\to{\mathfrak g}$). Let $H\subset G$ be a closed connected subgroup. Is $\log(H)$ necessarily a subspace? And if $H$ is not necessarily connected, is $\log H$ an additive subgroup of ${\mathfrak g}$?