  We study some properties of the coset poset associated with the family of
subgroups of class $\leq 2$ of a nilpotent group of class $\leq 3$. We prove
that under certain assumptions on the group the coset poset is simply-connected
if and only if the group is $2$-Engel, and $2$-connected if and only if the
group is nilpotent of class $2$ or less. We determine the homotopy type of the
coset poset for the group of $4\times 4$ upper unitriangular matrices over
$\mathbb{F}_p$, and for the Burnside groups of exponent $3$.
