  Let $\lambda(G)$ be the maximum number of subgroups in an irredundant
covering of the finite group $G$. We prove that if $G$ is a group with
$\lambda(G) \leqslant 6$, then $G$ is supersolvable. We also describe the
structure of the groups $G$ with $\lambda(G)=6$. Moreover, we show that if $G$
is a group with $\lambda(G) \leqslant 30$, then $G$ is solvable.
