We prove that a nonperfect locally gradedminimal non-metahamiltonian group $G$ is a soluble group withderived length of at most 4. On the other hand, if $G$ is perfect,then $G/\Phi (G)$ is isomorphic to $A_{5}$, where $\Phi (G)$ isthe Frattini subgroup of $G$ and $A_{5}$ is the alternating group.Moreover, we show that under some conditions,if G is a $p$-group, then G is metabelian, where $p$ is a prime integer.