Some minimal conditions in certain extremely large classes of groups

Let $\mathfrak{L}$ (respectively $\mathfrak{T}$) be the minimal local in the sense of D. Robinson class of groups, containing the class of weakly graded (respectively primitive graded) groups and closed with respect to forming subgroups and series. In the present paper, we completely describe: the $\mathfrak{L}$-groups with the minimal conditions for non-abelian subgroups and for non-abelian non-normal subgroups; the $\mathfrak{T}$-groups with the minimal conditions for (all) subgroups and for non-normal subgroups. By the way, we establish that every $\overline{IH}$-group, belonging to $\mathfrak{L}$, is solvable.