  A nonempty class $\mathfrak{X}$ of finite groups is called complete if it is
closed under taking subgroups, homomorphic images and extensions. We deal with
a classical problem of determining $\mathfrak{X}$-maximal subgroups. We
consider two definitions of submaximal $\mathfrak{X}$-subgroups suggested by
Wielandt and discuss which one better suits our task. We prove that these
definitions are not equivalent yet Wielandt-Hartley's theorem holds true for
either definition of $\mathfrak{X}$-submaximality. We also give some
applications of the strong version of Wielandt-Hartley's theorem.
