  We study the electronic properties of the confined honeycomb lattice in the
presence of the intrinsic spin-orbit (ISO) interaction and perpendicular
magnetic field, and report on uncommon aspects of the quantum spin Hall
conductance corroborated by peculiar properties of the edge states. The ISO
interaction induces two specific gaps in the Hofstadter spectrum, namely the
'weak' topological gap defined by Beugeling et al [Phys. Rev. B 86, 075118
(2012)], and spin-imbalanced gaps in the relativistic range of the energy
spectrum. We analyze the evolution of the helical states with the magnetic
field and with increasing Anderson disorder. The 'edge' localization of the
spin-dependent states and its dependence on the disorder strength is shown. The
quantum transport, treated in the Landauer-B\"{u}ttiker formalism, reveals
interesting new plateaus of the quantum spin Hall effect (QSHE), and also of
the integer quantum Hall effect (IQHE), in the energy ranges corresponding to
the spin-imbalanced gaps. The properties of the spin-dependent transmittance
matrix that determine the symmetries with respect to the spin, energy and
magnetic field of the longitudinal and transverse resistance are shown.
