  We study a three-dimensional chiral second order topological insulator (SOTI)
subject to a magnetic field. Via its gauge field, the applied magnetic field
influences the electronic motion on the lattice, and via the Zeeman effect, the
field influences the electronic spin. We compare two approaches to the problem:
an effective surface theory, and a full lattice calculation. The surface theory
predicts a massive Dirac spectrum on each of the gapped surfaces, giving rise
to Landau levels once the surfaces are pierced by magnetic flux. The surface
theory qualitatively agrees with our lattice calculations, accurately
predicting the surface gap as well as the spin and orbital components of the
states at the edges of the surface Dirac bands. In the context of the lattice
theory, we calculate the spectrum with and without magnetic field and find a
deviation from the surface theory when a gauge field is applied. The energy of
the lowest-lying Landau level is found closer to zero than is predicted by the
surface theory, which leads to an observable magnetotransport signature: inside
the surface gap, there exist different energy regions where either one or two
chiral hinge modes propagate in either direction, quantizing the differential
conductance to either one or two conductance quanta.
