  We show that a finitely generated subgroup of the genus two handlebody group
is stable if and only if the orbit map to the disk graph is a quasi-isometric
embedding. To this end, we prove that the genus two handlebody group is a
hierarchically hyperbolic group, and that the maximal hyperbolic space in the
hierarchy is quasi-isometric to the disk graph of a genus two handlebody by
appealing to a construction of Hamenst\"adt-Hensel. We then utilize the
characterization of stable subgroups of hierarchically hyperbolic groups
provided by Abbott-Behrstock-Berlyne-Durham-Russell. We also present several
applications of the main theorems, and show that the higher genus analogues of
the genus two results do not hold.

Geometric Topology;Group Theory