In this talk, we explore the geometry of the handlebody group, i.e. the mapping class group of a handlebody. This talk will include a heuristic description of hierarchically hyperbolic spaces, and using this description, we will see that the handlebody group of genus two is a hierarchically hyperbolic group (HHG). Then, by analyzing the structure of the maximal hyperbolic space associated to the handlebody group and utilizing the characterization of stable subgroups of HHGs, I will show that the stable subgroups of the genus two handlebody group are precisely those subgroups whose orbit maps are quasi-isometric embeddings into the disk graph. Lastly, we will see that various properties of the genus two handlebody group do not hold for higher genus handlebody groups.