A galaxy group (or group of galaxies or GrG) is a group of (on the order of) fifty or fewer "significant" galaxies (e.g., with the luminosity of the Milky Way or brighter) that are gravitationally bound, perhaps along with some smaller galaxies, e.g., dwarf galaxies. The Milky Way is within such a group termed the Local Group. A galaxy group can be part of a galaxy cluster. A compact group is a galaxy group within a small volume. A loose group is the other extreme.
The term subgroup (for galaxy subgroup) is definitely used within the Local Group, essentially meaning a significant galaxy plus its satellite galaxies, but likely is sometimes based upon mere proximity. An example is the Milky Way subgroup.