There are 3 levels of groups in mage society.
Examples of factions would be Council of Nine, Disparate Alliance (Mage20), Technocracy, Marauders, Nephandi.
A group inside the Technocracy is a Convention. A group inside the Council of 9 is a Tradition. Examples of groups would be Order of Hermes, New World Order, Wu Lung.
Subgroups are firmly within a group. The Friends of Courage in the Technocracy are not inside one Convention so they can be called a society but not a subgroup. The Statisticians are inside Iteration X so they are a subgroup. Subgroups are called Methodologies in the Technocracy. The Council of 9 doesn’t specify so they have sects, houses, etc. Examples of subgroups would be Genegineers (Progenitors) & House Bonisagus (Order of Hermes).