The standard invariant of a subfactor can be viewed in different ways as a discrete group like'' mathematical structure - a lambda-lattice in the sense of Popa, a Jones planar algebra, or a C*-tensor category of bimodules. This discrete group point of view will be the guiding theme of the mini course. After an introduction to different approaches to the standard invariant, I will present joint work with Popa and Shlyakhtenko on the unitary representation theory of these structures, on approximation and rigidity properties like amenability, the Haagerup property or property (T), on (co)homology and $L^2$-Betti numbers. I will present several examples and also discuss a number of open problems on the realization of standard invariants through hyperfinite subfactors.