$SO(4,2)$ is called the full dynamical group of the Kepler (or Hydrogen atom problem). The $SO(4)$ , $SO(3,2)$ and $SO(4,1)$ subgroups of $SO(4,2)$ are called partial dynamical groups.
Unlike symmetry groups which commute with the Hamiltonian, dynamical groups do not. They have the following properties:
The system's phase space is a coadjoint orbit of the group, or equivalently,
The system's Hilbert space is spanned by an irreducible representation of the group.
In many cases, although, it is not necessary, the Hamiltonian itself is a generator of the dynamical group.
The equivalence of the points 1. and 2. above stems from the fact that in the case under study there is a correspondence between coadjoint orbits and irreducible representations.
The partial dynamical groups span only part of the Hydrogen atom spectrum through their irreducible representations:
An $SO(4)$ irreducible representation spans the state vectors corresponding to a single energy shell of a bound state (fixed (and quantized) $n$ and varying $l$, $m$) .
An $SO(3,2)$ irreducible representation spans the full continuous spectrum and an irreducible representation of $SO(4,1)$ spans the full bound spectrum.
$SO(4,2)$ is the smallest group whose irreducible representations span both the continuous and the discrete spectrum.
The use of dynamical group representations reduces the problem of finding the Hamiltonian spectrum to an algebraic problem, instead of a solution of differential equations.