  In this paper, we present the quadratic associative symmetry algebra of the
3D nondegenerate maximally quantum superintegrable system. This is the complete
symmetry algebra of the system. It is demonstrated that the symmetry algebra
contains suitable quadratic subalgebras, each of which is generated by three
generators with relevant structure constants, which may depend on central
elements. We construct corresponding Casimir operators and present
finite-dimensional unirreps and structure functions via the realizations of
these subalgebras in the context of deformed oscillators. By imposing
constraints on the structure functions, we obtain the spectrum of the 3D
nondegenerate superintegrable system. We also show that this model is
multiseparable and admits separation of variables in cylindrical polar and
paraboloidal coordinates. We derive the physical spectrum by solving the
Schr\"{o}dinger equation of the system and compare the result with those
obtained from algebraic derivations.
