Numerical simulations of the equations of motion of 300 charged particles confined to a plane with an additional magnetic field orthogonal to the plane reproduce recently observed self-organization of non-neutral plasmas into a small number of interacting vortices. In the presence of damping we observe crystallized vortices, i.e. vortices with regular internal structure. We also observe crystallized vortex crystals, i.e. geometric patterns of crystallized vortices. Fractal vortex arrangements are investigated and found to be stable. Our results are relevant for quantum dots and artificial atoms.