A multi-vortex system nested in a Gaussian beam forms complex paths when it propagates in a free space. We consider a configuration of vortex quadrupole and present a systematic way to control the creation and annihilation of vortex pairs using only one parameter. It is shown that three classes of quadrupole vortex beams can be identified by the number of vortex pairs created or annihilated. The beam symmetry, asymptotic behavior, and stability are analyzed. The main theoretical results are verified experimentally.