  The goal of this paper is to give a conjectural census of complex hyperbolic
sporadic groups. We prove that only finitely many of these sporadic groups are
lattices. We also give a conjectural list of all lattices among sporadic
groups, and for each group in the list we give a conjectural presentation, as
well as a list of cusps and generators for their stabilisers. We describe
strong evidence for these conjectural statements, showing that their validity
depends on the solution of reasonably small systems of quadratic inequalities
in four variables.
