The scrambling rate lambda(L) associated with the exponential growth of out-of-time-ordered correlators can be used to characterize quantum chaos. Here we use a particular Majorana fermion representation of spin-1/2 systems to study quantum chaos in the Dicke model. We take the system to be in thermal equilibrium and compute lambda(L) throughout the phase diagram to leading order in 1/N. We find that the chaotic behavior is strongest close to the critical point. At high temperatures lambda(L) is nonzero over an extended region that includes both the normal and superradiant phases. At low temperatures lambda(L) is nonzero in (a) close vicinity of the critical point and (b) a region within the superradiant phase. In the process we also derive an effective theory for the superradiant phase at finite temperatures Our formalism does not rely on the assumption of total spin conservation.