A quandle is an algebraic structure whose axioms are related to the Reidemeister moves used in knot theory. In this paper, we investigate the conjugate quandle of the orientation-preserving isometry group \(\textrm{PSL}(2, \mathbb {C})\) of hyperbolic 3-space and its subquandles. We introduce a quandle, denoted by \(Q(\varGamma , \gamma )\) , associated with a pair \((\varGamma , \gamma )\) . Here, \(\varGamma \) is a Kleinian group, and \(\gamma \) is a non-trivial element of \(\varGamma \) . This construction can be regarded as a generalization of knot quandles to hyperbolic knots. Moreover, for pairs \((\varGamma , \gamma )\) satisfying certain conditions, we construct the canonical map from \(Q(\varGamma , \gamma )\) to the conjugate quandle of \(\textrm{PSL}(2,\mathbb {C})\) , which is an injective quandle homomorphism with a discrete image.
