  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 $\mathrm{PSL}(2,
\mathbb{C})$ of hyperbolic 3-space and its subquandles. We introduce a quandle,
denoted by $Q(\Gamma, \gamma)$, associated with a pair $(\Gamma, \gamma)$.
Here, $\Gamma$ is a Kleinian group, and $\gamma$ is a non-trivial element of
$\Gamma$. This construction can be regarded as a generalization of knot
quandles to hyperbolic knots. Moreover, for pairs $(\Gamma, \gamma)$ satisfying
certain conditions, we construct the canonical map from $Q(\Gamma, \gamma)$ to
the conjugate quandle of $\mathrm{PSL}(2, \mathbb{C})$, which is an injective
quandle homomorphism with a discrete image.
