  A paratopism is an action on a Latin hypercube of dimension d and order n
which is an element of the wreath product $S_n \wr S_{d+1}$. A paratopism is
said to be an autoparatopism if there is at least one Latin hypercube which is
mapped to itself under the action of the paratopism. In this paper we classify
autoparatopisms of Latin cubes given d = 3 and $n \in \mathbb{Z^+}$, upto the
conjugacy in $S_n \wr S_4$. In order to achieve this objective, we prove that
given an autoparatopism ${\sigma} \in S_n \wr S_{d+1}$, all the conjugates of
${\sigma}$ are autoparatpisms. Also, an important condition is presented, which
states that the cycle structure of the ${\delta}$ in the element ${\sigma}=
({\alpha}_1,{\alpha}_2,{\alpha}_3,{\alpha}_4;{\delta})$ of $S_n \wr S_4$
determines the conjugacy.
