  We consider the nonlinear Schr\"odinger equation \begin{equation*} \Delta u =
\big( 1 +\varepsilon V_1(|y|)\big)u - |u|^{p-1}u \quad \text{in} \quad
\mathbb{R}^N, \quad N\ge 3, \quad p \in \left(1,
\frac{N+2}{N-2}\right).\end{equation*} The phenomenon of pattern formation has
been a central theme in the study of nonlinear Schr\"odinger equations.
However, the following nonexistence of $O(N)$ symmetry breaking solution is
well-known: if the potential function is radial and radially nondecreasing, any
positive solution must be radial. Therefore, solutions of interesting patterns,
such as those with symmetry in a discrete subgroup of $O(N)$, can only exist
after violating the assumptions. For a potential function that is radial but
asymptotically decreasing, a solution with symmetry merely in a discrete
subgroup of $O(2)$ has been presented. These observations pose the question of
whether patterns of higher dimensions can appear. In this study, the existence
of nonradial solutions whose symmetry group is a discrete subgroup of $O(3)$,
more precisely, the orientation-preserving regular tetrahedral group is shown.
