The bending-dominated honeycomb-like cores are known for their weak out-of-plane strength and high directional dependency. This paper proposes a new strategy for controlling anisotropy in triply periodic minimal surface-based lattices (e.g., Schwartz Primitive, Pp) to enhance control of the deformation behavior of cores in sandwich panels subjected to impact and crushing loads. It considered creating a hierarchy in the topology of the P lattice through a multi-stage variation in the periodicity along the tested direction to have Php-latticed core. Furthermore, the relative density of the Php-latticed core was graded in a bi-linear manner with minimum relative density at the center, which yields Phpg lattice, to control the collapse behavior of the topology (honeycomb-like Pp i.e., Pc) at the center of the hybrid lattice. Additionally, this study designed a new type of minimal surface-based lattice (Pmp) from an implicit synthesis of Pp and Pc topologies. From the elasto-plastic-damage simulations, the Phpg-latticed core resisted perforation, while the Pmp-latticed core showed significantly improved toughness to the crushing load at quasi-static conditions compared to the other designs. The findings provide valuable insights into controlling the deformation of the core of sandwich panels subjected to impact and crushing loads.