The sheet-based triply periodic minimal surface (TPMS) architecture exhibits effective properties below the stiffness Hashin-Shtrikman upper bound and the strength Suquet upper bound (SU). However, some plate-based architectures have achieved these upper bounds, but limited by their inability to effectively eliminate stress concentrations and manufacturing difficulty. This study introduces a new class of isotropic sheet/shell-based lattice-based metamaterials by reinforcing the Schwartz Primitive TPMS architecture, due to its inherent high shear properties, with minimal surface-based (Schwartz Diamond and F-Rhombic Dodecahedron) and plate-based (simple cubic) architectures, inspired by their known high compressive properties. The aim is to design cubic symmetrical and isotropic lattice-based metamaterials capable of providing a combined high stiffness, strength, and specific energy absorption (SEA), rare in the literature. Effective mechanical properties are estimated using quasi-static finite element simulations validated by compression testing of 3D printed lattices. Reinforcing the Schwartz Primitive architecture with the simple cubic plate structure resulted in an open-cell isotropic lattice material providing a 47% increase in combined stiffness over the parent structure, effective yield strength reaching the SU bound in uniaxial loading, and high SEA beyond regular sheet-based TPMS architectures. This work has designed mechanically-efficient isotropic metamaterials that harness the advantages of minimal surface-based structures.