Architected metamaterials can exhibit advanced and controllable mechanical and dynamic properties, ranging from high stiffness to weight ratios, preferential energy absorption, band gaps, and negative refraction. In terms of their dynamic properties, periodicity on the size scale of the wavelength can lead to band gaps, or ranges of frequencies that cannot propagate. However, to achieve low frequency band gaps, which is of interest for structural vibration applications, requires impractically large unit cells. Locally resonant microstructures can lead to low frequency and narrowband band gaps but typically require large masses and soft materials, which limits structural applications that have strict weight and stiffness/strength requirements. Our recent work on elastic metastructures has numerically and experimentally shown that the locally resonant inclusions embedded in a 3D printed periodic lattice can lower and widen Bragg scattering band gaps, addressing limitations of purely periodic or purely locally resonant structures. In the present work, we numerically study the influence of different truss lattice topologies on the dynamic and static behavior of metastructures that combine a periodic lattice geometry with locally resonant inclusions. Results are presented in terms of efficiency parameters that shed light on the combined mechanical and dynamic properties of the metastructures, in terms of relative density, stiffness, and band gap width and frequency. These metastructures can inform the design of tailored materials that have desired mechanical and dynamic properties for applications in e.g. aerospace components and energy infrastructure.