We study magnetic multipoles on lattices that descend from a cubic diamond structure using a symmetry informed methodology. They include axial (parity-even) and polar (Dirac) quadrupoles and octupoles on two different magnetic lattices, one centrosymmetric and one polar. The signature absence of (2, 2, 2)-type Bragg reflections from axial entities on a cubic diamond structure is preserved for both low-symmetry magnetic lattices. Collinear magnetic order is established by time-reversal invariance, and body-centre anti-translation. A nonlinear magnetoelectric effect is allowed by the polar symmetry, and any kind of axial effect is forbidden in the centrosymmetric symmetry. The piezomagnetic effect and altermagnetism are forbidden by both magnetic symmetries. A realization of the lattice symmetries is afforded by the compound EuTi2Al20 that has a cubic diamond parent structure. Ruderman-Kittel-Kasuya-Yoshida interactions apparently quell magnetic frustration and install long-range magnetic order of Eu axial dipoles below a temperature ≈ 3.3 K [Kawamata et al., J. Phys. Soc. Jpn. 95, 024701 (2026)]. The signature absence condition for the diamond structure is replaced by Dirac multipoles of d-and f-atomic states at europium positions, and they are different in the centrosymmetric and polar lattices.