The three-dimensional charge density wave (CDW) compound Lu2βIr3βSi5β undergoes a first-order CDW phase transition at around 200 K. An atypical CDW state is found, that is characterized by an incommensurate CDW with q=[0.2499(3),0.4843(4),0.2386(2)] at 60 K, and a large orthorhombic-to-triclinic lattice distortion with Ξ²=91.945(2)β. We present the modulated crystal structure of the incommensurate CDW state. Structural analysis shows that the CDW resides on the zigzag chains of iridium atoms along c. The structural distortions are completely similar between nonmagnetic Lu2βIr3βSi5β and previously studied isostructural magnetic Er2βIr3βSi5β5 with the small differences explained by the different values of the atomic radii of Lu and Er. Such a similarity is unique to R2βIr3βSi5β (R=rareearth). It differs from, for example, the rare-earth CDW compounds R5βIr4βSi10β for which Lu5βIr4βSi10β and Er5Ir4Si10 possess entirely different CDW states. We argue that the mechanism of CDW formation, thus, is different for R2βIr3βSi5β and R5βIr4βSi10β