Ternary alloys Co-Zn-Mn with β-Mn-type structure exhibit helimagnetic order and skyrmion lattice states in a low-Mn-concentration region. Recent studies on Co_10−x/2Zn_10−x/2Mn_x (CZM) revealed that the magnetic modulation period and the paramagnetic-to-helimagnetic transition temperature are reduced with increasing Mn concentration x [Tokunaga et al., Nat. Commun. 6, 7638 (2015)] and that the helimagnetic order is replaced with a spin-glass state in a high-x region [Karube et al., Sci. Adv. 4, eaar7043 (2018)]. To understand the microscopic mechanism of the Mn substitution effect, we have performed crystal structure analyses of a series of CZM samples with x=4,5,6,7,8,10,12, and 16 by neutron powder diffraction. We present a model for the site occupancies of Co, Zn, and Mn in the unit cell, which has two independent crystallographic sites, specifically 8c and 12d sites. Zn atoms are always accommodated in the 12d sites, and Co atoms prefer the 8c sites, although some of them also exist in the 12d sites in a low-x region. Mn atoms mainly prefer the 12d sites but also share the 8c site with Co atoms in the region of x≥4 . A comparison between the site occupancies and the x−T magnetic phase diagram suggests that the amount of Mn in the 8c sites is relevant to the emergence of the spin-glass state.