We determine the exact value of the $η$-constant and the multiwise Davenport constants for finite abelian groups of rank three having the form $G \simeq C_2 \oplus C_{n_2} \oplus C_{n_3}$ with $2 \mid n_2 \mid n_3$. Moreover, we determine the Erdős-Ginzburg-Ziv constant of these groups under the assumption that $n_2/2$ has Property D or $n_2 = n_3$.