<s>[INST] A finite group $G$ has four conjugacy classes, numbered $1$ through $4$, with the following property: $$|Cl_{1}|=1,|Cl_{2}|=4,|Cl_{3}|=4,|Cl_4|=4.$$ Since it is a finite group, the number of elements of the group is given by $|G|=\sum\limits_{k=1}^{4} |Cl_{k}|$. Using Lagrange's theorem, we determine that the only possible orders for elements of this group are $1,2,3,4,6,8,12$.
Our question is: What is the maximum number of elements of order 8 in $G$?[/INST]