Suppose we have a group $G$ of order $24$, and we know apriori that this group has $5$ conjugacy classes, or sizes $1$, $3$, $6$, $6$, and $8$.
The question I'm presented with is whether or not one can conclude that $G$ has exactly two normal subgroups.
I know that normal subgroups are the union of conjugacy classes. Moreover, this union must include the singleton conjugacy class for the identity, and the order of the union should divide the order of the group. It seems plausible in this case then that we could take the conjugacy classes and obtain two normal subgroups of order $1 + 3 = 4$ and $1 + 3 + 8 = 12$. But is this guaranteed?