This question came up today in my exam, I said:
Splitting my group up will give me $C_8 \times C_3 \times C_8 \times C_5$.
We also have $24 = 8 \cdot 3$, so I want element of $8$ and $3$ in the group I've split up. So the number of elements of $24$ is going to be
$$\varphi(8) \times \varphi(3) \times \varphi(8) = 32$$
Is this correct?