The statement of the fundamental theorem of finite abelian groups I have is as follows:
Let $G$ be a finite abelian group. Then G can be written as $$\mathbb{Z}_{n_1} \oplus \cdots \oplus \mathbb{Z}_{n_s}$$ where each $n_{i+1} | n_i$ for $1 \le i \le s-1 $ and each $n_i \ge 2$. Moreover, this decomposition is unique.
I am unsure exactly what the uniqueness part means. For example, take $G$ to be a group of order 72. I can write down two compositions $$\mathbb{Z}_{24} \oplus \mathbb{Z}_{3}$$ $$\mathbb{Z}_{12} \oplus \mathbb{Z}_{6}$$ which satisfy the theorem, but do not appear to be unique, i.e. I have two distinct integer sequences $\{24, 3\}$ and $\{12, 6\}$. What am I missing?