Let $|G| = p_1^{k_1} p_2^{k_2} \cdots p_n^{k_n}$ with $n>1$.
Let $P$ be a Sylow $p_1$-subgroup of $G$. It is a proper subgroup of $G$. Then, we can enlarge the group to form a maximal subgroup containing $P$. Its order divides the order of $G$ by Lagrange. However, its order is not equal to $|G|$, since it is a proper subgroup of $G$. Hence, there is a prime power $p_i^{k_i}$ with $i \ne 1$ that does not divide $|P|$. By Sylow theory, we can construct a proper Sylow $p_i$-subgroup of $G$ and expand it to another maximal subgroup, creating another maximal subgroup.