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Consider the direct product $G=A_4\times \mathbb{Z}_3$, which is a group of order 36. Note that 36 has for divisors 1, 2, 3 , 4, 6, 9, 12, 18, 36.

Which of the following statements is FALSE about existence of subgroups of $G$ and the given order?

$G$ has a subgroup of order $3$.

$G$ has a subgroup of order $4$.

$G$ has a subgroup of order $6$.

$G$ has a subgroup of order $9$.

$G$ has a subgroup of order $18$.