Countable abelian Group Vs Non-Abelian countable Group

Question

Countable abelian Group Vs Non-Abelian countable Group

which of the following statement is True/false ?

$1$. Countable abelian group can have only countably many distinct subgroup

$2.$ Countable Non-abelian group can have only countably many distinct subgroup

My Try : For $1.$ i got false by countable group, uncountably many distinct subgroup?

Im confused about $2$

Answer 1

Let $G$ be a countable abelian group with uncountably many subgroups. Then $G\times S_3$ is a countable non-abelian group with uncountably many subgroups (even with uncountably many of the form $H\times S_3$, i.e., also non-abelian).