Unique subgroup of index 2 in a finite abelian group. 
Suppose $G$ is a finite abelian group, all elements of which are their own inverse. If the order of $G$ is greater than $2$, then prove or disprove that the subgroups of index $2$ in $G$ are not unique.  

Having looked at the example of groups like $G=\{e, a,b,ab\}$ or $G=\{e,a,b,c,ab,ac,bc,abc\}$, it seems that the subgroups of index $2$ are not unique. Is this true in general?

The groups you mention, called elementary abelian $\;2$ - group, are of the form 
$$C_2\times C_2\times\ldots\;,\;\;\text{ with}\;\;C_p\;\;\text{the cyclic group of order two}$$
If the group has order greater than two then there are at least two factors $\;C_p\;$ in the above, and from here there are at least two different subgroups of index two.
Observe that such a group is a vector space over the field $\;\Bbb F_2\cong C_2\;$, and from here, I think, Andreas comment.
