I am studying lattice diagrame of subgroups of groups. and I came to know about the lattices of $C_4\times C_2$ and $C_8\times C_2$ over here and here.
But the problem is: I am unable to understand what does it mean by the powers in the diagramme? I mean to say, you can find $C_2^8, C_2^4, C_2^2$ etc in the lattice of $C_8\times C_2$ . Can you please help me to understand the meaning of it?
It looks like $C_8^1$ and $C_8^2$ are the two cyclic subgroups of $C_8 \times C_2$ of order $8$, so the superscripts $1$ and $2$ are just labels to distinguish them.
In element lattice on the right we see that $C_8^1$ is generated by $(a,1)$ and $C_8^2$ is generated by $(a,b)$, where $a$ and $b$ are the generators for $C_8$ and $C_2$ respectively.