Let $G$ be finite nonabelian group. According to Aalipour et al. (2016), they defined the enhanced power graph of $G$ as a simple undirected graph where the vertices are all elements of $G$ and two vertices, $x$ and $y$, are adjacent if $\langle x,y \rangle$ is belong to the same cyclic subgroup of $G$.
If I consider dihedral group, $D_3=\{e,a,a^2,b,ab,{a^2}b\}$. Let $x=a,y=a^2$, then $\langle x,y \rangle = \langle a,a^2 \rangle = \{e,a,a^2\} =\langle a \rangle $, therefore it is belong to the same cyclic subgroup of $G$ and there is an edge between $a$ and $a^2$.
Now, I would like to extend this definition to the subgroup of the group: a simple undirected graph and two subgroups of $G$ that is $H$ and $K$, are adjacent if $\langle H,K \rangle$ is belong to the same cyclic subgroup of $G$.
If we look at the $D_3$, the subgroup of $G$ are $\langle e \rangle ,\langle a \rangle, \langle b \rangle, \langle ab \rangle, \langle a^2b \rangle $.
The question is it is possible to find $\langle H,K \rangle$ that is belong to the same cyclic subgroup of $G$? It is true if I do it like this: Let $\langle H,K \rangle=\langle \langle a \rangle, \langle a^2 \rangle \rangle=\langle a \rangle $, therefore $\langle H,K \rangle$ is belong to the same cyclic subgroup of $G$ and there is an edge between $\langle a \rangle$ and $\langle a^2 \rangle$ ?