Subgroup lattice of $\mathbb{Z}/6\mathbb{Z}$

Subgroups.
- Order $1$: $\{\bar0\}$.
- Order $2$: $\{\bar0,\bar3\}$ (generated by $\bar3$).
- Order $3$: $\{\bar0,\bar2,\bar4\}$ (generated by $\bar2$).
- Order $6$: the whole group.

Lattice.
$$\{\bar0\}\subset\{\bar0,\bar3\}\subset\mathbb{Z}/6\mathbb{Z},$$
$$\{\bar0\}\subset\{\bar0,\bar2,\bar4\}\subset\mathbb{Z}/6\mathbb{Z}.$$
The subgroups of orders $2$ and $3$ are incomparable.

Conclusion. Four subgroups, forming a diamond lattice.