When studying about groups, we can often grasp the structure of a small group "internally". For instance, we can "see" that $\mathbb Z/3\mathbb Z$ as a three-fold symmetric shape;more complicated examples include perceiving the difference in structure of $\mathbb Z/4\mathbb Z$ and $(\mathbb Z/2\mathbb Z)^2=K_4$ intuitively.
So the question is:
Is there a graphical way to convey the structural information of small groups intuitively?
I realize that it is sort of vague, but any answer is appreciated. You can just focus on specific examples of groups, but a general method would be great. Thanks.