Does the order, lattice of subgroups, and lattice of factor groups, uniquely determine a group up to isomorphism?

Question

If we have a two lattices (partially ordered) - one for subgroups, one for factor groups, and we know order of the group we want to have these subgroup and factor group lattices, is such a group unique up to isomorphism (if exists)? Or is there a counterexample?

If that's true, are sufficient conditions on the order and subgroup lattices to guarantee uniqueness? Another way - what if we now lattice for subgroup and group of automorphism of group; is that group uniquely determined by that information?

Thanks for help. (sorry for English)

Answer 1

No, the lattice of subgroups, the lattice of normal subgroups, the order of the group, and the automorphism group do not (even taken together) determine the isomorphism type of a finite group.

Take $G = \mathrm{SmallGroup}(243, 19)$ and $H = \mathrm{SmallGroup}(243, 20)$. There is a bijection $f \colon L(G) \to L(H)$ between their lattices of subgroups such that:

• $|X| = |f(X)|$,
• $X ≅ f(X)$ unless $X = G$,
• $X ≤ Y$ iff $f(X) ≤ f(Y)$,
• $X ⊴ G$ iff $f(X) ⊴ f(G) = H$,
• $G/X ≅ H/f(X)$ whenever $X ≠ 1$ is normal.

Additionally, $\operatorname{Aut}(G) ≅ \operatorname{Aut}(H)$. The fourth bullet shows in particular, that $f$ induces an isomorphism between the lattice of quotient groups of $G$ and the lattice of quotient groups of $H$. The second and fifth bullets show the isomorphism respects everything about the subgroups’ properties as abstract groups.

The groups $G$ and $H$ have presentations

\begin{align*} G &= \bigl\langle a, b, c \mid a^{27} = b^3 = c^3 = 1,\ ba = abc,\ ca = acz,\ cb = bcz \bigr\rangle\text{ where $z = a^9$} \,, \\ H &= \bigl\langle a, b, c \mid a^{27} = b^3 = c^3 = 1,\ ba = abc,\ ca = acz,\ cb = bcz \bigr\rangle\text{ where $z = a^{-9}$} \,. \end{align*}

The function $f$ is induced by a bijection of the underlying sets:

• $f(a^i b^j c^k) = a^i b^j c^k$.

There are no such groups of order dividing $64$ (even just having an isomorphism of subgroup lattices respecting normal subgroups).