Let $G$ be a finite group. We say that the lattice of subgroups of $G$ is graded if it is possible to assign a non-negative integer rank $r(H)$ to each subgroup $H$ in such a way that the following two properties hold.
- If $H \leq K$, then $r(H) \leq r(K)$.
- If $H \leq K$ and no group is strictly between $H$ and $K$, then $r(K)=r(H)+1$.
Examples of groups with graded lattices include
- Abelian groups ($r(H)$ is the number of prime factors in $|H|$, counted with multiplicity).
- The symmetric group $S_3$ (the identity has rank $0$, the whole group has rank $2$, all other subgroups have rank $1$)
- The quaternion group (identity has rank $0$, $\{1, -1\}$ has rank $1$, the subgroups generated by $i$, $j$, and $k$ have rank $2$, and the whole group has rank $3$).
An example of a group without a graded lattice is $A_4$, the alternating group on $4$ elements. One way to see this is to note that you have the two chains $$\{e\} \leq \{e, (12)(34)\} \leq \{e, (12)(34), (13)(24), (14)(23)\} \leq A_4$$ and $$\{e\} \leq \{e, (123), (132)\} \leq A_4$$ The two chains are both maximal ($A_4$ has no subgroups of order $6$), but looking at the two chains give conflicting values for the rank of the whole group vs. that of the identity.
Is there a general description of which groups do or do not have graded subgroup lattices?
I feel like this should have been studied somewhere before, but can't find anything on it.