Subgroup structure of quaternion group

The quaternion group is an 8-element group on the set:

$$\{ 1,-1,i,-i,j,-j,k,-k \}$$.

See the multiplication table of the group at quaternion group and more on the element structure at element structure of quaternion group.

Tables classifying subgroups up to automorphisms
Note that all subgroups are normal subgroups, so the quotient group exists in all cases.

Table classifying isomorphism types of subgroups
The first part of the GAP ID is the order of the subgroup.

Table classifying number of subgroups by order
Note that these orders satisfy the congruence condition on number of subgroups of given prime power order: the number of subgroups of order $$p^r$$ is congruent to $$1$$ modulo $$p$$. Here, $$p = 2$$, so this means that the number of subgroups of any given order is odd.

The entire lattice
The lattice of subgroups of the quaternion group has the following interesting features:


 * Since all subgroups are normal, but the group is not abelian, the inner automorphism group is a nontrivial group of automorphisms that fixes all elements of the lattice.
 * The outer automorphism group is isomorphic to the symmetric group of degree three. This group fixes each of the three characteristic subgroups: the trivial subgroup, the whole group, and the two-element center $$\{ -1, 1 \}$$. The three normal subgroups of order four are not characteristic and the elements of the outer automorphism group give rise to permutations on this set of subgroups.
 * The lattice does not enjoy reverse symmetry, in the sense that it is not isomorphic to its reverse lattice. This is because there are three maximal subgroups while there is only one minimal subgroup.

The sublattice of normal subgroups
Note that since all subgroups are normal, the lattice of subgroups coincides with the lattice of normal subgroups. The lattice of normal subgroups of the quaternion group is isomorphic as a lattice to the lattice of normal subgroups of the dihedral group:D8. However, the lattice of all subgroups of the dihedral group is substantially bigger.

This is in the Hall-Senior family (up to isocliny) $$\Gamma_2$$ and has the Hall-Senior genus as the dihedral group:D8. The general picture of the lattice of normal subgroups of that Hall-Senior genus is given below:



The sublattice of characteristic subgroups
The lattice of characteristic subgroups of the quaternion group is a totally ordered lattice with three elements: the trivial subgroup, the unique subgroup of order two, and the whole group. These subgroups are also fully characteristic, in fact verbal.

2-automorphism-invariance and 2-core-automorphism-invariance
A subgroup of a $$p$$-group is termed a p-automorphism-invariant subgroup if it is invariant under all automorphisms of the whole group whose order is a power of $$p$$, while it is termed a p-core-automorphism-invariant subgroup if it is invariant under all automorphisms in the $$p$$-core of the automorphism group. We have:

Characteristic $$\implies$$ $$p$$-automorphism-invariant $$\implies$$ $$p$$-core-automorphism-invariant $$\implies$$ normal

In the case of the quaternion group, we have the following:


 * The characteristic subgroups are the same as the $$2$$-automorphism-invariant subgroups, namely: the whole group, the trivial subgroup, and the center. Thus, the only subgroups of the quaternion group that are normal in every $$2$$-group containing it are the whole group, the trivial subgroup, and the center. In other words, for each of the subgroups of order four, we can find bigger $$2$$-groups containing the quaternion group in which these are not normal.
 * The $$2$$-core-automorphism-invariant subgroups are the same as the normal subgroups, which are the same as all subgroups.

Coprime automorphism-invariance
The coprime automorphism-invariant subgroups are the same as the coprime automorphism-invariant normal subgroups, which are the same as the characteristic subgroups. In other words, these are only the trivial subgroup, the whole group, and the center. In particular, this means that for any of the subgroups of order four, we can find a bigger group in which the quaternion group is Sylow, but that particular subgroup is not a normalizer-relatively normal subgroup.

Abelian subgroups of maximum order
There are three abelian subgroups of maximum order: the three cyclic normal subgroups generated by $$i,j,k$$ respectively. These are all automorphic subgroups. Together, they generate the whole group. These are also the only subgroups maximal among abelian subgroups.

In particular, the join of abelian subgroups of maximum order, sometimes called the Thompson subgroup and denoted by $$J$$, is the whole group. Thus, the ZJ-subgroup, which is defined as the center of this Thompson subgroup, equals the center of the whole group.

Abelian subgroups of maximum rank
The quaternion group has rank one: every abelian subgroup is cyclic. Thus, the abelian subgroups of maximum rank are the center and the three subgroups of order four. The join of these, i.e., the join of abelian subgroups of maximum rank, is thus the whole group.

Elementary abelian subgroups of maximum order
There is exactly one elementary abelian subgroup of maximum order: the center.