Subgroup structure of quaternion group

This article gives specific information, namely, subgroup structure, about a particular group, namely: quaternion group.
View subgroup structure of particular groups | View other specific information about 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 for quick information

Tables classifying subgroups up to automorphisms

Note that all subgroups are normal subgroups, so the quotient group exists in all cases.

Automorphism class of subgroups	List of subgroups	Isomorphism class	Order of subgroups	Index of subgroups	Number of conjugacy classes	Size of each conjugacy class	Total number of subgroups	Isomorphism class of quotient (if exists)	Characteristic subgroup?	Nilpotency class
trivial subgroup	$\{1\}$	trivial subgroup	1	8	1	1	1	quaternion group	Yes	0
center of quaternion group	$\{1,-1\}$	cyclic group:Z2	2	4	1	1	1	Klein four-group	Yes	1
cyclic maximal subgroups of quaternion group	$\{1,-1,i,-i\}$ $\{1,-1,j,-j\}$ $\{1,-1,k,-k\}$	cyclic group:Z4	4	2	3	1	3	cyclic group:Z2	No	1
whole group	$\{1,-1,i,-i,j,-j,k,-k\}$	quaternion group	8	1	1	1	1	trivial group	Yes	2
Total (4 rows)	--	--	--	--	6	--	6	--	--	--

Table classifying isomorphism types of subgroups

Group name	GAP ID	Occurrences as subgroup	Conjugacy classes of occurrence as subgroup	Automorphism classes of occurrence as subgroup	Occurrences as normal subgroup	Occurrences as characteristic subgroup
Trivial group	$(1,1)$	1	1	1	1	1
Cyclic group:Z2	$(2,1)$	1	1	1	1	1
Cyclic group:Z4	$(4,1)$	3	3	1	3	0
Quaternion group	$(8,4)$	1	1	1	1	1
Total	--	6	6	4	6	3

Lattice of subgroups

Lattice of subgroups

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) ${\mathrm{\Gamma }}_2$ 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.

Aspects of subgroup structure relevant for embeddings in bigger groups

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 $⟹$ $p$-automorphism-invariant $⟹$ $p$-core-automorphism-invariant $⟹$ 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

Further information: Coprime automorphism-invariant normal subgroup of Hall subgroup is normalizer-relatively normal, isomorph-normal coprime automorphism-invariant of Sylow implies weakly closed

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.

Maximality notions related to abelianness

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.