This article gives specific information, namely, fusion systems, about a particular group, namely: quaternion group.
View fusion systems for particular groups | View other specific information about quaternion group
This article discusses possible fusion systems for the quaternion group. 
Summary
| Item | Value | 
| Total number of saturated fusion systems on a concrete instance of the group (strict, not up to isomorphism of fusion systems) | 2 | 
| Total number of saturated fusion systems up to isomorphism | 2 | 
| List of saturated fusion systems with orbit sizes | inner fusion system (orbit size 1 under isomorphisms), non-inner fusion system for quaternion group (orbit size 1 under isomorphisms) | 
| Number of simple fusion systems | None (the reason is that in every fusion system, the fusion subsystem induced on the center of quaternion group is normal) | 
| Number of maximal fusion systems, i.e., fusion systems not contained in bigger fusion systems | 1 (the non-inner case) | 
Description of saturated fusion systems
| Isomorphism type of fusion system | Number of such fusion systems under strict counting | Can the fusion system be realized using a Sylow subgroup of a finite group? | Does the identity functor control strong fusion? This would mean that all fusion occurs in the normalizer | Is the fusion system simple? | Smallest size embedding realizing this fusion system (if any) | 
| inner fusion system | 1 | Yes | Yes | No | as a subgroup of itself | 
| non-inner fusion system for quaternion group | 1 | Yes | Yes | No | Q8 in SL(2,3) | 
Inner fusion system
| Equivalence class under isomorphisms | Subgroups involved | Order | Index | Total number of subgroups | Are all group automorphisms of each subgroup included? | Size of automorphism group from the fusion system | Total number of isomorphisms (including automorphisms and others) = (number of automorphisms)  (number of subgroups)   | 
| trivial subgroup | trivial subgroup | 1 | 8 | 1 | Yes | 1 | 1 | 
|  | center of quaternion group | 2 | 4 | 1 | Yes | 1 | 1 | 
|  | cyclic maximal subgroups of quaternion group | 4 | 2 | 1 | Yes | 2 | 2 | 
|  | cyclic maximal subgroups of quaternion group | 4 | 2 | 1 | Yes | 2 | 2 | 
|  | cyclic maximal subgroups of quaternion group | 4 | 2 | 1 | Yes | 2 | 2 | 
| whole group | whole group | 8 | 1 | 1 | No | 4 | 4 | 
Sylow subgroups realizing this fusion system
This fusion system is realized by a group having quaternion group as its 2-Sylow subgroup if and if it possesses a normal complement, so the 2-Sylow subgroup is a retract of the group and the group is a semidirect product of a normal  -subgroup and the quaternion group, or equivalently the group is a 2-nilpotent group.
-subgroup and the quaternion group, or equivalently the group is a 2-nilpotent group. 
Some examples are below:
Fusion system using an outer automorphism of order three
| Equivalence class under isomorphisms | Subgroups involved | Order | Index | Number of subgroups (=1 iff weakly closed subgroup) | Are all group automorphisms of each subgroup included? | Size of automorphism group from the fusion system | Total number of isomorphisms (including automorphisms and others) = (number of automorphisms)  (number of subgroups)   | 
| trivial subgroup | trivial subgroup | 1 | 8 | 1 | Yes | 1 | 1 | 
|  | center of quaternion group | 2 | 4 | 1 | Yes | 1 | 1 | 
|  ,  ,  | cyclic maximal subgroups of quaternion group | 4 | 2 | 3 | Yes | 2 | 18 | 
| whole group | whole group | 8 | 1 | 1 | Yes | 24 | 24 | 
Sylow subgroups realizing this fusion system
Any situation where quaternion group arises as a 2-Sylow subgroup that is not a retract, i.e., does not have a normal complement. Any such example must admit special linear group:SL(2,3) as a subquotient. Examples are given below: