Finite Group DatabasebetaFinite Group DatabasebetaOrder 9216.
At least 11491 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has at least 11491 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #72226446 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #72226400 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #72218837 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #72218415 | C22 | 1 | 1 | yes | characteristic, complemented |
| 5 | #72209718 | D3 | 1 | 1 | yes | characteristic, complemented |
| 6 | #72209700 | D3 | 1 | 1 | yes | characteristic, complemented |
| 7 | C24×C22⋊A4 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 8 | C22⋊(C22⋊(C22⋊A4)) | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 9 | C22×#1020 | S4 | 1 | 1 | yes | characteristic, complemented |
| 10 | C28 | D32 | 1 | 1 | yes | characteristic, complemented |
| 11 | C22×C22⋊A4 | C2×S4 | 1 | 1 | yes | characteristic |
| 12 | C26 | D3×S4 | 1 | 1 | yes | characteristic |
| 13 | C22⋊A4 | #1020⋊C2 | 1 | 1 | yes | characteristic, complemented |
| 14 | C24 | #10593244 | 1 | 1 | yes | characteristic, complemented |
| 15 | C24 | (#1020⋊C3)⋊C2 | 1 | 1 | yes | characteristic, complemented |
| 16 | C22 | #72215889 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #72218415 | C22 |
| Frattini | C22 | #72215889 |
| Fitting | C28 | D32 |
| socle | C26 | D3×S4 |
| LCS | G ▷ #72218415 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72226446 | C2 | semidirect |
| 2. #72226400 | C2 | semidirect |
| 3. #72218837 | C2 | semidirect |
| 4. #72218415 | C22 | semidirect |
| 5. #72209718 | D3 | semidirect |
| 6. #72209700 | D3 | semidirect |
| 7. C24×C22⋊A4 | C2×D3 | semidirect |
| 8. C22⋊(C22⋊(C22⋊A4)) | C2×D3 | semidirect |
| 9. C22×#1020 | S4 | semidirect |
| 10. C28 | D32 | semidirect |
| 13. C22⋊A4 | #1020⋊C2 | semidirect |
| 14. C24 | #10593244 | semidirect |
| 15. C24 | (#1020⋊C3)⋊C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72226446 | 2 |
| #72226400 | 2 |
| #72220400 | 3 |
| #72218837 | 2 |
| #72216034 | 3 |
| #72215889 | 4 |
| #10593244 | 16 |