Finite Group DatabasebetaFinite Group DatabasebetaOrder 26816.
334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D419 |
| commutator, socle, cosocle | C22×C419 | C24 |
| Fitting | C4×D4×C419 | C2 |
| hypercenter | C4×D4 | D419 |
| hypocenter | C419 | #546 |
| LCS | G ▷ C22×C419 ▷ C419 | |
| UCS | C1 ◁ C22 ◁ C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#20269411 | C2 | semidirect |
| 2. C22×D1676 | C2 | semidirect |
| 3. #27211706 | C2 | semidirect |
| 4. #27211703 | C2 | semidirect |
| 5. #27211697 | C2 | semidirect |
| 6. #27211692 | C2 | semidirect |
| 7. #27211691 | C2 | semidirect |
| 8. #27211686 | C2 | semidirect |
| 9. C4×D1676 | C2 | semidirect |
| 10. C22⋊C4×D419 | C2 | semidirect |
| 11. C4×D4×C419 | C2 | semidirect |
| 14. C2×Dic838 | C22 | semidirect |
| 19. C2×C22⋊D419 | C22 | semidirect |
| 20. C2×C22⋊D419 | C22 | semidirect |
| 21. C2×C4×D419 | C22 | semidirect |
| 22. C2×C4×D419 | C22 | semidirect |
| 23. C4⋊Dic419 | C22 | semidirect |
| 24. Dic419⋊C4 | C22 | semidirect |
| 26. (C2×D419)⋊C4 | C22 | semidirect |
| 27. (C2×D419)⋊C4 | C22 | semidirect |
| 28. (C2×D419)⋊C4 | C22 | semidirect |
| 29. (C2×D419)⋊C4 | C22 | semidirect |
| 32. C22×C4×C419 | C22 | semidirect |
| 33. C4⋊C4×C419 | C22 | semidirect |
| 35. C22⋊C4×C419 | C22 | semidirect |
| 36. C42×C419 | C22 | semidirect |
| 45. D1676 | D4 | semidirect |
| 67. C22×C419 | #41 | semidirect |
| 90. C419 | #546 | semidirect |
| 101. C4×D4 | D419 | semidirect |
| 116. C22×C4 | C2×D419 | semidirect |
| 119. C4⋊C4 | C2×D419 | semidirect |
| 126. C22⋊C4 | C2×D419 | semidirect |
| 127. C42 | C2×D419 | semidirect |
| 165. C22 | #20269411 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#20269411 | 2 |
| C22×D1676 | 2 |
| #27211706 | 2 |
| #27211703 | 2 |
| #27211697 | 2 |
| #27211692 | 2 |
| #27211691 | 2 |
| #27211686 | 2 |
| C4×D1676 | 2 |
| C22⋊C4×D419 | 2 |
| C4×D4×C419 | 2 |
| #546 | 419 |