Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1332 conjugacy classes of subgroups; only the 449 normal subgroups are shown.
This group has 1332 conjugacy classes of subgroups — too many to draw. Only its 449 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×H(3,𝔽4) | C2 | semidirect |
| 2. #2847 | C2 | semidirect |
| 3. #2812 | C2 | semidirect |
| 4. #2712 | C2 | semidirect |
| 5. #2684 | C2 | semidirect |
| 6. #2547 | C2 | semidirect |
| 7. #2491 | C2 | semidirect |
| 8. #2408 | C2 | semidirect |
| 9. C4×#120 | C2 | semidirect |
| 10. C23×D4 | C22 | semidirect |
| 11. C24×C4 | C22 | semidirect |
| 13. C2×C4⋊Q8 | C22 | semidirect |
| 15. C2×#124 | C22 | semidirect |
| 19. C2×#120 | C22 | semidirect |
| 21. #397 | C22 | semidirect |
| 22. #397 | C22 | semidirect |
| 23. #394 | C22 | semidirect |
| 24. #394 | C22 | semidirect |
| 25. #393 | C22 | semidirect |
| 27. #391 | C22 | semidirect |
| 28. #390 | C22 | semidirect |
| 29. #388 | C22 | semidirect |
| 31. #386 | C22 | semidirect |
| 32. #385 | C22 | semidirect |
| 33. #380 | C22 | semidirect |
| 34. #380 | C22 | semidirect |
| 35. #379 | C22 | semidirect |
| 36. #379 | C22 | semidirect |
| 37. C4×C4⋊C4 | C22 | semidirect |
| 38. C4×C22⋊C4 | C22 | semidirect |
| 57. #120 | D4 | semidirect |
| 82. C24 | C2×D4 | semidirect |
| 121. C22⋊C4 | C2×D4 | semidirect |
| 183. C4 | H(3,𝔽4) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×H(3,𝔽4) | 2 |
| #2847 | 2 |
| #2812 | 2 |
| #2712 | 2 |
| #2684 | 2 |
| #2547 | 2 |
| #2491 | 2 |
| #2408 | 2 |
| C4×#120 | 2 |