Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2716 conjugacy classes of subgroups; only the 569 normal subgroups are shown.
This group has 2716 conjugacy classes of subgroups — too many to draw. Only its 569 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. C22×H(5,𝔽2) | C2 | semidirect |
| 2. C2×#548 | C2 | semidirect |
| 3. #2616 | C2 | semidirect |
| 4. #2435 | C2 | semidirect |
| 5. #2408 | C2 | semidirect |
| 6. #2388 | C2 | semidirect |
| 7. #2341 | C2 | semidirect |
| 11. C23×Q8 | C22 | semidirect |
| 14. C2×#124 | C22 | semidirect |
| 15. C2×C22⋊Q8 | C22 | semidirect |
| 16. C2×#121 | C22 | semidirect |
| 17. C2×#120 | C22 | semidirect |
| 18. C2×C4×Q8 | C22 | semidirect |
| 19. C2×C4×D4 | C22 | semidirect |
| 20. #395 | C22 | semidirect |
| 21. #394 | C22 | semidirect |
| 22. #394 | C22 | semidirect |
| 23. #393 | C22 | semidirect |
| 24. #392 | C22 | semidirect |
| 25. #390 | C22 | semidirect |
| 26. #380 | C22 | semidirect |
| 27. #379 | C22 | semidirect |
| 28. C4×C22⋊C4 | C22 | semidirect |
| 33. C2×#41 | D4 | semidirect |
| 106. C23 | H(5,𝔽2) | semidirect |
| 110. Q8 | #120 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×H(5,𝔽2) | 2 |
| C2×#548 | 2 |
| #2616 | 2 |
| #2435 | 2 |
| #2408 | 2 |
| #2388 | 2 |
| #2341 | 2 |