Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
678 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 678 conjugacy classes of subgroups — too many to draw. Only its 181 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#99 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×D4 | C24 |
| prime radical | #3498 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×D4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2883 | C2 | semidirect |
| 3. #2128 | C2 | semidirect |
| 4. #2127 | C2 | semidirect |
| 5. #2126 | C2 | semidirect |
| 6. C2×#353 | C2 | semidirect |
| 7. C2×H(5,𝔽2) | C4 | semidirect |
| 12. #535 | C4 | semidirect |
| 13. #535 | C4 | semidirect |
| 16. #532 | C4 | semidirect |
| 18. C2×#127 | C4 | semidirect |
| 21. C4×#41 | C4 | semidirect |
| 23. #410 | C22 | semidirect |
| 25. C2×#99 | C22 | semidirect |
| 26. #354 | C22 | semidirect |
| 27. #353 | C22 | semidirect |
| 28. #353 | C22 | semidirect |
| 34. C2×#41 | C2×C4 | semidirect |
| 35. C2×#41 | C2×C4 | semidirect |
| 43. C22×D4 | C2×C4 | semidirect |
| 44. C22×D4 | C2×C4 | semidirect |
| 48. C4⋊Q8 | C2×C4 | semidirect |
| 51. #127 | C2×C4 | semidirect |
| 52. #127 | C2×C4 | semidirect |
| 53. #127 | C2×C4 | semidirect |
| 56. #124 | C2×C4 | semidirect |
| 57. #124 | C2×C4 | semidirect |
| 58. #124 | C2×C4 | semidirect |
| 77. #117 | C2×C4 | semidirect |
| 78. #117 | C2×C4 | semidirect |
| 80. C2×C42 | C2×C4 | semidirect |
| 81. C2×C42 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3498 | 2 |
| #2883 | 2 |
| #2128 | 2 |
| #2127 | 2 |
| #2126 | 2 |
| C2×#353 | 2 |