Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
626 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 626 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 | #3500 | 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. #2129 | C2 | semidirect |
| 4. #2127 | C2 | semidirect |
| 5. #2126 | C2 | semidirect |
| 6. C2×#354 | C2 | semidirect |
| 7. C2×2−1+4 | C4 | semidirect |
| 12. #536 | C4 | semidirect |
| 14. #535 | C4 | semidirect |
| 16. #532 | C4 | semidirect |
| 18. C2×#124 | C4 | semidirect |
| 21. C4×#41 | C4 | semidirect |
| 26. #354 | C22 | semidirect |
| 27. #354 | C22 | semidirect |
| 28. #353 | C22 | semidirect |
| 35. C2×#41 | C2×C4 | semidirect |
| 36. C2×#41 | C2×C4 | semidirect |
| 40. C22×Q8 | C2×C4 | semidirect |
| 41. C22×Q8 | C2×C4 | semidirect |
| 46. C4⋊Q8 | C2×C4 | semidirect |
| 47. C4⋊Q8 | C2×C4 | semidirect |
| 48. #127 | C2×C4 | semidirect |
| 52. #124 | C2×C4 | semidirect |
| 53. #124 | C2×C4 | semidirect |
| 54. #124 | C2×C4 | semidirect |
| 55. #124 | C2×C4 | semidirect |
| 75. #117 | C2×C4 | semidirect |
| 76. #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 |
|---|---|
| #3500 | 2 |
| #2883 | 2 |
| #2129 | 2 |
| #2127 | 2 |
| #2126 | 2 |
| C2×#354 | 2 |