Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
691 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 691 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 | #2308 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3498 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2888 | C2 | semidirect |
| 3. #2883 | C2 | semidirect |
| 4. #1911 | C2 | semidirect |
| 5. #1910 | C2 | semidirect |
| 6. #1887 | C2 | semidirect |
| 7. #1884 | C2 | semidirect |
| 8. #1827 | C2 | semidirect |
| 9. #1795 | C2 | semidirect |
| 12. #568 | C22 | semidirect |
| 16. #535 | C4 | semidirect |
| 17. #535 | C4 | semidirect |
| 20. #532 | C4 | semidirect |
| 22. C2×#127 | C4 | semidirect |
| 24. #518 | C22 | semidirect |
| 26. #421 | C22 | semidirect |
| 27. #418 | C22 | semidirect |
| 31. C2×#100 | C22 | semidirect |
| 33. #410 | C22 | semidirect |
| 35. #339 | C22 | semidirect |
| 36. H(3,ℤ/4) | C22 | semidirect |
| 41. C2×#41 | C2×C4 | semidirect |
| 51. C22×D4 | C2×C4 | semidirect |
| 59. C4⋊Q8 | C2×C4 | semidirect |
| 65. #127 | C2×C4 | semidirect |
| 66. #127 | C2×C4 | semidirect |
| 68. #124 | C2×C4 | semidirect |
| 69. #124 | C2×C4 | semidirect |
| 70. #124 | C2×C4 | semidirect |
| 87. #117 | C2×C4 | semidirect |
| 91. 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 |
| #2888 | 2 |
| #2883 | 2 |
| #1911 | 2 |
| #1910 | 2 |
| #1887 | 2 |
| #1884 | 2 |
| #1827 | 2 |
| #1795 | 2 |