Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1521 conjugacy classes of subgroups; only the 495 normal subgroups are shown.
This group has 1521 conjugacy classes of subgroups — too many to draw. Only its 495 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#121 | C2 | semidirect |
| 2. #3186 | C2 | direct (N × H) |
| 3. C2×#480 | C2 | semidirect |
| 4. C2×#466 | C2 | semidirect |
| 5. C2×#447 | C2 | semidirect |
| 6. C22×C22⋊C8 | C2 | semidirect |
| 8. C23×D4 | C22 | semidirect |
| 10. C22×D8 | C22 | semidirect |
| 11. C23×C8 | C22 | semidirect |
| 16. C2×#121 | C22 | semidirect |
| 17. C2×#121 | C22 | semidirect |
| 18. C22×C4⋊C4 | C22 | semidirect |
| 20. #480 | C22 | semidirect |
| 21. #466 | C22 | semidirect |
| 22. #447 | C22 | semidirect |
| 23. C2×#107 | C22 | semidirect |
| 24. C2×#102 | C22 | semidirect |
| 25. C2×C22⋊C8 | C22 | semidirect |
| 26. C2×C22⋊C8 | C22 | semidirect |
| 37. C22×D4 | C23 | semidirect |
| 47. C2×D8 | C23 | semidirect |
| 50. C22×C8 | C23 | semidirect |
| 56. #121 | C23 | semidirect |
| 60. C2×C4⋊C4 | C23 | semidirect |
| 64. #107 | C23 | semidirect |
| 65. #102 | C23 | semidirect |
| 66. C22⋊C8 | C23 | semidirect |
| 74. C24 | D8 | semidirect |
| 134. C23 | C2×D8 | semidirect |
| 186. C2 | #3186 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#121 | 2 |
| #3186 | 2 |
| C2×#480 | 2 |
| C2×#466 | 2 |
| C2×#447 | 2 |
| C22×C22⋊C8 | 2 |