Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2577 conjugacy classes of subgroups; only the 991 normal subgroups are shown.
This group has 2577 conjugacy classes of subgroups — too many to draw. Only its 991 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| prime radical | C22×H(5,𝔽2) | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#568 | C2 | semidirect |
| 3. #3428 | C2 | semidirect |
| 4. #2897 | C2 | semidirect |
| 5. #2896 | C2 | semidirect |
| 6. C2×#418 | C2 | semidirect |
| 9. C2×H(5,𝔽2) | C4 | semidirect |
| 17. C2×#131 | C22 | semidirect |
| 18. C22×M16 | C22 | semidirect |
| 19. #520 | C22 | semidirect |
| 23. C2×#117 | C22 | semidirect |
| 24. #419 | C22 | semidirect |
| 25. #418 | C22 | semidirect |
| 26. #418 | C22 | semidirect |
| 27. #417 | C22 | semidirect |
| 28. #416 | C22 | semidirect |
| 29. C2×#102 | C22 | semidirect |
| 33. H(5,𝔽2) | C2×C4 | semidirect |
| 39. C2×#41 | C2×C4 | semidirect |
| 52. C22×D4 | C2×C4 | semidirect |
| 61. C2×M16 | C23 | semidirect |
| 62. C22×C8 | C23 | semidirect |
| 68. #117 | C23 | semidirect |
| 69. C2×C4⋊C4 | C23 | semidirect |
| 72. #103 | C23 | semidirect |
| 73. #102 | C23 | 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×#568 | 2 |
| #3428 | 2 |
| #2897 | 2 |
| #2896 | 2 |
| C2×#418 | 2 |