Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1379 conjugacy classes of subgroups; only the 527 normal subgroups are shown.
This group has 1379 conjugacy classes of subgroups — too many to draw. Only its 527 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 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×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×#137 | C2 | semidirect |
| 3. C23×SD16 | C2 | semidirect |
| 4. C22×#131 | C2 | semidirect |
| 5. C2×#471 | C2 | direct (N × H) |
| 6. C22×#108 | C2 | semidirect |
| 7. C22×#101 | C2 | semidirect |
| 8. C22×#100 | C2 | semidirect |
| 13. C2×#137 | C22 | semidirect |
| 22. C22×SD16 | C22 | semidirect |
| 25. C2×#131 | C22 | semidirect |
| 32. #471 | C22 | direct (N × H) |
| 33. C2×#108 | C22 | semidirect |
| 34. C2×#101 | C22 | semidirect |
| 35. C2×#100 | C22 | semidirect |
| 60. #137 | C23 | semidirect |
| 69. C2×SD16 | C23 | semidirect |
| 72. #131 | C23 | semidirect |
| 88. #108 | C23 | semidirect |
| 89. #101 | C23 | semidirect |
| 90. #100 | C23 | semidirect |
| 213. C22 | #471 | direct (N × H) |
| 227. C2 | C2×#471 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#137 | 2 |
| C22×AGL(1,ℤ/8) | 2 |
| C23×SD16 | 2 |
| C22×#131 | 2 |
| C2×#471 | 2 |
| C22×#108 | 2 |
| C22×#101 | 2 |
| C22×#100 | 2 |