Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1173 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 1173 conjugacy classes of subgroups — too many to draw. Only its 487 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 |
| prime radical | C2×#525 | C2 |
| 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 |
|---|---|---|
| 2. C22×#117 | C2 | semidirect |
| 3. #3093 | C2 | direct (N × H) |
| 4. C2×#485 | C2 | semidirect |
| 5. C2×#482 | C2 | semidirect |
| 6. C2×#428 | C2 | semidirect |
| 7. C2×#417 | C2 | semidirect |
| 8. C2×#407 | C2 | semidirect |
| 23. C2×#117 | C22 | semidirect |
| 27. #485 | C22 | semidirect |
| 28. #482 | C22 | semidirect |
| 29. #428 | C22 | semidirect |
| 30. C2×#107 | C22 | semidirect |
| 31. C2×#106 | C22 | semidirect |
| 32. #417 | C22 | semidirect |
| 33. C2×#103 | C22 | semidirect |
| 34. C2×#102 | C22 | semidirect |
| 35. #407 | C22 | semidirect |
| 83. #107 | C23 | semidirect |
| 84. #106 | C23 | semidirect |
| 85. #103 | C23 | semidirect |
| 86. #102 | C23 | semidirect |
| 209. C2 | #3093 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#525 | 2 |
| C22×#117 | 2 |
| #3093 | 2 |
| C2×#485 | 2 |
| C2×#482 | 2 |
| C2×#428 | 2 |
| C2×#417 | 2 |
| C2×#407 | 2 |