Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
326 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #442 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C2×#442 |
| prime radical | #3414 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#444 | C2 | semidirect |
| 3. #2943 | C2 | semidirect |
| 4. AGL(1,ℤ/16) | C2 | semidirect |
| 5. #2180 | C2 | semidirect |
| 6. #2155 | C2 | semidirect |
| 7. #2148 | C2 | semidirect |
| 8. #2145 | C2 | semidirect |
| 9. C2×#347 | C2 | semidirect |
| 14. #510 | C4 | semidirect |
| 16. #509 | C4 | semidirect |
| 18. #508 | C4 | semidirect |
| 23. #442 | C22 | semidirect |
| 24. #441 | C22 | semidirect |
| 31. #365 | C22 | semidirect |
| 32. #362 | C22 | semidirect |
| 33. #361 | C22 | semidirect |
| 35. #347 | C22 | semidirect |
| 44. #135 | C2×C4 | semidirect |
| 46. C2×Q16 | C2×C4 | semidirect |
| 50. C2×D8 | C2×C4 | semidirect |
| 66. Q32 | C2×C4 | semidirect |
| 67. SD32 | C2×C4 | semidirect |
| 68. SD32 | C2×C4 | semidirect |
| 69. D16 | C2×C4 | semidirect |
| 71. M32 | C2×C4 | semidirect |
| 75. C2×C16 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3414 | 2 |
| C2×#444 | 2 |
| #2943 | 2 |
| AGL(1,ℤ/16) | 2 |
| #2180 | 2 |
| #2155 | 2 |
| #2148 | 2 |
| #2145 | 2 |
| C2×#347 | 2 |