Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
323 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 323 conjugacy classes of subgroups — too many to draw. Only its 151 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 | #442 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ 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 |
|---|---|---|
| 3. C2×#441 | C2 | semidirect |
| 4. #2180 | C2 | direct (N × H) |
| 5. C2×#365 | C2 | semidirect |
| 6. C2×#362 | C2 | semidirect |
| 8. C2×#347 | C2 | semidirect |
| 13. #510 | C4 | semidirect |
| 15. C2×Q32 | C4 | semidirect |
| 16. C2×SD32 | C4 | semidirect |
| 22. #441 | C22 | semidirect |
| 28. #365 | C22 | semidirect |
| 29. #362 | C22 | semidirect |
| 31. #347 | C22 | semidirect |
| 60. Q32 | C2×C4 | semidirect |
| 61. SD32 | C2×C4 | semidirect |
| 177. C2 | #2180 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#510 | 2 |
| C2×#444 | 2 |
| C2×#441 | 2 |
| #2180 | 2 |
| C2×#365 | 2 |
| C2×#362 | 2 |
| C2×#360 | 2 |
| C2×#347 | 2 |