Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
3119 conjugacy classes of subgroups; only the 639 normal subgroups are shown.
This group has 3119 conjugacy classes of subgroups — too many to draw. Only its 639 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C23 |
| commutator, Frattini, cosocle | C23 | C25 |
| socle | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#41 | C2 | semidirect |
| 2. C22×C4×D4 | C2 | semidirect |
| 3. #2388 | C2 | direct (N × H) |
| 4. C2×#395 | C2 | semidirect |
| 5. C2×#394 | C2 | semidirect |
| 6. C2×#393 | C2 | semidirect |
| 7. C2×#392 | C2 | semidirect |
| 8. C2×C4×C22⋊C4 | C2 | semidirect |
| 10. C22×#41 | C22 | semidirect |
| 17. C2×C4×D4 | C22 | semidirect |
| 23. #395 | C22 | semidirect |
| 24. #394 | C22 | semidirect |
| 25. #393 | C22 | semidirect |
| 26. #392 | C22 | semidirect |
| 27. C4×C22⋊C4 | C22 | semidirect |
| 45. C23×C4 | D4 | semidirect |
| 73. C24 | #41 | semidirect |
| 99. C22×C4 | C2×D4 | semidirect |
| 100. C22×C4 | #41 | semidirect |
| 129. C23 | C2×#41 | semidirect |
| 149. C2×C4 | C2×#41 | semidirect |
| 173. C2 | #2388 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#41 | 2 |
| C22×C4×D4 | 2 |
| #2388 | 2 |
| C2×#395 | 2 |
| C2×#394 | 2 |
| C2×#393 | 2 |
| C2×#392 | 2 |
| C2×C4×C22⋊C4 | 2 |