Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
3803 conjugacy classes of subgroups; only the 647 normal subgroups are shown.
This group has 3803 conjugacy classes of subgroups — too many to draw. Only its 647 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| prime radical | C24×D4 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×#124 | C2 | semidirect |
| 3. C22×C22⋊Q8 | C2 | semidirect |
| 4. #2435 | C2 | direct (N × H) |
| 5. C2×#394 | C2 | semidirect |
| 6. C2×#390 | C2 | semidirect |
| 7. C2×#379 | C2 | semidirect |
| 8. C2×C4×C22⋊C4 | C2 | semidirect |
| 10. C23×Q8 | C22 | semidirect |
| 18. C2×#124 | C22 | semidirect |
| 20. C2×C22⋊Q8 | C22 | semidirect |
| 21. C22×C4⋊C4 | C22 | semidirect |
| 22. C22×C4⋊C4 | C22 | semidirect |
| 25. C22×C22⋊C4 | C22 | semidirect |
| 26. C22×C22⋊C4 | C22 | semidirect |
| 27. C22×C42 | C22 | semidirect |
| 28. #394 | C22 | semidirect |
| 29. #390 | C22 | semidirect |
| 30. #379 | C22 | semidirect |
| 31. C4×C22⋊C4 | C22 | semidirect |
| 32. C2×#95 | C22 | semidirect |
| 42. C22×Q8 | C23 | semidirect |
| 65. C2×C4⋊C4 | C23 | semidirect |
| 66. C2×C4⋊C4 | C23 | semidirect |
| 73. C2×C22⋊C4 | C23 | semidirect |
| 74. C2×C22⋊C4 | C23 | semidirect |
| 76. C2×C42 | C23 | semidirect |
| 77. #95 | C23 | semidirect |
| 154. C23 | #124 | semidirect |
| 196. C22 | C2×#124 | semidirect |
| 210. C2 | #2435 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×D4 | 2 |
| C22×#124 | 2 |
| C22×C22⋊Q8 | 2 |
| #2435 | 2 |
| C2×#394 | 2 |
| C2×#390 | 2 |
| C2×#379 | 2 |
| C2×C4×C22⋊C4 | 2 |