Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1467 conjugacy classes of subgroups; only the 743 normal subgroups are shown.
This group has 1467 conjugacy classes of subgroups — too many to draw. Only its 743 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 | C22 | C24×C4 |
| Frattini, cosocle | C23 | C25 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22 ▷ 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. #2379 | C2 | direct (N × H) |
| 3. C2×#391 | C2 | semidirect |
| 4. C2×#388 | C2 | semidirect |
| 6. C2×#385 | C2 | semidirect |
| 7. C2×#382 | C2 | semidirect |
| 11. C2×#124 | C4 | semidirect |
| 21. #391 | C22 | semidirect |
| 22. #388 | C22 | semidirect |
| 24. #385 | C22 | semidirect |
| 25. #382 | C22 | semidirect |
| 32. C22×Q8 | C2×C4 | semidirect |
| 46. #124 | C2×C4 | semidirect |
| 55. C2×C22⋊C4 | C2×C4 | semidirect |
| 60. C2×C42 | C2×C4 | semidirect |
| 76. C2×Q8 | C22×C4 | semidirect |
| 109. C22⋊C4 | C22×C4 | semidirect |
| 111. C42 | C22×C4 | semidirect |
| 173. C2 | #2379 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#124 | 2 |
| #2379 | 2 |
| C2×#391 | 2 |
| C2×#388 | 2 |
| C2×#386 | 2 |
| C2×#385 | 2 |
| C2×#382 | 2 |