Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1459 conjugacy classes of subgroups; only the 567 normal subgroups are shown.
This group has 1459 conjugacy classes of subgroups — too many to draw. Only its 567 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 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×Q16 | C2 | semidirect |
| 4. C22×C4×Q8 | C2 | semidirect |
| 5. C2×#464 | C2 | direct (N × H) |
| 6. C22×C4⋊C8 | C2 | semidirect |
| 7. C22×#103 | C2 | semidirect |
| 13. C22×Q16 | C22 | semidirect |
| 21. C2×C4×Q8 | C22 | semidirect |
| 28. #464 | C22 | direct (N × H) |
| 29. C2×C4⋊C8 | C22 | semidirect |
| 30. C2×#103 | C22 | semidirect |
| 52. C2×Q16 | C23 | semidirect |
| 62. C4×Q8 | C23 | semidirect |
| 71. C4⋊C8 | C23 | semidirect |
| 72. #103 | C23 | semidirect |
| 161. C22 | #464 | direct (N × H) |
| 175. C2 | C2×#464 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q16 | 2 |
| C23×SD16 | 2 |
| C22×#124 | 2 |
| C22×C4×Q8 | 2 |
| C2×#464 | 2 |
| C22×C4⋊C8 | 2 |
| C22×#103 | 2 |
| C22×#102 | 2 |