Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
4119 conjugacy classes of subgroups; only the 3263 normal subgroups are shown.
This group has 4119 conjugacy classes of subgroups — too many to draw. Only its 3263 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 | C22 | C26 |
| socle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#533 | C2 | direct (N × H) |
| 2. C22×C4⋊Q8 | C2 | semidirect |
| 3. C22×#125 | C2 | semidirect |
| 5. C22×C4×Q8 | C2 | semidirect |
| 11. #533 | C22 | direct (N × H) |
| 12. C2×C4⋊Q8 | C22 | semidirect |
| 13. C2×#125 | C22 | semidirect |
| 15. C2×C4×Q8 | C22 | semidirect |
| 30. C23×C4 | Q8 | semidirect |
| 38. C4⋊Q8 | C23 | semidirect |
| 39. #125 | C23 | semidirect |
| 41. C4×Q8 | C23 | semidirect |
| 63. C22×C4 | C2×Q8 | semidirect |
| 93. C2×C4 | C22×Q8 | semidirect |
| 102. C22 | #533 | direct (N × H) |
| 113. C2 | C2×#533 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#533 | 2 |
| C22×C4⋊Q8 | 2 |
| C22×#125 | 2 |
| C22×C22⋊Q8 | 2 |
| C22×C4×Q8 | 2 |
| C22×#117 | 2 |
| C23×C42 | 2 |