Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
351 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 351 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C24 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Q32 | C2 | semidirect |
| 4. #2216 | C2 | direct (N × H) |
| 5. C2×#366 | C2 | semidirect |
| 6. C2×#358 | C2 | semidirect |
| 11. C2×Q32 | C22 | semidirect |
| 20. #366 | C22 | semidirect |
| 21. #358 | C22 | semidirect |
| 81. C23 | Q32 | semidirect |
| 109. C22 | C2×Q32 | semidirect |
| 119. C2 | #2216 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q32 | 2 |
| C23×C16 | 2 |
| C2×#467 | 2 |
| #2216 | 2 |
| C2×#366 | 2 |
| C2×#358 | 2 |