Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
705 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 705 conjugacy classes of subgroups — too many to draw. Only its 159 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 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×Q16 | C2 | semidirect |
| 2. C2×#467 | C2 | semidirect |
| 5. #2056 | C2 | semidirect |
| 6. #2001 | C2 | semidirect |
| 7. #1969 | C2 | semidirect |
| 8. #1808 | C2 | semidirect |
| 9. #1787 | C2 | semidirect |
| 50. C2×Q16 | D4 | semidirect |
| 96. C2×D4 | Q16 | semidirect |
| 144. C23 | C2×Q16 | semidirect |
| 174. C2×C4 | C2×Q16 | semidirect |
| 192. C22 | #467 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q16 | 2 |
| C2×#467 | 2 |
| C2×C8×D4 | 2 |
| #2409 | 2 |
| #2056 | 2 |
| #2001 | 2 |
| #1969 | 2 |
| #1808 | 2 |
| #1787 | 2 |