Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
692 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 692 conjugacy classes of subgroups — too many to draw. Only its 151 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×#470 | C2 | semidirect |
| 5. #2056 | C2 | semidirect |
| 6. #2001 | C2 | semidirect |
| 7. #1971 | C2 | semidirect |
| 8. #1807 | C2 | semidirect |
| 9. #1787 | C2 | semidirect |
| 50. C2×Q16 | D4 | semidirect |
| 186. C22 | #470 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q16 | 2 |
| C2×#470 | 2 |
| C2×#435 | 2 |
| #2409 | 2 |
| #2056 | 2 |
| #2001 | 2 |
| #1971 | 2 |
| #1807 | 2 |
| #1787 | 2 |