Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
617 conjugacy classes of subgroups; only the 207 normal subgroups are shown.
This group has 617 conjugacy classes of subgroups — too many to draw. Only its 207 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 | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#405 | C2 | semidirect |
| 4. #2316 | C2 | semidirect |
| 6. #1936 | C2 | semidirect |
| 7. #1934 | C2 | semidirect |
| 8. #1736 | C2 | semidirect |
| 9. #1735 | C2 | semidirect |
| 18. C2×#102 | C4 | semidirect |
| 23. C2×#97 | C22 | semidirect |
| 24. C2×C4×C8 | C22 | semidirect |
| 28. #389 | C22 | semidirect |
| 31. #340 | C22 | semidirect |
| 32. #340 | C22 | semidirect |
| 33. #340 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#418 | 2 |
| C22×#102 | 2 |
| C2×#405 | 2 |
| #2316 | 2 |
| #2309 | 2 |
| #1936 | 2 |
| #1934 | 2 |
| #1736 | 2 |
| #1735 | 2 |