Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
645 conjugacy classes of subgroups; only the 265 normal subgroups are shown.
This group has 645 conjugacy classes of subgroups — too many to draw. Only its 265 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C4 | C23×C8 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C2×H(5,𝔽2) | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2988 | C2 | semidirect |
| 3. C8×#41 | C2 | semidirect |
| 4. #1478 | C2 | semidirect |
| 5. #1477 | C2 | semidirect |
| 6. #1476 | C2 | semidirect |
| 7. C2×D4⋊C8 | C2 | semidirect |
| 23. #431 | C22 | semidirect |
| 24. #431 | C22 | semidirect |
| 25. C2×C4⋊C8 | C22 | semidirect |
| 26. C2×C4×C8 | C22 | semidirect |
| 27. Q8⋊C8 | C22 | semidirect |
| 28. Q8⋊C8 | C22 | semidirect |
| 29. D4⋊C8 | C22 | semidirect |
| 30. D4⋊C8 | C22 | semidirect |
| 31. D4⋊C8 | C22 | semidirect |
| 33. H(5,𝔽2) | C8 | semidirect |
| 84. #41 | C2×C8 | semidirect |
| 93. C2×D4 | C2×C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×H(5,𝔽2) | 2 |
| #2988 | 2 |
| C8×#41 | 2 |
| #1478 | 2 |
| #1477 | 2 |
| #1476 | 2 |
| C2×D4⋊C8 | 2 |