Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
343 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 75 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 | #121 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #1994 | C2 | semidirect |
| 4. #1503 | C2 | semidirect |
| 6. #1303 | C2 | semidirect |
| 7. #1276 | C2 | semidirect |
| 13. #476 | C4 | semidirect |
| 14. #461 | C4 | semidirect |
| 34. C4⋊Q8 | C2×C4 | semidirect |
| 35. C4⋊Q8 | C2×C4 | semidirect |
| 64. C4⋊C8 | C2×C4 | semidirect |
| 123. C42 | SD16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3101 | 2 |
| C42×D4 | 2 |
| #1994 | 2 |
| #1503 | 2 |
| #1486 | 2 |
| #1303 | 2 |
| #1276 | 2 |