Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
361 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 361 conjugacy classes of subgroups — too many to draw. Only its 85 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 | #120 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | C2×#127 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ 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 |
|---|---|---|
| 1. #2972 | C2 | semidirect |
| 3. #1478 | C2 | semidirect |
| 4. #1460 | C2 | semidirect |
| 5. #1276 | C2 | semidirect |
| 28. #127 | C8 | semidirect |
| 47. C4⋊C8 | D4 | semidirect |
| 64. C2×D4 | C2×C8 | semidirect |
| 93. C42 | C2×C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2972 | 2 |
| C4×#127 | 2 |
| #1478 | 2 |
| #1460 | 2 |
| #1276 | 2 |