Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
367 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 367 conjugacy classes of subgroups — too many to draw. Only its 123 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 | #127 |
| commutator, Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×#127 |
| prime radical | #3169 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3170 | C2 | semidirect |
| 3. #3167 | C2 | semidirect |
| 4. #3131 | C2 | semidirect |
| 5. #1720 | C2 | semidirect |
| 6. #1719 | C2 | semidirect |
| 7. #1718 | C2 | semidirect |
| 8. #1717 | C2 | semidirect |
| 9. C2×#322 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3170 | 2 |
| #3169 | 2 |
| #3167 | 2 |
| #3131 | 2 |
| #1720 | 2 |
| #1719 | 2 |
| #1718 | 2 |
| #1717 | 2 |
| C2×#322 | 2 |