Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
520 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 135 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, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#469 | C2 | semidirect |
| 2. C2×#466 | C2 | semidirect |
| 3. C2×#435 | C2 | semidirect |
| 4. C2×#427 | C2 | semidirect |
| 5. #2414 | C2 | semidirect |
| 6. #2067 | C2 | semidirect |
| 7. #2033 | C2 | semidirect |
| 8. #1972 | C2 | semidirect |
| 9. #1944 | C2 | semidirect |
| 10. #1875 | C2 | semidirect |
| 11. #1809 | C2 | semidirect |
| 57. C22×C8 | D4 | semidirect |
| 88. #106 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#469 | 2 |
| C2×#466 | 2 |
| C2×#435 | 2 |
| C2×#427 | 2 |
| #2414 | 2 |
| #2067 | 2 |
| #2033 | 2 |
| #1972 | 2 |
| #1944 | 2 |
| #1875 | 2 |
| #1809 | 2 |