Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
515 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 515 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 | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3154 | C2 | semidirect |
| 2. C2×#495 | C2 | semidirect |
| 3. C2×#473 | C2 | semidirect |
| 4. C2×#472 | C2 | semidirect |
| 5. #3059 | C2 | semidirect |
| 6. #3058 | C2 | semidirect |
| 7. C2×#443 | C2 | semidirect |
| 8. #2004 | C2 | semidirect |
| 9. #2003 | C2 | semidirect |
| 10. #1977 | C2 | semidirect |
| 11. #1772 | C2 | semidirect |
| 104. #108 | D4 | semidirect |
| 115. C4×C8 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3154 | 2 |
| C2×#495 | 2 |
| C2×#473 | 2 |
| C2×#472 | 2 |
| #3059 | 2 |
| #3058 | 2 |
| C2×#443 | 2 |
| #2004 | 2 |
| #2003 | 2 |
| #1977 | 2 |
| #1772 | 2 |