Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
446 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 446 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2428 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ 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. #3155 | C2 | semidirect |
| 2. #3068 | C2 | semidirect |
| 3. #3067 | C2 | semidirect |
| 4. C2×#472 | C2 | semidirect |
| 5. #2957 | C2 | semidirect |
| 6. #2048 | C2 | semidirect |
| 7. #2008 | C2 | semidirect |
| 8. #1978 | C2 | semidirect |
| 9. #1953 | C2 | semidirect |
| 10. #1914 | C2 | semidirect |
| 11. #1757 | C2 | semidirect |
| 100. #108 | D4 | semidirect |
| 101. #108 | D4 | semidirect |
| 105. #101 | D4 | semidirect |
| 109. #100 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3155 | 2 |
| #3068 | 2 |
| #3067 | 2 |
| C2×#472 | 2 |
| #2957 | 2 |
| #2048 | 2 |
| #2008 | 2 |
| #1978 | 2 |
| #1953 | 2 |
| #1914 | 2 |
| #1757 | 2 |