Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
634 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 634 conjugacy classes of subgroups — too many to draw. Only its 163 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 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#496 | C2 | semidirect |
| 2. C2×#466 | C2 | semidirect |
| 3. C2×#465 | C2 | semidirect |
| 4. C22×#97 | C2 | semidirect |
| 5. #2856 | C2 | semidirect |
| 6. #2850 | C2 | semidirect |
| 7. #1973 | C2 | semidirect |
| 8. #1972 | C2 | semidirect |
| 9. #1961 | C2 | semidirect |
| 10. #1959 | C2 | semidirect |
| 11. #1847 | C2 | semidirect |
| 59. C22×C8 | D4 | semidirect |
| 95. #97 | D4 | semidirect |
| 216. C8 | #121 | semidirect |
| 231. C22 | #496 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#496 | 2 |
| C2×#466 | 2 |
| C2×#465 | 2 |
| C22×#97 | 2 |
| #2856 | 2 |
| #2850 | 2 |
| #1973 | 2 |
| #1972 | 2 |
| #1961 | 2 |
| #1959 | 2 |
| #1847 | 2 |