Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
204 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | #6493 |
| commutator, cosocle | C42×C7 | C23 |
| Frattini | C42 | C22×D7 |
| Fitting | C7×C8×D4 | C2 |
| socle | C22×C7 | #121 |
| hypercenter | C8×D4 | D7 |
| hypocenter | C7 | #1667 |
| LCS | G ▷ C42×C7 ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×C8×D4 | C2 | semidirect |
| 2. #89787 | C2 | semidirect |
| 3. #89467 | C2 | semidirect |
| 4. #89280 | C2 | semidirect |
| 5. #89253 | C2 | semidirect |
| 36. C8×D4 | D7 | semidirect |
| 42. C7×D4 | D8 | semidirect |
| 56. C7×C8 | D8 | semidirect |
| 111. D4 | D56 | semidirect |
| 121. C8 | #1141 | semidirect |
| 122. C7 | #1667 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×C8×D4 | 2 |
| #89787 | 2 |
| #89467 | 2 |
| #89280 | 2 |
| #89253 | 2 |
| #1667 | 7 |