Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
560 conjugacy classes of subgroups; only the 193 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 193 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D97 |
| commutator | C22×C97 | C23×C4 |
| Fitting | #382×C97 | C2 |
| socle, cosocle | C23×C97 | C24 |
| prime radical | C22×D388 | C22 |
| hypercenter | #382 | D97 |
| hypocenter | C97 | #2357 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ #382 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #20260627 | C2 | semidirect |
| 4. #20260597 | C2 | semidirect |
| 5. C4⋊C4×Dic97 | C2 | semidirect |
| 6. C4×C4⋊Dic97 | C2 | semidirect |
| 7. #20260582 | C2 | semidirect |
| 8. #382×C97 | C2 | semidirect |
| 11. C4×D388 | C4 | semidirect |
| 98. C97 | #2357 | semidirect |
| 102. #382 | D97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D388 | 2 |
| #20260636 | 2 |
| #20260627 | 2 |
| #20260597 | 2 |
| C4⋊C4×Dic97 | 2 |
| C4×C4⋊Dic97 | 2 |
| #20260582 | 2 |
| #382×C97 | 2 |
| #2357 | 97 |