Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
656 conjugacy classes of subgroups; only the 261 normal subgroups are shown.
This group has 656 conjugacy classes of subgroups — too many to draw. Only its 261 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 | #384×C97 | C2 |
| socle, cosocle | C23×C97 | C24 |
| prime radical | C23×D97 | C23 |
| hypercenter | #384 | D97 |
| hypocenter | C97 | #2285 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ #384 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #20260605 | C2 | semidirect |
| 4. #20260600 | C2 | semidirect |
| 6. C4⋊C4×Dic97 | C2 | semidirect |
| 7. #384×C97 | C2 | semidirect |
| 17. C42×D97 | C4 | semidirect |
| 67. C4×D97 | C4⋊C4 | semidirect |
| 80. C4×C97 | #117 | semidirect |
| 91. C97 | #2285 | semidirect |
| 95. #384 | D97 | semidirect |
| 174. C4 | #15690768 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15690768 | 2 |
| C2×C42×D97 | 2 |
| #20260605 | 2 |
| #20260600 | 2 |
| #20260592 | 2 |
| C4⋊C4×Dic97 | 2 |
| #384×C97 | 2 |
| #2285 | 97 |