Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
924 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 237 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, socle, cosocle | C23×C97 | C24 |
| Fitting | C22×C42×C97 | C2 |
| hypercenter | C22×C42 | D97 |
| hypocenter | C97 | #2856 |
| LCS | G ▷ C23×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ C22×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15690807 | C2 | semidirect |
| 2. C2×#15690784 | C2 | semidirect |
| 3. #20260637 | C2 | semidirect |
| 4. #20260605 | C2 | semidirect |
| 5. C22×C42×C97 | C2 | semidirect |
| 27. C22×C4×C97 | D4 | semidirect |
| 31. C42×C97 | D4 | semidirect |
| 51. C22×C97 | #127 | semidirect |
| 59. C4×C97 | #121 | semidirect |
| 67. C97 | #2856 | semidirect |
| 70. C22×C42 | D97 | semidirect |
| 92. C22×C4 | D388 | semidirect |
| 98. C42 | C22⋊D97 | semidirect |
| 121. C22 | #15690807 | semidirect |
| 127. C4 | #15690784 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15690807 | 2 |
| C2×#15690784 | 2 |
| #20260637 | 2 |
| #20260605 | 2 |
| C22×C42×C97 | 2 |
| #2856 | 97 |