Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
1224 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 1224 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 | C24×C4×C97 | C2 |
| hypercenter | C24×C4 | D97 |
| hypocenter | C97 | #2847 |
| LCS | G ▷ C23×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ C24×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15690784 | C2 | semidirect |
| 2. C2×#15690779 | C2 | semidirect |
| 3. #20260614 | C2 | semidirect |
| 4. #20260595 | C2 | semidirect |
| 5. C24×C4×C97 | C2 | semidirect |
| 6. C22×D388 | C22 | semidirect |
| 10. C2×C4⋊Dic97 | C22 | semidirect |
| 12. C2×#13095488 | C22 | semidirect |
| 13. #15690717 | C22 | semidirect |
| 29. C24×C97 | D4 | semidirect |
| 35. C22×C4×C97 | D4 | semidirect |
| 65. C22×C97 | #121 | semidirect |
| 73. C4×C97 | #120 | semidirect |
| 81. C97 | #2847 | semidirect |
| 82. C24×C4 | D97 | semidirect |
| 102. C24 | D388 | semidirect |
| 113. C22×C4 | C22⋊D97 | semidirect |
| 149. C22 | #15690784 | semidirect |
| 155. C4 | #15690779 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15690784 | 2 |
| C2×#15690779 | 2 |
| #20260614 | 2 |
| #20260595 | 2 |
| C24×C4×C97 | 2 |
| #2847 | 97 |