Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
1640 conjugacy classes of subgroups; only the 565 normal subgroups are shown.
This group has 1640 conjugacy classes of subgroups — too many to draw. Only its 565 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D97 |
| commutator, cosocle | C22×C97 | C25 |
| Frattini | C22 | C24×D97 |
| Fitting | C22×C4⋊C4×C97 | C2 |
| socle | C23×C97 | C24 |
| hypercenter | C22×C4⋊C4 | D97 |
| hypocenter | C97 | C2×#537 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ C22×C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#13095511 | C2 | semidirect |
| 2. #20261316 | C2 | direct (N × H) |
| 3. C2×#15690790 | C2 | semidirect |
| 4. C2×#15690785 | C2 | semidirect |
| 5. C2×#15690784 | C2 | semidirect |
| 6. C22×C4⋊C4×C97 | C2 | semidirect |
| 8. C2×#13095511 | C22 | semidirect |
| 14. #15690790 | C22 | semidirect |
| 15. #15690785 | C22 | semidirect |
| 16. #15690784 | C22 | semidirect |
| 22. C2×C4⋊C4×C97 | C22 | semidirect |
| 40. C22×C4×C97 | D4 | semidirect |
| 58. C2×C4×C97 | C2×D4 | semidirect |
| 84. C2×C97 | #537 | semidirect |
| 90. C97 | C2×#537 | semidirect |
| 96. C22×C4⋊C4 | D97 | semidirect |
| 111. C2×C4⋊C4 | C2×D97 | semidirect |
| 128. C22×C4 | D388 | semidirect |
| 155. C2×C4 | C2×D388 | semidirect |
| 175. C2 | #20261316 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#13095511 | 2 |
| #20261316 | 2 |
| C2×#15690790 | 2 |
| C2×#15690785 | 2 |
| C2×#15690784 | 2 |
| C22×C4⋊C4×C97 | 2 |
| C2×#537 | 97 |