Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
760 conjugacy classes of subgroups; only the 264 normal subgroups are shown.
This group has 760 conjugacy classes of subgroups — too many to draw. Only its 264 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C97 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C97 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C97 ◁ C2×C42×C97 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D8×C97 | C2 | semidirect |
| 2. C2×#127×C97 | C2 | semidirect |
| 3. #493×C97 | C2 | direct (N × H) |
| 4. C2×C4×C8×C97 | C2 | semidirect |
| 8. C2×D8×C97 | C22 | semidirect |
| 11. #127×C97 | C22 | semidirect |
| 13. C4×C8×C97 | C22 | semidirect |
| 23. C2×C8×C97 | D4 | semidirect |
| 34. C2×C4×C97 | D8 | semidirect |
| 39. C8×C97 | C2×D4 | semidirect |
| 47. C4×C97 | C2×D8 | semidirect |
| 50. C2×C97 | #493 | direct (N × H) |
| 53. C2×#493 | C97 | direct (N × H) |
| 54. C97 | C2×#493 | direct (N × H) |
| 55. C22×D8 | C2×C97 | semidirect |
| 56. C2×#127 | C2×C97 | semidirect |
| 57. #493 | C2×C97 | direct (N × H) |
| 58. C2×C4×C8 | C2×C97 | semidirect |
| 62. C2×D8 | C22×C97 | semidirect |
| 65. #127 | C22×C97 | semidirect |
| 67. C4×C8 | C22×C97 | semidirect |
| 77. C2×C8 | D4×C97 | semidirect |
| 88. C2×C4 | D8×C97 | semidirect |
| 93. C8 | C2×D4×C97 | semidirect |
| 101. C4 | C2×D8×C97 | semidirect |
| 104. C2 | #493×C97 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D8×C97 | 2 |
| C2×#127×C97 | 2 |
| #493×C97 | 2 |
| C2×C4×C8×C97 | 2 |
| C2×#493 | 97 |