Finite Group DatabasebetaFinite Group DatabasebetaOrder 12736.
342 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 178 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×C199 | C23 |
| commutator, Frattini, cosocle | C22 | C24×C199 |
| prime radical | C22×D4×C199 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C199 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×C4×C199 | C2 | semidirect |
| 3. #123×C199 | C2 | direct (N × H) |
| 4. C2×C4⋊C4×C199 | C2 | semidirect |
| 5. C2×C22⋊C4×C199 | C2 | semidirect |
| 6. C2×C22⋊C4×C199 | C2 | semidirect |
| 15. C22×C4×C199 | C22 | semidirect |
| 16. C4⋊C4×C199 | C22 | semidirect |
| 17. C22⋊C4×C199 | C22 | semidirect |
| 18. C22⋊C4×C199 | C22 | semidirect |
| 48. C2×C199 | #123 | direct (N × H) |
| 51. C199 | C2×#123 | direct (N × H) |
| 52. C2×#123 | C199 | direct (N × H) |
| 54. C23×C4 | C2×C199 | semidirect |
| 55. #123 | C2×C199 | direct (N × H) |
| 56. C2×C4⋊C4 | C2×C199 | semidirect |
| 57. C2×C22⋊C4 | C2×C199 | semidirect |
| 58. C2×C22⋊C4 | C2×C199 | semidirect |
| 67. C22×C4 | C22×C199 | semidirect |
| 68. C4⋊C4 | C22×C199 | semidirect |
| 69. C22⋊C4 | C22×C199 | semidirect |
| 70. C22⋊C4 | C22×C199 | semidirect |
| 100. C2 | #123×C199 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D4×C199 | 2 |
| C23×C4×C199 | 2 |
| #123×C199 | 2 |
| C2×C4⋊C4×C199 | 2 |
| C2×C22⋊C4×C199 | 2 |
| C2×#123 | 199 |