Finite Group DatabasebetaFinite Group DatabasebetaOrder 12736.
310 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C199 | C23 |
| commutator, Frattini, cosocle | C22 | C24×C199 |
| socle | C22×C199 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C199 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#41×C199 | C2 | semidirect |
| 2. C4⋊Q8×C199 | C2 | semidirect |
| 3. #127×C199 | C2 | semidirect |
| 4. #124×C199 | C2 | semidirect |
| 5. #121×C199 | C2 | semidirect |
| 6. C4×D4×C199 | C2 | semidirect |
| 7. C2×C42×C199 | C2 | semidirect |
| 29. C2×C4×C199 | D4 | semidirect |
| 44. C4×C199 | #41 | semidirect |
| 49. C199 | #532 | direct (N × H) |
| 50. #532 | C199 | direct (N × H) |
| 51. C2×#41 | C2×C199 | semidirect |
| 52. C4⋊Q8 | C2×C199 | semidirect |
| 53. #127 | C2×C199 | semidirect |
| 54. #124 | C2×C199 | semidirect |
| 55. #121 | C2×C199 | semidirect |
| 56. C4×D4 | C2×C199 | semidirect |
| 57. C2×C42 | C2×C199 | semidirect |
| 79. C2×C4 | D4×C199 | semidirect |
| 94. C4 | #41×C199 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#41×C199 | 2 |
| C4⋊Q8×C199 | 2 |
| #127×C199 | 2 |
| #124×C199 | 2 |
| #121×C199 | 2 |
| C4×D4×C199 | 2 |
| C2×C42×C199 | 2 |
| #532 | 199 |