Finite Group DatabasebetaFinite Group DatabasebetaOrder 16256.
804 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 804 conjugacy classes of subgroups — too many to draw. Only its 232 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×C127 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C127 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#121×C127 | C2 | semidirect |
| 2. #399×C127 | C2 | semidirect |
| 3. #392×C127 | C2 | semidirect |
| 4. C2×#95×C127 | C2 | semidirect |
| 18. C22×C4×C127 | D4 | semidirect |
| 52. #2656 | C127 | direct (N × H) |
| 53. C127 | #2656 | direct (N × H) |
| 54. C2×#121 | C2×C127 | semidirect |
| 55. #399 | C2×C127 | semidirect |
| 56. #392 | C2×C127 | semidirect |
| 57. C2×#95 | C2×C127 | semidirect |
| 71. C22×C4 | D4×C127 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#121×C127 | 2 |
| #399×C127 | 2 |
| #392×C127 | 2 |
| C2×#95×C127 | 2 |
| #2656 | 127 |