Finite Group DatabasebetaFinite Group DatabasebetaOrder 16256.
364 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 180 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4×C127 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C24×C127 |
| socle | C2×C127 | C22×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C127 ◁ C22×C4×C127 ◁ C22×C8×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#135×C127 | C2 | semidirect |
| 2. #510×C127 | C2 | semidirect |
| 3. #509×C127 | C2 | semidirect |
| 4. #508×C127 | C2 | semidirect |
| 5. C2×M32×C127 | C2 | semidirect |
| 10. #135×C127 | C22 | semidirect |
| 11. C2×Q16×C127 | C22 | semidirect |
| 13. C2×D8×C127 | C22 | semidirect |
| 15. Q32×C127 | C22 | semidirect |
| 16. SD32×C127 | C22 | semidirect |
| 17. D16×C127 | C22 | semidirect |
| 18. M32×C127 | C22 | semidirect |
| 19. C2×C16×C127 | C22 | semidirect |
| 71. #3414 | C127 | direct (N × H) |
| 72. C127 | #3414 | direct (N × H) |
| 73. C2×#135 | C2×C127 | semidirect |
| 74. #510 | C2×C127 | semidirect |
| 75. #509 | C2×C127 | semidirect |
| 76. #508 | C2×C127 | semidirect |
| 77. C2×M32 | C2×C127 | semidirect |
| 82. #135 | C22×C127 | semidirect |
| 83. C2×Q16 | C22×C127 | semidirect |
| 85. C2×D8 | C22×C127 | semidirect |
| 87. Q32 | C22×C127 | semidirect |
| 88. SD32 | C22×C127 | semidirect |
| 89. D16 | C22×C127 | semidirect |
| 90. M32 | C22×C127 | semidirect |
| 91. C2×C16 | C22×C127 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#135×C127 | 2 |
| #510×C127 | 2 |
| #509×C127 | 2 |
| #508×C127 | 2 |
| C2×M32×C127 | 2 |
| #3414 | 127 |