Finite Group DatabasebetaFinite Group DatabasebetaOrder 16256.
370 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 208 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C127 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C127 |
| prime radical | C2×D4×C127 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C127 ◁ C4×Q8×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q82×C127 | C2 | semidirect |
| 3. #498×C127 | C2 | semidirect |
| 5. #475×C127 | C2 | semidirect |
| 6. C8×Q8×C127 | C2 | semidirect |
| 31. SD16×C127 | Q8 | direct (N × H) |
| 47. Q8×C127 | SD16 | direct (N × H) |
| 48. Q8×C127 | C2×Q8 | semidirect |
| 59. C8×C127 | C2×Q8 | semidirect |
| 75. Q8×SD16 | C127 | direct (N × H) |
| 76. C127 | Q8×SD16 | direct (N × H) |
| 77. Q82 | C2×C127 | semidirect |
| 79. #498 | C2×C127 | semidirect |
| 81. #475 | C2×C127 | semidirect |
| 82. C8×Q8 | C2×C127 | semidirect |
| 107. SD16 | Q8×C127 | direct (N × H) |
| 123. Q8 | SD16×C127 | direct (N × H) |
| 124. Q8 | C2×Q8×C127 | semidirect |
| 135. C8 | C2×Q8×C127 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q82×C127 | 2 |
| D4×Q8×C127 | 2 |
| #498×C127 | 2 |
| #476×C127 | 2 |
| #475×C127 | 2 |
| C8×Q8×C127 | 2 |
| C4×SD16×C127 | 2 |
| Q8×SD16 | 127 |