Finite Group DatabasebetaFinite Group DatabasebetaOrder 16256.
492 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 216 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 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C127 |
| prime radical | C24×C127 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C127 ◁ C23×C4×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Q16×C127 | C2 | semidirect |
| 4. #470×C127 | C2 | direct (N × H) |
| 5. C2×#106×C127 | C2 | semidirect |
| 6. C2×#103×C127 | C2 | semidirect |
| 10. C2×Q16×C127 | C22 | semidirect |
| 20. #106×C127 | C22 | semidirect |
| 21. #103×C127 | C22 | semidirect |
| 80. C2×C127 | #470 | direct (N × H) |
| 83. C2×#470 | C127 | direct (N × H) |
| 84. C127 | C2×#470 | direct (N × H) |
| 85. C22×Q16 | C2×C127 | semidirect |
| 88. #470 | C2×C127 | direct (N × H) |
| 89. C2×#106 | C2×C127 | semidirect |
| 90. C2×#103 | C2×C127 | semidirect |
| 94. C2×Q16 | C22×C127 | semidirect |
| 104. #106 | C22×C127 | semidirect |
| 105. #103 | C22×C127 | semidirect |
| 164. C2 | #470×C127 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q16×C127 | 2 |
| C22×M16×C127 | 2 |
| C2×C22⋊Q8×C127 | 2 |
| #470×C127 | 2 |
| C2×#106×C127 | 2 |
| C2×#103×C127 | 2 |
| C2×#470 | 127 |