Finite Group DatabasebetaFinite Group DatabasebetaOrder 1408.
38 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C11×Q64 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C11×D32 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C11×C64 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C11×Q32 | — | 2 | 1 | no | — |
| 5 | C11×D16 | — | 2 | 1 | no | — |
| 6 | C11×C32 | C22 | 1 | 1 | yes | characteristic |
| 7 | C11×Q16 | — | 4 | 1 | no | — |
| 8 | C11×D8 | — | 4 | 1 | no | — |
| 9 | C11×C16 | D4 | 1 | 1 | yes | characteristic |
| 10 | SD128 | C11 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 11 | Q8×C11 | — | 8 | 1 | no | — |
| 12 | D4×C11 | — | 8 | 1 | no | — |
| 13 | C8×C11 | D8 | 1 | 1 | yes | characteristic |
| 14 | Q64 | C2×C11 | 1 | 1 | yes | characteristic, complemented |
| 15 | D32 | C2×C11 | 1 | 1 | yes | characteristic |
| 16 | C64 | C2×C11 | 1 | 1 | yes | characteristic, complemented |
| 17 | C22×C11 | — | 16 | 1 | no | — |
| 18 | C4×C11 | — | 16 | 1 | no | — |
| 19 | C4×C11 | D16 | 1 | 1 | yes | characteristic |
| 20 | Q32 | — | 2 | 1 | no | — |
| 21 | D16 | — | 2 | 1 | no | — |
| 22 | C32 | C22×C11 | 1 | 1 | yes | characteristic |
| 23 | C2×C11 | — | 32 | 1 | no | — |
| 24 | C2×C11 | D32 | 1 | 1 | yes | characteristic |
| 25 | Q16 | — | 4 | 1 | no | — |
| 26 | D8 | — | 4 | 1 | no | — |
| 27 | C16 | D4×C11 | 1 | 1 | yes | characteristic |
| 28 | C11 | SD128 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 29 | Q8 | — | 8 | 1 | no | — |
| 30 | D4 | — | 8 | 1 | no | — |
| 31 | C8 | C11×D8 | 1 | 1 | yes | characteristic |
| 32 | C22 | — | 16 | 1 | no | — |
| 33 | C4 | — | 16 | 1 | no | — |
| 34 | C4 | C11×D16 | 1 | 1 | yes | characteristic |
| 35 | C2 | — | 32 | 1 | no | — |
| 36 | C2 | C11×D32 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | D32 |
| commutator, Frattini, cosocle | C32 | C22×C11 |
| prime radical | C11×D32 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C4×C11 ◁ C8×C11 ◁ C11×C16 ◁ C11×C32 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C11×Q64 | C2 | semidirect |
| 3. C11×C64 | C2 | semidirect |
| 10. SD128 | C11 | direct (N × H) |
| 14. Q64 | C2×C11 | semidirect |
| 16. C64 | C2×C11 | semidirect |
| 28. C11 | SD128 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C11×Q64 | 2 |
| C11×D32 | 2 |
| C11×C64 | 2 |
| SD128 | 11 |