Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
19 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Q64 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | D32 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C64 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | Q32 | — | 2 | 1 | no | — |
| 5 | D16 | — | 2 | 1 | no | — |
| 6 | C32 | C22 | 1 | 1 | yes | characteristic |
| 7 | Q16 | — | 4 | 1 | no | — |
| 8 | D8 | — | 4 | 1 | no | — |
| 9 | C16 | D4 | 1 | 1 | yes | characteristic |
| 10 | Q8 | — | 8 | 1 | no | — |
| 11 | D4 | — | 8 | 1 | no | — |
| 12 | C8 | D8 | 1 | 1 | yes | characteristic |
| 13 | C22 | — | 16 | 1 | no | — |
| 14 | C4 | — | 16 | 1 | no | — |
| 15 | C4 | D16 | 1 | 1 | yes | characteristic |
| 16 | C2 | — | 32 | 1 | no | — |
| 17 | C2 | D32 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D32 |
| commutator, Frattini, cosocle | C32 | C22 |
| prime radical | D32 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ G | |
| LECS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q64 | C2 | semidirect |
| 3. C64 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q64 | 2 |
| D32 | 2 |
| C64 | 2 |