Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
91 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #564 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #490 | C2 | 1 | 3 | yes | maximal |
| 3 | #330 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #127 | C22 | 1 | 1 | yes | characteristic |
| 5 | #125 | — | 2 | 6 | no | — |
| 6 | #125 | C22 | 1 | 3 | yes | — |
| 7 | #102 | — | 2 | 6 | no | — |
| 8 | #97 | C22 | 1 | 3 | yes | — |
| 9 | C2×D4 | — | 2 | 3 | no | — |
| 10 | C2×C8 | — | 2 | 3 | no | — |
| 11 | C4⋊C4 | — | 2 | 6 | no | — |
| 12 | C4⋊C4 | — | 2 | 6 | no | — |
| 13 | C4⋊C4 | D4 | 1 | 6 | yes | — |
| 14 | C42 | — | 2 | 2 | no | — |
| 15 | C42 | C23 | 1 | 1 | yes | characteristic |
| 16 | C23 | — | 4 | 1 | no | — |
| 17 | D4 | — | 4 | 6 | no | — |
| 18 | C2×C4 | — | 2 | 6 | no | — |
| 19 | C2×C4 | C2×D4 | 1 | 3 | yes | — |
| 20 | C8 | — | 4 | 3 | no | — |
| 21 | C22 | — | 8 | 3 | no | — |
| 22 | C22 | #120 | 1 | 1 | yes | characteristic |
| 23 | C4 | — | 2 | 3 | no | — |
| 24 | C4 | — | 4 | 6 | no | — |
| 25 | C2 | — | 16 | 1 | no | — |
| 26 | C2 | #454 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C42 | C23 |
| prime radical | #127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #564 | C2 | semidirect |
| 3. #330 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #564 | 2 |
| #490 | 2 |
| #330 | 2 |