Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
45 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #424 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #349 | C2 | 1 | 2 | yes | maximal |
| 3 | C2×M16 | C22 | 1 | 1 | yes | characteristic |
| 4 | C22×C8 | C4 | 1 | 1 | yes | characteristic |
| 5 | #117 | C4 | 1 | 1 | yes | characteristic |
| 6 | M32 | — | 2 | 2 | no | — |
| 7 | C4⋊C8 | — | 2 | 2 | no | — |
| 8 | C22×C4 | C2×C4 | 1 | 1 | yes | characteristic |
| 9 | M16 | — | 4 | 1 | no | — |
| 10 | C2×C8 | — | 2 | 1 | no | — |
| 11 | C2×C8 | — | 4 | 1 | no | — |
| 12 | C2×C8 | D4 | 1 | 2 | yes | — |
| 13 | C4⋊C4 | — | 2 | 1 | no | — |
| 14 | C22⋊C4 | C8 | 1 | 2 | yes | — |
| 15 | C42 | — | 2 | 1 | no | — |
| 16 | C16 | — | 4 | 2 | no | — |
| 17 | C23 | C2×C8 | 1 | 1 | yes | characteristic |
| 18 | C2×C4 | — | 2 | 1 | no | — |
| 19 | C2×C4 | — | 2 | 1 | no | — |
| 20 | C2×C4 | — | 2 | 2 | no | — |
| 21 | C2×C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 22 | C2×C4 | M16 | 1 | 1 | yes | characteristic |
| 23 | C8 | — | 2 | 2 | no | — |
| 24 | C8 | — | 4 | 1 | no | — |
| 25 | C22 | — | 2 | 1 | no | — |
| 26 | C22 | — | 4 | 1 | no | — |
| 27 | C22 | C22⋊C8 | 1 | 1 | yes | characteristic |
| 28 | C4 | — | 2 | 1 | no | — |
| 29 | C4 | — | 4 | 2 | no | — |
| 30 | C4 | #99 | 1 | 1 | yes | characteristic |
| 31 | C4 | #101 | 1 | 1 | yes | characteristic |
| 32 | C2 | — | 2 | 1 | no | — |
| 33 | C2 | — | 4 | 1 | no | — |
| 34 | C2 | #324 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #99 |
| commutator, prime radical | C23 | C2×C8 |
| Frattini, cosocle | C2×M16 | C22 |
| socle | C2 | #324 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #424 | 2 |
| #349 | 2 |