Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
98 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #430 | C2 | 1 | 3 | yes | maximal |
| 2 | C2×#101 | C2 | 1 | 3 | yes | maximal |
| 3 | #338 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C22×Q8 | C22 | 1 | 1 | yes | characteristic |
| 5 | C2×M16 | C22 | 1 | 3 | yes | — |
| 6 | C2×M16 | C22 | 1 | 3 | yes | — |
| 7 | #108 | — | 2 | 6 | no | — |
| 8 | #101 | — | 2 | 6 | no | — |
| 9 | C2×Q8 | — | 2 | 3 | no | — |
| 10 | C2×Q8 | — | 2 | 3 | no | — |
| 11 | C22×C4 | Q8 | 1 | 2 | yes | — |
| 12 | C22×C4 | C23 | 1 | 1 | yes | characteristic |
| 13 | M16 | — | 2 | 3 | no | — |
| 14 | M16 | — | 2 | 3 | no | — |
| 15 | M16 | — | 2 | 3 | no | — |
| 16 | M16 | D4 | 1 | 6 | yes | — |
| 17 | C2×C8 | — | 2 | 3 | no | — |
| 18 | C2×C8 | — | 2 | 3 | no | — |
| 19 | C23 | C2×Q8 | 1 | 1 | yes | characteristic |
| 20 | Q8 | — | 4 | 3 | no | — |
| 21 | Q8 | — | 4 | 1 | no | — |
| 22 | C2×C4 | — | 2 | 6 | no | — |
| 23 | C2×C4 | C2×D4 | 1 | 3 | yes | — |
| 24 | C2×C4 | #41 | 1 | 3 | yes | — |
| 25 | C8 | — | 2 | 6 | no | — |
| 26 | C8 | — | 4 | 3 | no | — |
| 27 | C22 | — | 4 | 1 | no | — |
| 28 | C22 | C22⋊Q8 | 1 | 3 | yes | — |
| 29 | C4 | — | 4 | 2 | no | — |
| 30 | C4 | #121 | 1 | 3 | yes | — |
| 31 | C4 | #126 | 1 | 1 | yes | characteristic |
| 32 | C2 | — | 2 | 3 | no | — |
| 33 | C2 | #396 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #396 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C23 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #430 | 2 |
| C2×#101 | 2 |
| #338 | 2 |