Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
198 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C4×Q8 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C8×Q8 | C2 | 1 | 8 | yes | maximal, complemented |
| 3 | #431 | C2 | 1 | 6 | yes | maximal |
| 4 | C22×Q8 | C4 | 1 | 1 | yes | characteristic |
| 5 | C4×Q8 | C4 | 1 | 4 | yes | — |
| 6 | C4×Q8 | C22 | 1 | 4 | yes | — |
| 7 | C2×C4⋊C4 | C4 | 1 | 3 | yes | — |
| 8 | C2×C42 | C22 | 1 | 3 | yes | — |
| 9 | C4⋊C8 | C22 | 1 | 12 | yes | — |
| 10 | C22⋊C8 | C22 | 1 | 4 | yes | — |
| 11 | C4×C8 | C22 | 1 | 12 | yes | — |
| 12 | C2×Q8 | C8 | 1 | 8 | yes | complemented |
| 13 | C2×Q8 | C2×C4 | 1 | 4 | yes | — |
| 14 | C22×C4 | C2×C4 | 1 | 3 | yes | — |
| 15 | C22×C4 | C2×C4 | 1 | 3 | yes | — |
| 16 | C22×C4 | C23 | 1 | 1 | yes | characteristic |
| 17 | C2×C8 | C23 | 1 | 8 | yes | — |
| 18 | C4⋊C4 | C2×C4 | 1 | 12 | yes | — |
| 19 | C42 | C2×C4 | 1 | 6 | yes | — |
| 20 | C42 | C23 | 1 | 6 | yes | — |
| 21 | C23 | C22×C4 | 1 | 1 | yes | characteristic |
| 22 | Q8 | C2×C8 | 1 | 16 | yes | — |
| 23 | C2×C4 | — | 2 | 2 | no | — |
| 24 | C2×C4 | — | 4 | 3 | no | — |
| 25 | C2×C4 | C2×C8 | 1 | 12 | yes | — |
| 26 | C2×C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 27 | C2×C4 | C22×C4 | 1 | 6 | yes | — |
| 28 | C2×C4 | C22×C4 | 1 | 6 | yes | — |
| 29 | C2×C4 | C24 | 1 | 1 | yes | characteristic |
| 30 | C8 | — | 2 | 8 | no | — |
| 31 | C22 | — | 2 | 1 | no | — |
| 32 | C22 | — | 2 | 1 | no | — |
| 33 | C22 | C22×C8 | 1 | 2 | yes | — |
| 34 | C22 | C23×C4 | 1 | 1 | yes | characteristic |
| 35 | C4 | — | 2 | 1 | no | — |
| 36 | C4 | — | 2 | 6 | no | — |
| 37 | C4 | C22×C8 | 1 | 12 | yes | — |
| 38 | C4 | 2−1+4 | 1 | 2 | yes | — |
| 39 | C2 | — | 2 | 2 | no | — |
| 40 | C2 | #519 | 1 | 1 | yes | characteristic |
| 41 | C2 | C23×C8 | 1 | 1 | yes | characteristic |
| 42 | C2 | #568 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C24 |
| commutator | C2 | C23×C8 |
| socle | C22 | C23×C4 |
| prime radical | C23 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C8×Q8 | C2 | semidirect |
| 12. C2×Q8 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×Q8 | 2 |
| C8×Q8 | 2 |
| #431 | 2 |