Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
744 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C23×Q8 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #519 | C2 | 1 | 16 | yes | maximal, direct product factor |
| 3 | C2×C4×Q8 | C2 | 1 | 8 | yes | maximal, complemented |
| 4 | C2×#117 | C2 | 1 | 6 | yes | maximal |
| 5 | C22×Q8 | C4 | 1 | 16 | yes | complemented |
| 6 | C22×Q8 | C22 | 1 | 4 | yes | — |
| 7 | C22×Q8 | C22 | 1 | 8 | yes | — |
| 8 | C23×C4 | C22 | 1 | 3 | yes | — |
| 9 | C4×Q8 | C22 | 1 | 64 | yes | complemented |
| 10 | #117 | C22 | 1 | 48 | yes | — |
| 11 | C2×C4⋊C4 | C22 | 1 | 12 | yes | — |
| 12 | C2×C22⋊C4 | C22 | 1 | 4 | yes | — |
| 13 | C2×C42 | C22 | 1 | 12 | yes | — |
| 14 | C24 | C23 | 1 | 1 | yes | characteristic |
| 15 | C2×Q8 | C2×C4 | 1 | 32 | yes | — |
| 16 | C2×Q8 | C2×C4 | 1 | 64 | yes | complemented |
| 17 | C2×Q8 | C23 | 1 | 16 | yes | — |
| 18 | C22×C4 | C2×C4 | 1 | 24 | yes | — |
| 19 | C22×C4 | C23 | 1 | 8 | yes | — |
| 20 | C22×C4 | C23 | 1 | 12 | yes | — |
| 21 | C22×C4 | C23 | 1 | 6 | yes | — |
| 22 | C4⋊C4 | C23 | 1 | 48 | yes | — |
| 23 | C22⋊C4 | C23 | 1 | 16 | yes | — |
| 24 | C42 | C23 | 1 | 48 | yes | — |
| 25 | C23 | — | 2 | 4 | no | — |
| 26 | C23 | C22×C4 | 1 | 4 | yes | — |
| 27 | C23 | C24 | 1 | 2 | yes | — |
| 28 | C23 | C24 | 1 | 1 | yes | characteristic |
| 29 | Q8 | C22×C4 | 1 | 64 | yes | — |
| 30 | C2×C4 | — | 2 | 16 | no | — |
| 31 | C2×C4 | C22×C4 | 1 | 24 | yes | — |
| 32 | C2×C4 | C22×C4 | 1 | 48 | yes | — |
| 33 | C2×C4 | C24 | 1 | 16 | yes | — |
| 34 | C2×C4 | C24 | 1 | 12 | yes | — |
| 35 | C22 | — | 2 | 8 | no | — |
| 36 | C22 | — | 2 | 4 | no | — |
| 37 | C22 | C23×C4 | 1 | 4 | yes | — |
| 38 | C22 | C23×C4 | 1 | 2 | yes | — |
| 39 | C22 | 2−1+4 | 1 | 4 | yes | — |
| 40 | C22 | C25 | 1 | 1 | yes | characteristic |
| 41 | C4 | — | 2 | 16 | no | — |
| 42 | C4 | C23×C4 | 1 | 24 | yes | — |
| 43 | C2 | — | 2 | 4 | no | — |
| 44 | C2 | #519 | 1 | 4 | yes | direct product factor |
| 45 | C2 | C24×C4 | 1 | 1 | yes | characteristic |
| 46 | C2 | C2×2−1+4 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C24 |
| commutator | C2 | C24×C4 |
| Frattini, cosocle | C22 | C25 |
| prime radical | C24 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #519 | C2 | direct (N × H) |
| 3. C2×C4×Q8 | C2 | semidirect |
| 5. C22×Q8 | C4 | semidirect |
| 9. C4×Q8 | C22 | semidirect |
| 16. C2×Q8 | C2×C4 | semidirect |
| 44. C2 | #519 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q8 | 2 |
| #519 | 2 |
| C2×C4×Q8 | 2 |
| C2×#117 | 2 |