Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
120 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #559 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #493 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #460 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #331 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | Q8⋊C8 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C2×SD16 | — | 2 | 2 | no | — |
| 7 | C2×D8 | — | 2 | 2 | no | — |
| 8 | #127 | C22 | 1 | 1 | yes | characteristic |
| 9 | #127 | C22 | 1 | 1 | yes | characteristic |
| 10 | #124 | — | 2 | 2 | no | — |
| 11 | #123 | — | 2 | 2 | no | — |
| 12 | C4×Q8 | C22 | 1 | 2 | yes | — |
| 13 | C4×D4 | — | 2 | 2 | no | — |
| 14 | C4⋊C8 | C22 | 1 | 2 | yes | — |
| 15 | #102 | — | 2 | 2 | no | — |
| 16 | C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×Q8 | D4 | 1 | 2 | yes | — |
| 18 | C2×D4 | — | 2 | 2 | no | — |
| 19 | C2×D4 | — | 2 | 2 | no | — |
| 20 | C2×D4 | — | 2 | 1 | no | — |
| 21 | C2×D4 | D4 | 1 | 2 | yes | — |
| 22 | C22×C4 | — | 2 | 2 | no | — |
| 23 | SD16 | — | 4 | 4 | no | — |
| 24 | D8 | — | 4 | 4 | no | — |
| 25 | C2×C8 | — | 2 | 1 | no | — |
| 26 | C2×C8 | — | 2 | 2 | no | — |
| 27 | C4⋊C4 | — | 2 | 2 | no | — |
| 28 | C4⋊C4 | D4 | 1 | 2 | yes | — |
| 29 | C22⋊C4 | — | 2 | 4 | no | — |
| 30 | C22⋊C4 | — | 2 | 2 | no | — |
| 31 | C42 | — | 2 | 2 | no | — |
| 32 | C42 | C23 | 1 | 1 | yes | characteristic |
| 33 | C23 | — | 2 | 2 | no | — |
| 34 | C23 | — | 4 | 1 | no | — |
| 35 | Q8 | — | 2 | 2 | no | — |
| 36 | Q8 | — | 2 | 2 | no | — |
| 37 | D4 | — | 4 | 2 | no | — |
| 38 | D4 | — | 4 | 2 | no | — |
| 39 | D4 | — | 4 | 2 | no | — |
| 40 | D4 | — | 4 | 2 | no | — |
| 41 | D4 | — | 4 | 2 | no | — |
| 42 | D4 | — | 4 | 2 | no | — |
| 43 | C2×C4 | — | 2 | 2 | no | — |
| 44 | C2×C4 | — | 2 | 2 | no | — |
| 45 | C2×C4 | — | 8 | 2 | no | — |
| 46 | C2×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 47 | C2×C4 | C2×D4 | 1 | 2 | yes | — |
| 48 | C8 | — | 2 | 2 | no | — |
| 49 | C8 | — | 4 | 2 | no | — |
| 50 | C22 | — | 4 | 4 | no | — |
| 51 | C22 | — | 4 | 2 | no | — |
| 52 | C22 | — | 8 | 2 | no | — |
| 53 | C22 | — | 8 | 1 | no | — |
| 54 | C22 | #120 | 1 | 1 | yes | characteristic |
| 55 | C4 | — | 2 | 1 | no | — |
| 56 | C4 | — | 4 | 2 | no | — |
| 57 | C4 | — | 4 | 2 | no | — |
| 58 | C4 | #135 | 1 | 2 | yes | — |
| 59 | C4 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 60 | C2 | — | 8 | 2 | no | — |
| 61 | C2 | — | 16 | 1 | no | — |
| 62 | C2 | #449 | 1 | 2 | yes | — |
| 63 | C2 | #453 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C42 | C23 |
| prime radical | #493 | C2 |
| 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. #559 | C2 | semidirect |
| 3. #460 | C2 | semidirect |
| 4. #331 | C2 | semidirect |
| 5. Q8⋊C8 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #559 | 2 |
| #493 | 2 |
| #460 | 2 |
| #331 | 2 |
| Q8⋊C8 | 2 |