Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
465 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #564 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #563 | C2 | 1 | 10 | yes | maximal, complemented |
| 3 | #562 | C2 | 1 | 5 | yes | maximal, complemented |
| 4 | #533 | C2 | 1 | 5 | yes | maximal, complemented |
| 5 | #529 | C2 | 1 | 10 | yes | maximal, complemented |
| 6 | C4⋊Q8 | C22 | 1 | 10 | yes | — |
| 7 | #126 | C22 | 1 | 20 | yes | — |
| 8 | #125 | C22 | 1 | 10 | yes | — |
| 9 | #125 | C22 | 1 | 5 | yes | — |
| 10 | #125 | C22 | 1 | 5 | yes | — |
| 11 | #124 | C22 | 1 | 10 | yes | — |
| 12 | #123 | C22 | 1 | 20 | yes | — |
| 13 | C22⋊Q8 | C22 | 1 | 10 | yes | — |
| 14 | C22⋊Q8 | C22 | 1 | 20 | yes | — |
| 15 | #121 | C22 | 1 | 10 | yes | — |
| 16 | C4×Q8 | C22 | 1 | 10 | yes | — |
| 17 | C4×D4 | C22 | 1 | 10 | yes | — |
| 18 | #117 | C22 | 1 | 10 | yes | — |
| 19 | C2×C42 | C22 | 1 | 5 | yes | — |
| 20 | C2×Q8 | C23 | 1 | 10 | yes | — |
| 21 | C2×D4 | C23 | 1 | 10 | yes | — |
| 22 | C22×C4 | C23 | 1 | 5 | yes | — |
| 23 | C22×C4 | C23 | 1 | 10 | yes | — |
| 24 | C4⋊C4 | C23 | 1 | 10 | yes | — |
| 25 | C4⋊C4 | C23 | 1 | 10 | yes | — |
| 26 | C4⋊C4 | C23 | 1 | 10 | yes | — |
| 27 | C4⋊C4 | C23 | 1 | 10 | yes | — |
| 28 | C4⋊C4 | C23 | 1 | 20 | yes | — |
| 29 | C22⋊C4 | C23 | 1 | 20 | yes | — |
| 30 | C22⋊C4 | C23 | 1 | 20 | yes | — |
| 31 | C42 | C23 | 1 | 5 | yes | — |
| 32 | C42 | C23 | 1 | 5 | yes | — |
| 33 | C42 | C23 | 1 | 10 | yes | — |
| 34 | C23 | C24 | 1 | 5 | yes | — |
| 35 | Q8 | — | 4 | 10 | no | — |
| 36 | D4 | — | 4 | 10 | no | — |
| 37 | C2×C4 | — | 2 | 10 | no | — |
| 38 | C2×C4 | — | 4 | 10 | no | — |
| 39 | C2×C4 | C24 | 1 | 5 | yes | — |
| 40 | C2×C4 | C24 | 1 | 10 | yes | — |
| 41 | C2×C4 | C24 | 1 | 10 | yes | — |
| 42 | C2×C4 | C24 | 1 | 1 | yes | characteristic |
| 43 | C22 | — | 2 | 5 | no | — |
| 44 | C22 | — | 2 | 10 | no | — |
| 45 | C22 | C25 | 1 | 1 | yes | characteristic |
| 46 | C4 | — | 2 | 5 | no | — |
| 47 | C4 | — | 2 | 10 | no | — |
| 48 | C4 | — | 2 | 10 | no | — |
| 49 | C4 | 2−1+4 | 1 | 2 | yes | — |
| 50 | C2 | — | 4 | 5 | no | — |
| 51 | C2 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 52 | C2 | #585 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C24 |
| commutator, Frattini, socle, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #564 | C2 | semidirect |
| 2. #563 | C2 | semidirect |
| 3. #562 | C2 | semidirect |
| 4. #533 | C2 | semidirect |
| 5. #529 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #564 | 2 |
| #563 | 2 |
| #562 | 2 |
| #533 | 2 |
| #529 | 2 |