Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
175 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #578 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #510 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | #508 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C2×Q32 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #504 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | 2−1+4 | — | 2 | 2 | no | — |
| 7 | #137 | — | 2 | 6 | no | — |
| 8 | #135 | C22 | 1 | 6 | yes | — |
| 9 | C2×Q16 | C22 | 1 | 6 | yes | — |
| 10 | #131 | C22 | 1 | 1 | yes | characteristic |
| 11 | Q32 | C22 | 1 | 3 | yes | — |
| 12 | Q32 | C22 | 1 | 6 | yes | — |
| 13 | SD32 | C22 | 1 | 6 | yes | — |
| 14 | D16 | C22 | 1 | 1 | yes | characteristic |
| 15 | M32 | C22 | 1 | 3 | yes | — |
| 16 | C2×C16 | C22 | 1 | 3 | yes | — |
| 17 | #41 | — | 2 | 6 | no | — |
| 18 | #41 | — | 4 | 6 | no | — |
| 19 | #41 | D4 | 1 | 1 | yes | characteristic |
| 20 | C2×Q8 | — | 2 | 6 | no | — |
| 21 | C2×Q8 | — | 4 | 2 | no | — |
| 22 | Q16 | — | 2 | 6 | no | — |
| 23 | Q16 | C23 | 1 | 6 | yes | — |
| 24 | SD16 | — | 2 | 6 | no | — |
| 25 | D8 | C23 | 1 | 2 | yes | — |
| 26 | M16 | D4 | 1 | 3 | yes | — |
| 27 | C2×C8 | C23 | 1 | 3 | yes | — |
| 28 | C16 | C23 | 1 | 1 | yes | characteristic |
| 29 | C16 | C23 | 1 | 3 | yes | — |
| 30 | Q8 | — | 2 | 6 | no | — |
| 31 | Q8 | — | 4 | 6 | no | — |
| 32 | Q8 | D8 | 1 | 1 | yes | characteristic |
| 33 | D4 | — | 2 | 2 | no | — |
| 34 | D4 | — | 4 | 6 | no | — |
| 35 | D4 | D8 | 1 | 3 | yes | — |
| 36 | C2×C4 | — | 4 | 6 | no | — |
| 37 | C2×C4 | — | 4 | 6 | no | — |
| 38 | C2×C4 | C2×D4 | 1 | 3 | yes | — |
| 39 | C8 | C2×D4 | 1 | 3 | yes | — |
| 40 | C8 | C24 | 1 | 1 | yes | characteristic |
| 41 | C22 | — | 4 | 2 | no | — |
| 42 | C22 | C2×D8 | 1 | 3 | yes | — |
| 43 | C4 | — | 4 | 6 | no | — |
| 44 | C4 | C2×D8 | 1 | 3 | yes | — |
| 45 | C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 46 | C2 | — | 2 | 3 | no | — |
| 47 | C2 | — | 8 | 2 | no | — |
| 48 | C2 | C22×D8 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C24 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #131 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #578 | C2 | semidirect |
| 2. #510 | C2 | semidirect |
| 3. #508 | C2 | semidirect |
| 4. C2×Q32 | C2 | semidirect |
| 5. #504 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #578 | 2 |
| #510 | 2 |
| #508 | 2 |
| C2×Q32 | 2 |
| #504 | 2 |