Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
500 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#120 | C2 | 1 | 6 | yes | maximal, complemented |
| 2 | #394 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | #390 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C25 | C22 | 1 | 2 | yes | complemented |
| 5 | C22×D4 | C22 | 1 | 6 | yes | — |
| 6 | C22×D4 | C22 | 1 | 3 | yes | — |
| 7 | #120 | — | 2 | 24 | no | — |
| 8 | C2×C4⋊C4 | C22 | 1 | 3 | yes | complemented |
| 9 | C2×C22⋊C4 | C22 | 1 | 12 | yes | complemented |
| 10 | C2×C22⋊C4 | C22 | 1 | 6 | yes | complemented |
| 11 | C2×C42 | C22 | 1 | 1 | yes | characteristic, complemented |
| 12 | #95 | C22 | 1 | 2 | yes | complemented |
| 13 | C24 | — | 2 | 12 | no | — |
| 14 | C24 | — | 4 | 6 | no | — |
| 15 | C24 | — | 4 | 2 | no | — |
| 16 | C24 | C23 | 1 | 6 | yes | — |
| 17 | C2×D4 | — | 2 | 24 | no | — |
| 18 | C2×D4 | — | 4 | 3 | no | — |
| 19 | C2×D4 | — | 4 | 3 | no | — |
| 20 | C2×D4 | — | 4 | 6 | no | — |
| 21 | C2×D4 | D4 | 1 | 12 | yes | complemented |
| 22 | C22×C4 | C23 | 1 | 6 | yes | — |
| 23 | C22×C4 | C23 | 1 | 3 | yes | — |
| 24 | C4⋊C4 | — | 4 | 3 | no | — |
| 25 | C22⋊C4 | — | 2 | 12 | no | — |
| 26 | C22⋊C4 | — | 2 | 24 | no | — |
| 27 | C42 | — | 4 | 1 | no | — |
| 28 | C23 | — | 2 | 12 | no | — |
| 29 | C23 | — | 2 | 6 | no | — |
| 30 | C23 | — | 2 | 6 | no | — |
| 31 | C23 | — | 2 | 12 | no | — |
| 32 | C23 | — | 4 | 24 | no | — |
| 33 | C23 | — | 4 | 24 | no | — |
| 34 | C23 | — | 4 | 8 | no | — |
| 35 | C23 | C2×D4 | 1 | 12 | yes | — |
| 36 | C23 | C24 | 1 | 1 | yes | characteristic |
| 37 | D4 | — | 4 | 12 | no | — |
| 38 | D4 | — | 4 | 24 | no | — |
| 39 | C2×C4 | — | 2 | 3 | no | — |
| 40 | C2×C4 | — | 2 | 3 | no | — |
| 41 | C2×C4 | — | 2 | 12 | no | — |
| 42 | C2×C4 | — | 2 | 6 | no | — |
| 43 | C2×C4 | C2×D4 | 1 | 6 | yes | — |
| 44 | C22 | — | 2 | 12 | no | — |
| 45 | C22 | — | 2 | 24 | no | — |
| 46 | C22 | — | 4 | 32 | no | — |
| 47 | C22 | — | 4 | 12 | no | — |
| 48 | C22 | — | 4 | 6 | no | — |
| 49 | C22 | — | 4 | 6 | no | — |
| 50 | C22 | C22×D4 | 1 | 3 | yes | — |
| 51 | C22 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 52 | C22 | H(5,𝔽2) | 1 | 3 | yes | — |
| 53 | C4 | — | 2 | 6 | no | — |
| 54 | C4 | — | 4 | 6 | no | — |
| 55 | C2 | — | 4 | 12 | no | — |
| 56 | C2 | #534 | 1 | 3 | yes | — |
| 57 | C2 | D42 | 1 | 3 | yes | — |
| 58 | C2 | H(3,𝔽4) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#120 | C2 | semidirect |
| 2. #394 | C2 | semidirect |
| 3. #390 | C2 | semidirect |
| 4. C25 | C22 | semidirect |
| 8. C2×C4⋊C4 | C22 | semidirect |
| 9. C2×C22⋊C4 | C22 | semidirect |
| 10. C2×C22⋊C4 | C22 | semidirect |
| 11. C2×C42 | C22 | semidirect |
| 12. #95 | C22 | semidirect |
| 21. C2×D4 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#120 | 2 |
| #394 | 2 |
| #390 | 2 |