Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
410 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C23×D4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#100 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #407 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C25 | C4 | 1 | 2 | yes | — |
| 5 | C22×D4 | — | 2 | 8 | no | — |
| 6 | C22×D4 | — | 2 | 4 | no | — |
| 7 | C22×D4 | C4 | 1 | 2 | yes | — |
| 8 | C22×D4 | C22 | 1 | 2 | yes | — |
| 9 | C23×C4 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×M16 | C22 | 1 | 4 | yes | complemented |
| 11 | #100 | — | 2 | 8 | no | — |
| 12 | C22⋊C8 | — | 2 | 4 | no | — |
| 13 | C24 | — | 2 | 8 | no | — |
| 14 | C24 | — | 2 | 4 | no | — |
| 15 | C24 | — | 4 | 8 | no | — |
| 16 | C24 | C2×C4 | 1 | 1 | yes | characteristic |
| 17 | C24 | C2×C4 | 1 | 4 | yes | — |
| 18 | C2×D4 | — | 2 | 32 | no | — |
| 19 | C2×D4 | — | 2 | 8 | no | — |
| 20 | C2×D4 | — | 2 | 4 | no | — |
| 21 | C2×D4 | — | 2 | 4 | no | — |
| 22 | C2×D4 | — | 2 | 4 | no | — |
| 23 | C2×D4 | D4 | 1 | 8 | yes | — |
| 24 | C22×C4 | — | 2 | 4 | no | — |
| 25 | C22×C4 | C2×C4 | 1 | 1 | yes | characteristic |
| 26 | C22×C4 | D4 | 1 | 4 | yes | — |
| 27 | C22×C4 | C23 | 1 | 1 | yes | characteristic |
| 28 | M16 | — | 2 | 8 | no | — |
| 29 | C2×C8 | — | 2 | 4 | no | — |
| 30 | C23 | — | 2 | 4 | no | — |
| 31 | C23 | — | 2 | 2 | no | — |
| 32 | C23 | — | 2 | 16 | no | — |
| 33 | C23 | — | 2 | 4 | no | — |
| 34 | C23 | — | 2 | 4 | no | — |
| 35 | C23 | — | 4 | 8 | no | — |
| 36 | C23 | — | 4 | 32 | no | — |
| 37 | C23 | — | 4 | 8 | no | — |
| 38 | C23 | — | 4 | 8 | no | — |
| 39 | C23 | C22⋊C4 | 1 | 2 | yes | — |
| 40 | C23 | C22⋊C4 | 1 | 8 | yes | — |
| 41 | C23 | C22×C4 | 1 | 1 | yes | characteristic |
| 42 | D4 | — | 2 | 16 | no | — |
| 43 | D4 | — | 2 | 16 | no | — |
| 44 | C2×C4 | — | 2 | 2 | no | — |
| 45 | C2×C4 | — | 2 | 8 | no | — |
| 46 | C2×C4 | C22⋊C4 | 1 | 2 | yes | — |
| 47 | C2×C4 | C2×D4 | 1 | 4 | yes | — |
| 48 | C2×C4 | C2×D4 | 1 | 2 | yes | — |
| 49 | C8 | — | 4 | 4 | no | — |
| 50 | C22 | — | 2 | 8 | no | — |
| 51 | C22 | — | 2 | 2 | no | — |
| 52 | C22 | — | 2 | 2 | no | — |
| 53 | C22 | — | 2 | 2 | no | — |
| 54 | C22 | — | 2 | 8 | no | — |
| 55 | C22 | — | 4 | 8 | no | — |
| 56 | C22 | — | 4 | 32 | no | — |
| 57 | C22 | — | 4 | 8 | no | — |
| 58 | C22 | — | 4 | 8 | no | — |
| 59 | C22 | #100 | 1 | 4 | yes | complemented |
| 60 | C22 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 61 | C22 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 62 | C4 | — | 2 | 2 | no | — |
| 63 | C4 | #120 | 1 | 4 | yes | — |
| 64 | C2 | — | 2 | 4 | no | — |
| 65 | C2 | — | 2 | 2 | no | — |
| 66 | C2 | — | 4 | 8 | no | — |
| 67 | C2 | #379 | 1 | 1 | yes | characteristic |
| 68 | C2 | C2×#100 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C23×D4 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#100 | C2 | semidirect |
| 3. #407 | C2 | semidirect |
| 10. C2×M16 | C22 | semidirect |
| 59. C22 | #100 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D4 | 2 |
| C2×#100 | 2 |
| #407 | 2 |