Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
444 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×C4⋊C4 | C2 | 1 | 6 | yes | maximal |
| 2 | C22×C42 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #384 | C2 | 1 | 8 | yes | maximal, direct product factor |
| 4 | C23×C4 | C22 | 1 | 4 | yes | — |
| 5 | C23×C4 | C22 | 1 | 3 | yes | — |
| 6 | C2×C4⋊C4 | — | 2 | 24 | no | — |
| 7 | C2×C4⋊C4 | C22 | 1 | 24 | yes | — |
| 8 | C2×C42 | C4 | 1 | 8 | yes | complemented |
| 9 | C2×C42 | C22 | 1 | 4 | yes | — |
| 10 | C24 | C23 | 1 | 1 | yes | characteristic |
| 11 | C22×C4 | — | 2 | 24 | no | — |
| 12 | C22×C4 | C2×C4 | 1 | 12 | yes | — |
| 13 | C22×C4 | D4 | 1 | 12 | yes | — |
| 14 | C22×C4 | Q8 | 1 | 12 | yes | — |
| 15 | C22×C4 | C23 | 1 | 8 | yes | — |
| 16 | C22×C4 | C23 | 1 | 6 | yes | — |
| 17 | C4⋊C4 | — | 2 | 48 | no | — |
| 18 | C42 | C2×C4 | 1 | 16 | yes | complemented |
| 19 | C23 | C22×C4 | 1 | 2 | yes | — |
| 20 | C23 | C2×D4 | 1 | 6 | yes | — |
| 21 | C23 | C2×Q8 | 1 | 6 | yes | — |
| 22 | C23 | C24 | 1 | 1 | yes | characteristic |
| 23 | C2×C4 | — | 2 | 24 | no | — |
| 24 | C2×C4 | — | 4 | 16 | no | — |
| 25 | C2×C4 | C4⋊C4 | 1 | 48 | yes | complemented |
| 26 | C2×C4 | C22×C4 | 1 | 12 | yes | — |
| 27 | C2×C4 | C2×D4 | 1 | 12 | yes | — |
| 28 | C2×C4 | C2×Q8 | 1 | 12 | yes | — |
| 29 | C22 | C2×C4⋊C4 | 1 | 12 | yes | — |
| 30 | C22 | #127 | 1 | 4 | yes | — |
| 31 | C22 | C4⋊Q8 | 1 | 12 | yes | — |
| 32 | C22 | C23×C4 | 1 | 1 | yes | characteristic |
| 33 | C22 | C22×D4 | 1 | 3 | yes | — |
| 34 | C22 | C22×Q8 | 1 | 3 | yes | — |
| 35 | C4 | — | 4 | 8 | no | — |
| 36 | C4 | C2×C4⋊C4 | 1 | 24 | yes | complemented |
| 37 | C2 | #384 | 1 | 8 | yes | direct product factor |
| 38 | C2 | C22×C4⋊C4 | 1 | 3 | yes | — |
| 39 | C2 | C2×#127 | 1 | 1 | yes | characteristic |
| 40 | C2 | C2×C4⋊Q8 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C23 |
| commutator | C22 | C23×C4 |
| Frattini, cosocle | C23 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #384 | C2 | direct (N × H) |
| 8. C2×C42 | C4 | semidirect |
| 18. C42 | C2×C4 | semidirect |
| 25. C2×C4 | C4⋊C4 | semidirect |
| 36. C4 | C2×C4⋊C4 | semidirect |
| 37. C2 | #384 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4⋊C4 | 2 |
| C22×C42 | 2 |
| #384 | 2 |