Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
100 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #405 | C2 | 1 | 3 | yes | maximal |
| 2 | #338 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | H(3,ℤ/4) | C2 | 1 | 3 | yes | maximal |
| 4 | C2×M16 | C22 | 1 | 3 | yes | — |
| 5 | C22×C8 | C22 | 1 | 1 | yes | characteristic |
| 6 | #117 | C22 | 1 | 3 | yes | — |
| 7 | #97 | C4 | 1 | 6 | yes | complemented |
| 8 | C4×C8 | C4 | 1 | 6 | yes | complemented |
| 9 | C22×C4 | C23 | 1 | 1 | yes | characteristic |
| 10 | M16 | — | 2 | 6 | no | — |
| 11 | C2×C8 | C2×C4 | 1 | 6 | yes | — |
| 12 | C2×C8 | C2×C4 | 1 | 3 | yes | — |
| 13 | C2×C8 | C2×C4 | 1 | 3 | yes | — |
| 14 | C4⋊C4 | — | 2 | 3 | no | — |
| 15 | C22⋊C4 | — | 2 | 3 | no | — |
| 16 | C42 | C2×C4 | 1 | 6 | yes | — |
| 17 | C23 | #41 | 1 | 1 | yes | characteristic |
| 18 | C2×C4 | — | 2 | 6 | no | — |
| 19 | C2×C4 | C22×C4 | 1 | 3 | yes | — |
| 20 | C2×C4 | #41 | 1 | 3 | yes | — |
| 21 | C8 | — | 2 | 6 | no | — |
| 22 | C8 | C42 | 1 | 3 | yes | — |
| 23 | C8 | C42 | 1 | 1 | yes | characteristic |
| 24 | C22 | — | 4 | 1 | no | — |
| 25 | C22 | #117 | 1 | 3 | yes | — |
| 26 | C4 | — | 4 | 6 | no | — |
| 27 | C4 | C2×C42 | 1 | 1 | yes | characteristic |
| 28 | C4 | #117 | 1 | 3 | yes | — |
| 29 | C2 | — | 2 | 3 | no | — |
| 30 | C2 | #376 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C8 | C42 |
| commutator | C4 | C2×C42 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #376 |
| prime radical | C23 | #41 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. #97 | C4 | semidirect |
| 8. C4×C8 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #405 | 2 |
| #338 | 2 |
| H(3,ℤ/4) | 2 |