Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
712 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #585 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #568 | C2 | 1 | 15 | yes | maximal, complemented |
| 3 | C2×#131 | C2 | 1 | 15 | yes | maximal, complemented |
| 4 | 2−1+4 | C4 | 1 | 6 | yes | — |
| 5 | H(5,𝔽2) | C4 | 1 | 10 | yes | — |
| 6 | C2×#41 | C22 | 1 | 15 | yes | — |
| 7 | #131 | C22 | 1 | 20 | yes | complemented |
| 8 | #131 | C22 | 1 | 60 | yes | complemented |
| 9 | C2×M16 | C22 | 1 | 45 | yes | complemented |
| 10 | C22×C8 | C22 | 1 | 15 | yes | complemented |
| 11 | #41 | C2×C4 | 1 | 60 | yes | — |
| 12 | #41 | C23 | 1 | 20 | yes | — |
| 13 | C2×Q8 | C2×C4 | 1 | 15 | yes | — |
| 14 | C2×D4 | C2×C4 | 1 | 45 | yes | — |
| 15 | C22×C4 | C23 | 1 | 15 | yes | — |
| 16 | M16 | C23 | 1 | 60 | yes | — |
| 17 | C2×C8 | C23 | 1 | 45 | yes | — |
| 18 | C2×C8 | C23 | 1 | 15 | yes | — |
| 19 | C23 | C22×C4 | 1 | 15 | yes | — |
| 20 | Q8 | C22×C4 | 1 | 20 | yes | — |
| 21 | D4 | C22×C4 | 1 | 60 | yes | — |
| 22 | C2×C4 | C22×C4 | 1 | 45 | yes | — |
| 23 | C2×C4 | C24 | 1 | 15 | yes | — |
| 24 | C8 | C24 | 1 | 15 | yes | — |
| 25 | C8 | C24 | 1 | 1 | yes | characteristic |
| 26 | C22 | — | 4 | 15 | no | — |
| 27 | C22 | C23×C4 | 1 | 15 | yes | — |
| 28 | C4 | C23×C4 | 1 | 15 | yes | — |
| 29 | C4 | C25 | 1 | 1 | yes | characteristic |
| 30 | C2 | — | 2 | 15 | no | — |
| 31 | C2 | C24×C4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C8 | C24 |
| commutator, socle | C2 | C24×C4 |
| Frattini, cosocle | C4 | C25 |
| prime radical | #585 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #568 | C2 | semidirect |
| 3. C2×#131 | C2 | semidirect |
| 7. #131 | C22 | semidirect |
| 8. #131 | C22 | semidirect |
| 9. C2×M16 | C22 | semidirect |
| 10. C22×C8 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #585 | 2 |
| #568 | 2 |
| C2×#131 | 2 |