Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
2996 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #585 | C2 | 1 | 28 | yes | maximal, complemented |
| 2 | C2×H(5,𝔽2) | C2 | 1 | 35 | yes | maximal, complemented |
| 3 | 2−1+4 | C22 | 1 | 56 | yes | complemented |
| 4 | H(5,𝔽2) | C22 | 1 | 280 | yes | complemented |
| 5 | C2×#41 | C22 | 1 | 210 | yes | complemented |
| 6 | C22×D4 | C22 | 1 | 105 | yes | complemented |
| 7 | C24 | C23 | 1 | 30 | yes | complemented |
| 8 | #41 | C23 | 1 | 560 | yes | complemented |
| 9 | C2×Q8 | C23 | 1 | 70 | yes | complemented |
| 10 | C2×D4 | C23 | 1 | 630 | yes | complemented |
| 11 | C22×C4 | C23 | 1 | 105 | yes | complemented |
| 12 | C23 | — | 8 | 30 | no | — |
| 13 | C23 | C24 | 1 | 105 | yes | — |
| 14 | Q8 | C24 | 1 | 56 | yes | — |
| 15 | D4 | C24 | 1 | 280 | yes | — |
| 16 | C2×C4 | C24 | 1 | 210 | yes | — |
| 17 | C22 | — | 4 | 105 | no | — |
| 18 | C22 | C25 | 1 | 35 | yes | — |
| 19 | C4 | C25 | 1 | 28 | yes | — |
| 20 | C2 | — | 2 | 35 | no | — |
| 21 | C2 | C26 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C2 | C26 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #585 | C2 | semidirect |
| 2. C2×H(5,𝔽2) | C2 | semidirect |
| 3. 2−1+4 | C22 | semidirect |
| 4. H(5,𝔽2) | C22 | semidirect |
| 5. C2×#41 | C22 | semidirect |
| 6. C22×D4 | C22 | semidirect |
| 7. C24 | C23 | semidirect |
| 8. #41 | C23 | semidirect |
| 9. C2×Q8 | C23 | semidirect |
| 10. C2×D4 | C23 | semidirect |
| 11. C22×C4 | C23 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #585 | 2 |
| C2×H(5,𝔽2) | 2 |