Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
3132 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C23×Q8 | C2 | 1 | 60 | yes | maximal, direct product factor |
| 2 | C24×C4 | C2 | 1 | 3 | yes | maximal |
| 3 | C25 | C22 | 1 | 1 | yes | characteristic |
| 4 | C22×Q8 | C22 | 1 | 560 | yes | direct product factor |
| 5 | C23×C4 | C22 | 1 | 90 | yes | — |
| 6 | C24 | Q8 | 1 | 16 | yes | direct product factor |
| 7 | C24 | C23 | 1 | 15 | yes | — |
| 8 | C2×Q8 | C23 | 1 | 960 | yes | direct product factor |
| 9 | C22×C4 | C23 | 1 | 420 | yes | — |
| 10 | C23 | C2×Q8 | 1 | 120 | yes | direct product factor |
| 11 | C23 | C24 | 1 | 35 | yes | — |
| 12 | Q8 | C24 | 1 | 256 | yes | direct product factor |
| 13 | C2×C4 | C24 | 1 | 360 | yes | — |
| 14 | C22 | C22×Q8 | 1 | 140 | yes | direct product factor |
| 15 | C22 | C25 | 1 | 15 | yes | — |
| 16 | C4 | C25 | 1 | 48 | yes | — |
| 17 | C2 | C23×Q8 | 1 | 30 | yes | direct product factor |
| 18 | C2 | C26 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C22 |
| commutator, Frattini, cosocle | C2 | C26 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×Q8 | C2 | direct (N × H) |
| 4. C22×Q8 | C22 | direct (N × H) |
| 6. C24 | Q8 | direct (N × H) |
| 8. C2×Q8 | C23 | direct (N × H) |
| 10. C23 | C2×Q8 | direct (N × H) |
| 12. Q8 | C24 | direct (N × H) |
| 14. C22 | C22×Q8 | direct (N × H) |
| 17. C2 | C23×Q8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q8 | 2 |
| C24×C4 | 2 |