Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
344 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C2×#121 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#123 | C2 | semidirect |
| 3. C22×C42 | C2 | semidirect |
| 4. #390 | C2 | semidirect |
| 5. #389 | C2 | semidirect |
| 6. #388 | C2 | semidirect |
| 7. #386 | C2 | semidirect |
| 8. #380 | C2 | semidirect |
| 41. C22×C4 | D4 | semidirect |
| 109. C4 | #123 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#123 | 2 |
| C2×#121 | 2 |
| C22×C42 | 2 |
| #390 | 2 |
| #389 | 2 |
| #388 | 2 |
| #386 | 2 |
| #380 | 2 |