Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
242 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | #532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#137 | C2 | semidirect |
| 3. C2×C4×Q8 | C2 | semidirect |
| 4. #464 | C2 | semidirect |
| 5. #462 | C2 | semidirect |
| 6. #460 | C2 | semidirect |
| 7. #423 | C2 | semidirect |
| 8. #417 | C2 | semidirect |
| 38. C2×Q8 | D4 | semidirect |
| 112. C4 | #137 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#137 | 2 |
| #532 | 2 |
| C2×C4×Q8 | 2 |
| #464 | 2 |
| #462 | 2 |
| #460 | 2 |
| #423 | 2 |
| #417 | 2 |