Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
187 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 187 conjugacy classes of subgroups — too many to draw. Only its 56 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 | #120 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C24 | C23 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #385 | C2 | semidirect |
| 5. #342 | C2 | semidirect |
| 17. #99 | C4 | semidirect |
| 45. C22⋊C4 | C2×C4 | semidirect |
| 53. C23 | C4⋊C4 | semidirect |
| 80. C2×C4 | C4⋊C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #518 | 2 |
| C2×#99 | 2 |
| #386 | 2 |
| #385 | 2 |
| #342 | 2 |