Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
213 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 213 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C22×D8 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×SD16 | C2 | semidirect |
| 3. C2×#124 | C2 | semidirect |
| 4. C2×#103 | C2 | semidirect |
| 5. C2×#102 | C2 | semidirect |
| 6. #390 | C2 | semidirect |
| 7. #336 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×SD16 | 2 |
| C22×D8 | 2 |
| C2×#124 | 2 |
| C2×#103 | 2 |
| C2×#102 | 2 |
| #390 | 2 |
| #336 | 2 |