Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
139 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 139 conjugacy classes of subgroups — too many to draw. Only its 36 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, Frattini, cosocle | C42 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #550 | C2 | semidirect |
| 2. #493 | C2 | semidirect |
| 3. #460 | C2 | semidirect |
| 4. #459 | C2 | semidirect |
| 5. #331 | C2 | semidirect |
| 6. Q8⋊C8 | C2 | semidirect |
| 7. D4⋊C8 | C2 | semidirect |
| 56. Q8 | D8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #550 | 2 |
| #493 | 2 |
| #460 | 2 |
| #459 | 2 |
| #331 | 2 |
| Q8⋊C8 | 2 |
| D4⋊C8 | 2 |