Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
310 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 68 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 | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C2×H(5,𝔽2) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#104 | C2 | semidirect |
| 3. #417 | C2 | semidirect |
| 4. C2×#100 | C2 | semidirect |
| 5. #390 | C2 | semidirect |
| 8. H(5,𝔽2) | C4 | semidirect |
| 14. C2×M16 | C22 | semidirect |
| 15. C2×C4⋊C4 | C22 | semidirect |
| 17. C2×C42 | C22 | semidirect |
| 64. Q8 | C22⋊C4 | semidirect |
| 71. D4 | C22⋊C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×H(5,𝔽2) | 2 |
| C2×#104 | 2 |
| #417 | 2 |
| C2×#100 | 2 |
| #390 | 2 |