Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
345 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 345 conjugacy classes of subgroups — too many to draw. Only its 135 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 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C22×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 |
|---|---|---|
| 1. #3164 | C2 | semidirect |
| 4. #2089 | C2 | semidirect |
| 5. #2060 | C2 | semidirect |
| 6. #1952 | C2 | semidirect |
| 7. #1945 | C2 | semidirect |
| 9. #1845 | C2 | semidirect |
| 47. #131 | Q8 | semidirect |
| 89. #104 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3164 | 2 |
| #3083 | 2 |
| #2913 | 2 |
| #2089 | 2 |
| #2060 | 2 |
| #1952 | 2 |
| #1945 | 2 |
| #1859 | 2 |
| #1845 | 2 |
| #1816 | 2 |
| #1764 | 2 |