Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
389 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 389 conjugacy classes of subgroups — too many to draw. Only its 103 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 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #2621 | C2 | semidirect |
| 3. #2074 | C2 | semidirect |
| 4. #2073 | C2 | semidirect |
| 5. #2072 | C2 | semidirect |
| 6. #2016 | C2 | semidirect |
| 7. #2015 | C2 | semidirect |
| 8. #2013 | C2 | semidirect |
| 9. #2001 | C2 | semidirect |
| 11. #1818 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2621 | 2 |
| #2620 | 2 |
| #2074 | 2 |
| #2073 | 2 |
| #2072 | 2 |
| #2016 | 2 |
| #2015 | 2 |
| #2013 | 2 |
| #2001 | 2 |
| #2000 | 2 |
| #1818 | 2 |