Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
561 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 561 conjugacy classes of subgroups — too many to draw. Only its 100 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 | #447 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #28485 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#2240 | C2 | semidirect |
| 3. C2×#1673 | C2 | semidirect |
| 4. #13275 | C2 | semidirect |
| 5. C2×#1361 | C2 | semidirect |
| 6. C2×D4⋊C16 | C2 | semidirect |
| 7. #7171 | C2 | semidirect |
| 43. #500 | D4 | semidirect |
| 72. C4×C16 | D4 | semidirect |
| 75. #335 | D4 | semidirect |
| 135. #107 | SD16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2240 | 2 |
| #28485 | 2 |
| C2×#1673 | 2 |
| #13275 | 2 |
| C2×#1361 | 2 |
| C2×D4⋊C16 | 2 |
| #7171 | 2 |