Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
560 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 102 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 | #28458 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ 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 |
| 2. C2×#2224 | C2 | semidirect |
| 4. C2×#1361 | C2 | semidirect |
| 5. C2×#1331 | C2 | semidirect |
| 6. #9731 | C2 | semidirect |
| 7. #7167 | C2 | semidirect |
| 45. #500 | D4 | semidirect |
| 73. C4⋊C16 | D4 | semidirect |
| 77. C4×C16 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2240 | 2 |
| C2×#2224 | 2 |
| #28458 | 2 |
| C2×#1361 | 2 |
| C2×#1331 | 2 |
| #9731 | 2 |
| #7167 | 2 |