Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1502 conjugacy classes of subgroups; only the 602 normal subgroups are shown.
This group has 1502 conjugacy classes of subgroups — too many to draw. Only its 602 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×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q8×D8 | C2 | semidirect |
| 2. Q8×D16 | C2 | direct (N × H) |
| 3. C2×#2252 | C2 | semidirect |
| 4. C2×#2224 | C2 | semidirect |
| 5. C2×Q8×C16 | C2 | semidirect |
| 10. Q8×D8 | C22 | semidirect |
| 15. #2252 | C22 | semidirect |
| 16. #2224 | C22 | semidirect |
| 17. Q8×C16 | C22 | semidirect |
| 32. C2×D16 | Q8 | direct (N × H) |
| 61. C2×D8 | C2×Q8 | semidirect |
| 77. D16 | C2×Q8 | direct (N × H) |
| 79. C2×C16 | C2×Q8 | semidirect |
| 93. C2×Q8 | D16 | direct (N × H) |
| 107. D8 | C22×Q8 | semidirect |
| 122. C16 | C22×Q8 | semidirect |
| 129. Q8 | C2×D16 | direct (N × H) |
| 163. C2 | Q8×D16 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×D8 | 2 |
| Q8×D16 | 2 |
| C2×#2252 | 2 |
| C2×#2224 | 2 |
| C2×Q8×C16 | 2 |
| C2×C4×D16 | 2 |