Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
768 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 768 conjugacy classes of subgroups — too many to draw. Only its 216 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×C32 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C32 |
| prime radical | C22×C32×D8 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C22×C32×C4 ◁ C22×C32×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C32×SD32 | C2 | semidirect |
| 3. C32×#467 | C2 | semidirect |
| 4. C32×#357 | C2 | semidirect |
| 5. C32×C22⋊C16 | C2 | semidirect |
| 6. C3×#2187 | C3 | direct (N × H) |
| 8. C2×C32×Q16 | C22 | semidirect |
| 15. C2×C32×C16 | C22 | semidirect |
| 16. C32×#107 | C22 | semidirect |
| 19. C2×C3×SD32 | C2×C3 | semidirect |
| 20. C3×#467 | C2×C3 | semidirect |
| 21. C3×#357 | C2×C3 | semidirect |
| 22. C3×C22⋊C16 | C2×C3 | semidirect |
| 41. #2187 | C32 | direct (N × H) |
| 43. C2×C3×Q16 | C22×C3 | semidirect |
| 50. C2×C3×C16 | C22×C3 | semidirect |
| 51. C3×#107 | C22×C3 | semidirect |
| 72. C2×SD32 | C2×C32 | semidirect |
| 73. #467 | C2×C32 | semidirect |
| 74. #357 | C2×C32 | semidirect |
| 75. C22⋊C16 | C2×C32 | semidirect |
| 100. C22×C32 | SD32 | semidirect |
| 108. C2×Q16 | C22×C32 | semidirect |
| 115. C2×C16 | C22×C32 | semidirect |
| 116. #107 | C22×C32 | semidirect |
| 165. C22×C3 | C3×SD32 | semidirect |
| 172. C32 | #2187 | direct (N × H) |
| 202. C22 | C32×SD32 | semidirect |
| 209. C3 | C3×#2187 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C32×D8 | 2 |
| C2×C32×SD32 | 2 |
| C32×#467 | 2 |
| C32×#357 | 2 |
| C32×C22⋊C16 | 2 |
| C3×#2187 | 3 |