Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
360 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 162 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×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C32 |
| prime radical | C32×#121 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C22×C32×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C32×SD16 | C2 | semidirect |
| 2. C2×C32×M16 | C2 | semidirect |
| 3. C32×C22⋊Q8 | C2 | semidirect |
| 5. C32×#107 | C2 | semidirect |
| 6. C32×#103 | C2 | semidirect |
| 7. C32×#102 | C2 | semidirect |
| 8. C3×#468 | C3 | direct (N × H) |
| 22. C2×C3×SD16 | C2×C3 | semidirect |
| 23. C2×C3×M16 | C2×C3 | semidirect |
| 24. C3×C22⋊Q8 | C2×C3 | semidirect |
| 26. C3×#107 | C2×C3 | semidirect |
| 27. C3×#103 | C2×C3 | semidirect |
| 28. C3×#102 | C2×C3 | semidirect |
| 44. C32×C8 | D4 | semidirect |
| 45. #468 | C32 | direct (N × H) |
| 72. C2×SD16 | C2×C32 | semidirect |
| 73. C2×M16 | C2×C32 | semidirect |
| 74. C22⋊Q8 | C2×C32 | semidirect |
| 76. #107 | C2×C32 | semidirect |
| 77. #103 | C2×C32 | semidirect |
| 78. #102 | C2×C32 | semidirect |
| 94. C3×C8 | C3×D4 | semidirect |
| 126. C32 | #468 | direct (N × H) |
| 142. C8 | C32×D4 | semidirect |
| 161. C3 | C3×#468 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C32×SD16 | 2 |
| C2×C32×M16 | 2 |
| C32×C22⋊Q8 | 2 |
| C32×#121 | 2 |
| C32×#107 | 2 |
| C32×#103 | 2 |
| C32×#102 | 2 |
| C3×#468 | 3 |