Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
288 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 150 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 | C23×C32 | D4 |
| 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 |
|---|---|---|
| 3. C32×#107 | C2 | semidirect |
| 4. C32×#106 | C2 | semidirect |
| 5. C32×#103 | C2 | semidirect |
| 6. C32×#103 | C2 | semidirect |
| 8. C3×#485 | C3 | direct (N × H) |
| 22. C3×#107 | C2×C3 | semidirect |
| 23. C3×#106 | C2×C3 | semidirect |
| 24. C3×#103 | C2×C3 | semidirect |
| 25. C3×#103 | C2×C3 | semidirect |
| 40. #485 | C32 | direct (N × H) |
| 65. #107 | C2×C32 | semidirect |
| 66. #106 | C2×C32 | semidirect |
| 67. #103 | C2×C32 | semidirect |
| 68. #103 | C2×C32 | semidirect |
| 109. C32 | #485 | direct (N × H) |
| 138. C3 | C3×#485 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×C22⋊Q8 | 2 |
| C32×#117 | 2 |
| C32×#107 | 2 |
| C32×#106 | 2 |
| C32×#103 | 2 |
| C32×C22⋊C8 | 2 |
| C3×#485 | 3 |