Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
576 conjugacy classes of subgroups; only the 438 normal subgroups are shown.
This group has 576 conjugacy classes of subgroups — too many to draw. Only its 438 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 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C32 |
| prime radical | C23×C32 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×C4⋊Q8 | C2 | semidirect |
| 3. C32×#125 | C2 | semidirect |
| 4. C32×#125 | C2 | semidirect |
| 6. C32×C4×Q8 | C2 | semidirect |
| 8. C3×#541 | C3 | direct (N × H) |
| 22. C3×C4⋊Q8 | C2×C3 | semidirect |
| 24. C3×#125 | C2×C3 | semidirect |
| 25. C3×#125 | C2×C3 | semidirect |
| 27. C3×C4×Q8 | C2×C3 | semidirect |
| 38. #541 | C32 | direct (N × H) |
| 61. C4⋊Q8 | C2×C32 | semidirect |
| 63. #125 | C2×C32 | semidirect |
| 64. #125 | C2×C32 | semidirect |
| 66. C4×Q8 | C2×C32 | semidirect |
| 102. C32 | #541 | direct (N × H) |
| 124. C3 | C3×#541 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×C4⋊Q8 | 2 |
| C32×#126 | 2 |
| C32×#125 | 2 |
| C32×C22⋊Q8 | 2 |
| C32×C4×Q8 | 2 |
| C32×#117 | 2 |
| C3×#541 | 3 |