Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
840 conjugacy classes of subgroups; only the 251 normal subgroups are shown.
This group has 840 conjugacy classes of subgroups — too many to draw. Only its 251 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C3⋊D3 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C22×C32×Q8 | C2 |
| prime radical | C2×#638 | C22 |
| hypercenter | C22×Q8 | C3⋊D3 |
| hypocenter | C32 | D4×Q8 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C32×Q8 | C2 | semidirect |
| 2. C2×Q8×C3⋊D3 | C2 | semidirect |
| 3. #64155 | C2 | semidirect |
| 4. Q8×C3⋊Dic3 | C2 | semidirect |
| 5. #64153 | C2 | semidirect |
| 7. #64133 | C2 | semidirect |
| 23. C22×C3×Q8 | D3 | semidirect |
| 33. C32×Q8 | D4 | semidirect |
| 37. #638 | Q8 | direct (N × H) |
| 60. C22×C32 | C2×Q8 | semidirect |
| 64. C2×C3⋊D3 | C2×Q8 | semidirect |
| 69. C3⋊Dic3 | C2×Q8 | semidirect |
| 71. C22×Q8 | C3⋊D3 | semidirect |
| 80. C3×Q8 | #67 | semidirect |
| 111. C22×C3 | D3×Q8 | semidirect |
| 121. C32 | D4×Q8 | semidirect |
| 126. Q8 | #638 | direct (N × H) |
| 145. C22 | Q8×C3⋊D3 | semidirect |
| 152. C3 | Q8×#67 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C32×Q8 | 2 |
| C2×Q8×C3⋊D3 | 2 |
| #64155 | 2 |
| Q8×C3⋊Dic3 | 2 |
| #64153 | 2 |
| C4×#638 | 2 |
| #64133 | 2 |
| Q8×#67 | 3 |