Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
820 conjugacy classes of subgroups; only the 392 normal subgroups are shown.
This group has 820 conjugacy classes of subgroups — too many to draw. Only its 392 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C211 | C24 |
| commutator | C22 | C23×C4×C211 |
| Frattini, cosocle | C23 | C24×C211 |
| prime radical | C24×C211 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C4×Q8×C211 | C2 | semidirect |
| 3. #391×C211 | C2 | semidirect |
| 7. C22×Q8×C211 | C4 | semidirect |
| 32. C22⋊C4×C211 | Q8 | direct (N × H) |
| 41. Q8×C211 | C22⋊C4 | direct (N × H) |
| 52. C2×C4×C211 | C2×Q8 | semidirect |
| 57. C22×C211 | C4×Q8 | semidirect |
| 73. C211 | Q8×C22⋊C4 | direct (N × H) |
| 74. Q8×C22⋊C4 | C211 | direct (N × H) |
| 76. C2×C4×Q8 | C2×C211 | semidirect |
| 77. #391 | C2×C211 | semidirect |
| 81. C22×Q8 | C4×C211 | semidirect |
| 106. C22⋊C4 | Q8×C211 | direct (N × H) |
| 115. Q8 | C22⋊C4×C211 | direct (N × H) |
| 126. C2×C4 | C2×Q8×C211 | semidirect |
| 131. C22 | C4×Q8×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q8×C211 | 2 |
| C2×C4×Q8×C211 | 2 |
| #391×C211 | 2 |
| #380×C211 | 2 |
| C4×C22⋊C4×C211 | 2 |
| Q8×C22⋊C4 | 211 |