Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
780 conjugacy classes of subgroups; only the 520 normal subgroups are shown.
This group has 780 conjugacy classes of subgroups — too many to draw. Only its 520 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 | C2 | C22×C42×C211 |
| Frattini, cosocle | C23 | C24×C211 |
| prime radical | C22×D4×C211 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×C4⋊C4×C211 | C2 | semidirect |
| 3. C4×C22⋊C4×C211 | C2 | semidirect |
| 4. #376×C211 | C2 | semidirect |
| 5. C2×#95×C211 | C2 | semidirect |
| 8. C4×D4×C211 | C4 | semidirect |
| 19. C22×C4×C211 | C2×C4 | semidirect |
| 24. C4⋊C4×C211 | C2×C4 | semidirect |
| 25. C22⋊C4×C211 | C2×C4 | semidirect |
| 27. C42×C211 | C2×C4 | semidirect |
| 32. D4×C211 | C42 | semidirect |
| 56. C211 | #2275 | direct (N × H) |
| 57. #2275 | C211 | direct (N × H) |
| 59. C4×C4⋊C4 | C2×C211 | semidirect |
| 60. C4×C22⋊C4 | C2×C211 | semidirect |
| 61. #376 | C2×C211 | semidirect |
| 62. C2×#95 | C2×C211 | semidirect |
| 65. C4×D4 | C4×C211 | semidirect |
| 76. C22×C4 | C2×C4×C211 | semidirect |
| 81. C4⋊C4 | C2×C4×C211 | semidirect |
| 82. C22⋊C4 | C2×C4×C211 | semidirect |
| 84. C42 | C2×C4×C211 | semidirect |
| 89. D4 | C42×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D4×C211 | 2 |
| C4×C4⋊C4×C211 | 2 |
| C4×C22⋊C4×C211 | 2 |
| #376×C211 | 2 |
| C2×#95×C211 | 2 |
| #2275 | 211 |