Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
396 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C211 | D4 |
| commutator | C4 | C2×C42×C211 |
| Frattini, cosocle | C22×C4 | C23×C211 |
| socle | C23×C211 | C2×D4 |
| prime radical | C22×D4×C211 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C211 ◁ C2×C42×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C4×C8×C211 | C2 | semidirect |
| 4. C4×C4⋊C4×C211 | C2 | semidirect |
| 5. #340×C211 | C2 | semidirect |
| 10. C4×D4×C211 | C4 | semidirect |
| 16. #102×C211 | C4 | direct (N × H) |
| 34. C2×C8×C211 | C2×C4 | semidirect |
| 38. C4⋊C4×C211 | C2×C4 | semidirect |
| 51. D4×C211 | C42 | semidirect |
| 82. C4×C211 | #102 | direct (N × H) |
| 92. C211 | C4×#102 | direct (N × H) |
| 93. C4×#102 | C211 | direct (N × H) |
| 96. C2×C4×C8 | C2×C211 | semidirect |
| 97. C4×C4⋊C4 | C2×C211 | semidirect |
| 98. #340 | C2×C211 | semidirect |
| 103. C4×D4 | C4×C211 | semidirect |
| 109. #102 | C4×C211 | direct (N × H) |
| 127. C2×C8 | C2×C4×C211 | semidirect |
| 131. C4⋊C4 | C2×C4×C211 | semidirect |
| 144. D4 | C42×C211 | semidirect |
| 175. C4 | #102×C211 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D4×C211 | 2 |
| C2×#102×C211 | 2 |
| C2×C4×C8×C211 | 2 |
| C4×C4⋊C4×C211 | 2 |
| #340×C211 | 2 |
| C4×#102 | 211 |