Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
320 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C211 | C2×D4 |
| commutator | C4 | C2×C42×C211 |
| Frattini, cosocle | C22×C4 | C23×C211 |
| socle | C22×C211 | C2×C22⋊C4 |
| prime radical | C2×#41×C211 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C211 ◁ C2×C42×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C4×M16×C211 | C2 | semidirect |
| 4. #376×C211 | C2 | semidirect |
| 5. #339×C211 | C2 | semidirect |
| 8. C4×Q8×C211 | C4 | semidirect |
| 10. C4×D4×C211 | C4 | semidirect |
| 16. #104×C211 | C4 | semidirect |
| 31. M16×C211 | C2×C4 | semidirect |
| 41. C42×C211 | C2×C4 | semidirect |
| 82. C211 | #1759 | direct (N × H) |
| 83. #1759 | C211 | direct (N × H) |
| 86. C4×M16 | C2×C211 | semidirect |
| 87. #376 | C2×C211 | semidirect |
| 88. #339 | C2×C211 | semidirect |
| 91. C4×Q8 | C4×C211 | semidirect |
| 93. C4×D4 | C4×C211 | semidirect |
| 99. #104 | C4×C211 | semidirect |
| 114. M16 | C2×C4×C211 | semidirect |
| 124. C42 | C2×C4×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#41×C211 | 2 |
| C2×#104×C211 | 2 |
| C4×M16×C211 | 2 |
| #376×C211 | 2 |
| #339×C211 | 2 |
| #1759 | 211 |