Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
320 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 112 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 | C2×C4 | C22×C4×C211 |
| Frattini, cosocle | C22×C4 | C23×C211 |
| socle | C22×C211 | C2×C22⋊C4 |
| prime radical | #532×C211 | C2 |
| LCS | G ▷ C2×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 |
|---|---|---|
| 2. #423×C211 | C2 | semidirect |
| 3. C2×#104×C211 | C2 | semidirect |
| 4. #376×C211 | C2 | semidirect |
| 7. C4⋊Q8×C211 | C4 | semidirect |
| 8. #127×C211 | C4 | semidirect |
| 9. #124×C211 | C4 | semidirect |
| 35. C42×C211 | D4 | semidirect |
| 71. C211 | #1963 | direct (N × H) |
| 72. #1963 | C211 | direct (N × H) |
| 74. #423 | C2×C211 | semidirect |
| 75. C2×#104 | C2×C211 | semidirect |
| 76. #376 | C2×C211 | semidirect |
| 79. C4⋊Q8 | C4×C211 | semidirect |
| 80. #127 | C4×C211 | semidirect |
| 81. #124 | C4×C211 | semidirect |
| 107. C42 | D4×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #532×C211 | 2 |
| #423×C211 | 2 |
| C2×#104×C211 | 2 |
| #376×C211 | 2 |
| #1963 | 211 |