Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
360 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 120 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. C43×C211 | C2 | semidirect |
| 7. C4⋊Q8×C211 | C4 | semidirect |
| 8. #127×C211 | C4 | semidirect |
| 9. #124×C211 | C4 | semidirect |
| 36. C42×C211 | D4 | semidirect |
| 64. C4×C211 | #104 | semidirect |
| 72. C211 | #1962 | direct (N × H) |
| 73. #1962 | C211 | direct (N × H) |
| 75. #423 | C2×C211 | semidirect |
| 76. C2×#104 | C2×C211 | semidirect |
| 77. C43 | C2×C211 | semidirect |
| 80. C4⋊Q8 | C4×C211 | semidirect |
| 81. #127 | C4×C211 | semidirect |
| 82. #124 | C4×C211 | semidirect |
| 109. C42 | D4×C211 | semidirect |
| 137. C4 | #104×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 |
| C43×C211 | 2 |
| #1962 | 211 |