Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
312 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 312 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, socle | C22×C211 | C2×C22⋊C4 |
| commutator | C23 | C22×C4×C211 |
| Frattini, cosocle | C22×C4 | C23×C211 |
| prime radical | C22×D4×C211 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×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×#101×C211 | C2 | semidirect |
| 5. C4×M16×C211 | C2 | semidirect |
| 80. C211 | #1967 | direct (N × H) |
| 81. #1967 | C211 | direct (N × H) |
| 83. #423 | C2×C211 | semidirect |
| 84. C2×#101 | C2×C211 | semidirect |
| 86. C4×M16 | C2×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#124×C211 | 2 |
| #423×C211 | 2 |
| C2×#101×C211 | 2 |
| C2×#100×C211 | 2 |
| C4×M16×C211 | 2 |
| #1967 | 211 |