Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
232 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 232 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, socle, prime radical | C23×C211 | D8 |
| commutator | C8 | C22×C4×C211 |
| Frattini, cosocle | C2×C8 | C23×C211 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C211 ◁ C22×C4×C211 ◁ C22×C8×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #358×C211 | C2 | direct (N × H) |
| 8. C2×Q16×C211 | C4 | semidirect |
| 25. Q16×C211 | C2×C4 | semidirect |
| 57. C2×C211 | #358 | direct (N × H) |
| 61. C211 | C2×#358 | direct (N × H) |
| 62. C2×#358 | C211 | direct (N × H) |
| 66. #358 | C2×C211 | direct (N × H) |
| 70. C2×Q16 | C4×C211 | semidirect |
| 87. Q16 | C2×C4×C211 | semidirect |
| 119. C2 | #358×C211 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q16×C211 | 2 |
| C22×C16×C211 | 2 |
| C2×#107×C211 | 2 |
| #358×C211 | 2 |
| C2×#358 | 211 |