Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
288 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 90 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 | C22×C109 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C23×C109 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C42×C109 ◁ C8×Q8×C109 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 90. Q8×C109 | Q32 | semidirect |
| 117. #12358 | C109 | direct (N × H) |
| 128. C109 | #12358 | direct (N × H) |
| 211. Q8 | Q32×C109 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×Q16×C109 | 2 |
| C109×#2241 | 2 |
| C109×#2209 | 2 |
| C109×#1670 | 2 |
| C109×#1365 | 2 |
| C109×Q8⋊C16 | 2 |
| C109×#1332 | 2 |
| #12358 | 109 |