Finite Group DatabasebetaFinite Group DatabasebetaOrder 312.
16 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3×Dic13 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #1057 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 3 | C2×C3×C13 | C4 | 1 | 1 | yes | characteristic |
| 4 | Dic13 | C2×C3 | 1 | 1 | yes | characteristic |
| 5 | C3×C13 | C8 | 1 | 1 | yes | characteristic, complemented, Hall |
| 6 | C2×C13 | C3×C4 | 1 | 1 | yes | characteristic |
| 7 | C3×C8 | — | 13 | 1 | no | maximal, Hall |
| 8 | C13 | C3×C8 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 9 | C3×C4 | — | 13 | 1 | no | — |
| 10 | C8 | — | 13 | 1 | no | Sylow |
| 11 | C2×C3 | #261 | 1 | 1 | yes | characteristic |
| 12 | C4 | — | 13 | 1 | no | — |
| 13 | C3 | #1057 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 14 | C2 | C3×#261 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #261 |
| commutator, hypocenter | C13 | C3×C8 |
| Frattini | C2 | C3×#261 |
| Fitting, socle, prime radical | C2×C3×C13 | C4 |
| cosocle | Dic13 | C2×C3 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C2×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #1057 | C3 | direct (N × H) |
| 5. C3×C13 | C8 | semidirect |
| 8. C13 | C3×C8 | semidirect |
| 13. C3 | #1057 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×Dic13 | 2 |
| #1057 | 3 |
| C3×C8 | 13 |