Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
135 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #2840 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 2 | C2×C22⋊A4 | — | 4 | 1 | no | maximal |
| 3 | C22⋊SL(2,𝔽3) | — | 4 | 1 | no | maximal |
| 4 | #95⋊C3 | — | 4 | 3 | no | maximal |
| 5 | #393 | — | 3 | 1 | no | — |
| 6 | #393 | — | 3 | 3 | no | — |
| 7 | #379 | — | 3 | 1 | no | — |
| 8 | C22⋊A4 | — | 4 | 1 | no | — |
| 9 | C25 | A4 | 1 | 1 | yes | characteristic |
| 10 | C22×Q8 | — | 3 | 1 | no | — |
| 11 | C22×Q8 | A4 | 1 | 1 | yes | characteristic, complemented |
| 12 | C2×C22⋊C4 | — | 3 | 3 | no | — |
| 13 | C2×C22⋊C4 | — | 3 | 3 | no | — |
| 14 | #95 | — | 3 | 3 | no | — |
| 15 | #95 | A4 | 1 | 3 | yes | complemented |
| 16 | C2×A4 | — | 16 | 4 | no | — |
| 17 | C2×A4 | — | 16 | 1 | no | — |
| 18 | SL(2,𝔽3) | — | 16 | 4 | no | — |
| 19 | C24 | — | 3 | 1 | no | — |
| 20 | C24 | — | 3 | 1 | no | — |
| 21 | C24 | — | 12 | 1 | no | — |
| 22 | C24 | — | 12 | 1 | no | — |
| 23 | C24 | SL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 24 | C2×Q8 | — | 12 | 1 | no | — |
| 25 | C2×Q8 | — | 12 | 2 | no | — |
| 26 | C2×Q8 | — | 12 | 1 | no | — |
| 27 | C22×C4 | — | 3 | 1 | no | — |
| 28 | C22×C4 | — | 3 | 3 | no | — |
| 29 | C22⋊C4 | — | 12 | 3 | no | — |
| 30 | C22⋊C4 | — | 12 | 3 | no | — |
| 31 | A4 | — | 16 | 4 | no | — |
| 32 | A4 | — | 16 | 1 | no | — |
| 33 | C23 | — | 3 | 1 | no | — |
| 34 | C23 | — | 3 | 1 | no | — |
| 35 | C23 | — | 4 | 4 | no | — |
| 36 | C23 | — | 6 | 3 | no | — |
| 37 | C23 | — | 6 | 3 | no | — |
| 38 | C23 | — | 12 | 1 | no | — |
| 39 | C23 | — | 12 | 4 | no | — |
| 40 | C23 | — | 12 | 3 | no | — |
| 41 | C23 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 42 | Q8 | — | 4 | 4 | no | — |
| 43 | Q8 | — | 12 | 4 | no | — |
| 44 | C2×C4 | — | 6 | 6 | no | — |
| 45 | C2×C4 | — | 6 | 3 | no | — |
| 46 | C2×C4 | — | 6 | 3 | no | — |
| 47 | C2×C3 | — | 64 | 1 | no | — |
| 48 | C22 | — | 3 | 1 | no | — |
| 49 | C22 | — | 3 | 1 | no | — |
| 50 | C22 | — | 4 | 4 | no | — |
| 51 | C22 | — | 6 | 3 | no | — |
| 52 | C22 | — | 6 | 3 | no | — |
| 53 | C22 | — | 12 | 3 | no | — |
| 54 | C22 | — | 12 | 4 | no | — |
| 55 | C22 | — | 12 | 1 | no | — |
| 56 | C22 | C22⋊SL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 57 | C4 | — | 12 | 3 | no | — |
| 58 | C4 | — | 12 | 1 | no | — |
| 59 | C3 | — | 64 | 1 | no | Sylow |
| 60 | C2 | — | 3 | 1 | no | — |
| 61 | C2 | — | 3 | 1 | no | — |
| 62 | C2 | — | 12 | 1 | no | — |
| 63 | C2 | — | 12 | 1 | no | — |
| 64 | C2 | #5464 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #5464 |
| commutator, Fitting, cosocle, hypocenter | #2840 | C3 |
| Frattini, socle | C23 | C22⋊A4 |
| LCS | G ▷ #2840 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #2840 | C3 | semidirect |
| 11. C22×Q8 | A4 | semidirect |
| 15. #95 | A4 | semidirect |
| 23. C24 | SL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2840 | 3 |
| C2×C22⋊A4 | 4 |
| C22⋊SL(2,𝔽3) | 4 |
| #95⋊C3 | 4 |