Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
83 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #543⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #2032 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 3 | D4×A4 | — | 4 | 1 | no | maximal |
| 4 | #543 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 5 | C2×#99 | — | 3 | 1 | no | — |
| 6 | #342 | — | 3 | 1 | no | — |
| 7 | C22×A4 | — | 4 | 1 | no | — |
| 8 | C22×A4 | — | 4 | 1 | no | — |
| 9 | C2×SL(2,𝔽3) | — | 8 | 2 | no | — |
| 10 | C4×A4 | — | 4 | 1 | no | — |
| 11 | C22×D4 | A4 | 1 | 1 | yes | characteristic |
| 12 | C22⋊Q8 | — | 6 | 2 | no | — |
| 13 | C2×C22⋊C4 | — | 3 | 1 | no | — |
| 14 | C2×C22⋊C4 | — | 3 | 1 | no | — |
| 15 | #99 | — | 6 | 2 | no | — |
| 16 | C2×A4 | — | 4 | 1 | no | — |
| 17 | C2×A4 | — | 8 | 1 | no | — |
| 18 | C2×A4 | — | 8 | 1 | no | — |
| 19 | C3×D4 | — | 16 | 1 | no | — |
| 20 | SL(2,𝔽3) | — | 8 | 2 | no | — |
| 21 | C24 | SL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 22 | C24 | C2×A4 | 1 | 1 | yes | characteristic |
| 23 | C2×Q8 | — | 2 | 2 | no | — |
| 24 | C2×D4 | — | 6 | 1 | no | — |
| 25 | C2×D4 | — | 6 | 1 | no | — |
| 26 | C22×C4 | — | 6 | 1 | no | — |
| 27 | C22×C4 | — | 6 | 1 | no | — |
| 28 | C22×C4 | SL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 29 | C4⋊C4 | — | 6 | 2 | no | — |
| 30 | C22⋊C4 | — | 3 | 2 | no | — |
| 31 | C22⋊C4 | — | 6 | 1 | no | — |
| 32 | C22⋊C4 | — | 6 | 1 | no | — |
| 33 | C22⋊C4 | — | 6 | 1 | no | — |
| 34 | C22×C3 | — | 16 | 1 | no | — |
| 35 | C22×C3 | — | 16 | 1 | no | — |
| 36 | A4 | — | 4 | 1 | no | — |
| 37 | C3×C4 | — | 16 | 1 | no | — |
| 38 | C23 | — | 3 | 1 | no | — |
| 39 | C23 | — | 3 | 1 | no | — |
| 40 | C23 | — | 6 | 1 | no | — |
| 41 | C23 | — | 8 | 1 | no | — |
| 42 | C23 | — | 8 | 1 | no | — |
| 43 | C23 | C2×SL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 44 | Q8 | — | 8 | 2 | no | — |
| 45 | D4 | — | 4 | 1 | no | — |
| 46 | D4 | — | 12 | 1 | no | — |
| 47 | C2×C4 | — | 6 | 1 | no | — |
| 48 | C2×C4 | — | 6 | 2 | no | — |
| 49 | C2×C4 | — | 12 | 1 | no | — |
| 50 | C2×C4 | — | 24 | 1 | no | — |
| 51 | C2×C4 | — | 24 | 1 | no | — |
| 52 | C2×C3 | — | 16 | 1 | no | — |
| 53 | C2×C3 | — | 32 | 1 | no | — |
| 54 | C2×C3 | — | 32 | 1 | no | — |
| 55 | C22 | — | 3 | 1 | no | — |
| 56 | C22 | — | 3 | 1 | no | — |
| 57 | C22 | — | 4 | 1 | no | — |
| 58 | C22 | — | 4 | 1 | no | — |
| 59 | C22 | — | 12 | 1 | no | — |
| 60 | C22 | — | 12 | 1 | no | — |
| 61 | C22 | — | 24 | 1 | no | — |
| 62 | C22 | #120⋊C3 | 1 | 1 | yes | characteristic |
| 63 | C4 | — | 4 | 1 | no | — |
| 64 | C4 | — | 12 | 2 | no | — |
| 65 | C4 | — | 24 | 1 | no | — |
| 66 | C3 | — | 16 | 1 | no | Sylow |
| 67 | C2 | — | 2 | 1 | no | — |
| 68 | C2 | — | 6 | 1 | no | — |
| 69 | C2 | — | 6 | 1 | no | — |
| 70 | C2 | — | 8 | 1 | no | — |
| 71 | C2 | #393⋊C3 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #393⋊C3 |
| commutator, cosocle, hypocenter | #543 | C2×C3 |
| Frattini | C24 | C2×A4 |
| Fitting | #2032 | C3 |
| hypercenter | C22 | #120⋊C3 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #543⋊C3 | C2 | semidirect |
| 2. #2032 | C3 | semidirect |
| 4. #543 | C2×C3 | semidirect |
| 21. C24 | SL(2,𝔽3) | semidirect |
| 28. C22×C4 | SL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #543⋊C3 | 2 |
| #2032 | 3 |
| D4×A4 | 4 |