Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
69 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Q8⋊SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #2205 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 3 | 2−1+4⋊C3 | — | 4 | 1 | no | maximal |
| 4 | Q82 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 5 | #456 | — | 3 | 1 | no | — |
| 6 | #356 | — | 3 | 1 | no | — |
| 7 | #41⋊C3 | — | 4 | 1 | no | — |
| 8 | C2×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 9 | C2×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 10 | C2×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 11 | C2×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 12 | 2−1+4 | A4 | 1 | 1 | yes | characteristic |
| 13 | #137 | — | 6 | 1 | no | — |
| 14 | C4⋊Q8 | — | 3 | 1 | no | — |
| 15 | C4⋊Q8 | — | 6 | 1 | no | — |
| 16 | C4×Q8 | — | 6 | 1 | no | — |
| 17 | #104 | — | 6 | 1 | no | — |
| 18 | #101 | — | 3 | 1 | no | — |
| 19 | C3×D4 | — | 16 | 1 | no | — |
| 20 | SL(2,𝔽3) | — | 4 | 1 | no | — |
| 21 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 22 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 23 | #41 | — | 4 | 1 | no | — |
| 24 | #41 | — | 6 | 1 | no | — |
| 25 | C2×Q8 | — | 2 | 1 | no | — |
| 26 | C2×Q8 | — | 2 | 1 | no | — |
| 27 | C2×Q8 | — | 3 | 1 | no | — |
| 28 | C2×Q8 | — | 4 | 1 | no | — |
| 29 | C2×Q8 | C2×A4 | 1 | 1 | yes | characteristic |
| 30 | Q16 | — | 12 | 1 | no | — |
| 31 | SD16 | — | 12 | 1 | no | — |
| 32 | M16 | — | 6 | 1 | no | — |
| 33 | C4⋊C4 | — | 6 | 2 | no | — |
| 34 | C4⋊C4 | — | 6 | 1 | no | — |
| 35 | C42 | — | 3 | 1 | no | — |
| 36 | C42 | — | 6 | 1 | no | — |
| 37 | C22×C3 | — | 16 | 1 | no | — |
| 38 | C22×C3 | — | 16 | 1 | no | — |
| 39 | C3×C4 | — | 16 | 1 | no | — |
| 40 | Q8 | — | 2 | 1 | no | — |
| 41 | Q8 | — | 4 | 1 | no | — |
| 42 | Q8 | — | 6 | 1 | no | — |
| 43 | Q8 | — | 6 | 1 | no | — |
| 44 | Q8 | — | 8 | 1 | no | — |
| 45 | Q8 | — | 12 | 1 | no | — |
| 46 | D4 | — | 4 | 1 | no | — |
| 47 | D4 | — | 6 | 1 | no | — |
| 48 | C2×C4 | — | 3 | 1 | no | — |
| 49 | C2×C4 | — | 6 | 1 | no | — |
| 50 | C2×C4 | — | 6 | 1 | no | — |
| 51 | C2×C4 | — | 12 | 1 | no | — |
| 52 | C8 | — | 12 | 1 | no | — |
| 53 | C2×C3 | — | 16 | 1 | no | — |
| 54 | C2×C3 | — | 32 | 1 | no | — |
| 55 | C2×C3 | — | 32 | 1 | no | — |
| 56 | C22 | — | 4 | 1 | no | — |
| 57 | C22 | #120⋊C3 | 1 | 1 | yes | characteristic |
| 58 | C4 | — | 4 | 1 | no | — |
| 59 | C4 | — | 6 | 1 | no | — |
| 60 | C4 | — | 6 | 1 | no | — |
| 61 | C4 | — | 6 | 1 | no | — |
| 62 | C4 | — | 12 | 1 | no | — |
| 63 | C3 | — | 16 | 1 | no | Sylow |
| 64 | C2 | — | 2 | 1 | no | — |
| 65 | C2 | — | 8 | 1 | no | — |
| 66 | C2 | B(4,𝔽2)⋊C3 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | B(4,𝔽2)⋊C3 |
| commutator, cosocle, hypocenter | Q82 | C2×C3 |
| Frattini | C2×Q8 | C2×A4 |
| Fitting | #2205 | C3 |
| hypercenter | C22 | #120⋊C3 |
| LCS | G ▷ Q82 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8⋊SL(2,𝔽3) | C2 | semidirect |
| 2. #2205 | C3 | semidirect |
| 4. Q82 | C2×C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8⋊SL(2,𝔽3) | 2 |
| #2205 | 3 |
| 2−1+4⋊C3 | 4 |