Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
78 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #5465 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #2202 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 3 | 2−1+4⋊C3 | — | 4 | 1 | no | maximal |
| 4 | H(3,𝔽4) | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 5 | #454 | — | 3 | 1 | no | — |
| 6 | #355 | — | 3 | 1 | no | — |
| 7 | C22×A4 | — | 8 | 1 | no | — |
| 8 | #41⋊C3 | — | 4 | 1 | no | — |
| 9 | C2×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 10 | C2×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 11 | C2×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 12 | 2−1+4 | A4 | 1 | 1 | yes | characteristic, complemented |
| 13 | AGL(1,ℤ/8) | — | 6 | 1 | no | — |
| 14 | #124 | — | 3 | 1 | no | — |
| 15 | #124 | — | 6 | 1 | no | — |
| 16 | #120 | — | 6 | 1 | no | — |
| 17 | #104 | — | 6 | 1 | no | — |
| 18 | #101 | — | 3 | 1 | no | — |
| 19 | C2×A4 | — | 8 | 1 | no | — |
| 20 | C2×A4 | — | 8 | 2 | no | — |
| 21 | C3×D4 | — | 16 | 1 | no | — |
| 22 | SL(2,𝔽3) | — | 4 | 1 | no | — |
| 23 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 24 | C24 | — | 2 | 1 | no | — |
| 25 | #41 | — | 4 | 1 | no | — |
| 26 | #41 | — | 6 | 1 | no | — |
| 27 | C2×Q8 | — | 2 | 1 | no | — |
| 28 | C2×Q8 | — | 4 | 1 | no | — |
| 29 | C2×Q8 | C2×A4 | 1 | 1 | yes | characteristic |
| 30 | C2×D4 | — | 3 | 1 | no | — |
| 31 | C2×D4 | — | 6 | 1 | no | — |
| 32 | SD16 | — | 12 | 1 | no | — |
| 33 | D8 | — | 12 | 1 | no | — |
| 34 | M16 | — | 6 | 1 | no | — |
| 35 | C22⋊C4 | — | 6 | 2 | no | — |
| 36 | C22⋊C4 | — | 6 | 1 | no | — |
| 37 | C42 | — | 3 | 1 | no | — |
| 38 | C22×C3 | — | 16 | 1 | no | — |
| 39 | C22×C3 | — | 16 | 1 | no | — |
| 40 | A4 | — | 8 | 1 | no | — |
| 41 | C3×C4 | — | 16 | 1 | no | — |
| 42 | C23 | — | 6 | 1 | no | — |
| 43 | C23 | — | 8 | 2 | no | — |
| 44 | C23 | — | 8 | 1 | no | — |
| 45 | Q8 | — | 4 | 1 | no | — |
| 46 | Q8 | — | 6 | 1 | no | — |
| 47 | Q8 | — | 8 | 1 | no | — |
| 48 | D4 | — | 4 | 1 | no | — |
| 49 | D4 | — | 6 | 1 | no | — |
| 50 | D4 | — | 12 | 1 | no | — |
| 51 | D4 | — | 24 | 1 | no | — |
| 52 | C2×C4 | — | 3 | 1 | no | — |
| 53 | C2×C4 | — | 6 | 1 | no | — |
| 54 | C2×C4 | — | 12 | 1 | no | — |
| 55 | C8 | — | 12 | 1 | no | — |
| 56 | C2×C3 | — | 16 | 1 | no | — |
| 57 | C2×C3 | — | 32 | 1 | no | — |
| 58 | C2×C3 | — | 32 | 1 | no | — |
| 59 | C22 | — | 4 | 1 | no | — |
| 60 | C22 | — | 8 | 1 | no | — |
| 61 | C22 | — | 12 | 2 | no | — |
| 62 | C22 | — | 12 | 1 | no | — |
| 63 | C22 | — | 24 | 1 | no | — |
| 64 | C22 | #120⋊C3 | 1 | 1 | yes | characteristic |
| 65 | C4 | — | 4 | 1 | no | — |
| 66 | C4 | — | 6 | 1 | no | — |
| 67 | C4 | — | 12 | 1 | no | — |
| 68 | C3 | — | 16 | 1 | no | Sylow |
| 69 | C2 | — | 2 | 1 | no | — |
| 70 | C2 | — | 8 | 1 | no | — |
| 71 | C2 | — | 24 | 1 | no | — |
| 72 | C2 | B(4,𝔽2)⋊C3 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | B(4,𝔽2)⋊C3 |
| commutator, cosocle, hypocenter | H(3,𝔽4) | C2×C3 |
| Frattini | C2×Q8 | C2×A4 |
| Fitting | #2202 | C3 |
| hypercenter | C22 | #120⋊C3 |
| LCS | G ▷ H(3,𝔽4) | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5465 | C2 | semidirect |
| 2. #2202 | C3 | semidirect |
| 4. H(3,𝔽4) | C2×C3 | semidirect |
| 12. 2−1+4 | A4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5465 | 2 |
| #2202 | 3 |
| 2−1+4⋊C3 | 4 |