Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
88 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #5407 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #504⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #2221 | — | 3 | 1 | no | maximal, Sylow |
| 4 | #986 | — | 2 | 2 | no | — |
| 5 | #131⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 6 | D48 | — | 4 | 1 | no | maximal |
| 7 | C2×D16 | — | 3 | 1 | no | — |
| 8 | #504 | D3 | 1 | 1 | yes | characteristic, complemented |
| 9 | #472 | — | 3 | 2 | no | — |
| 10 | #368 | — | 3 | 1 | no | — |
| 11 | #361 | — | 3 | 2 | no | — |
| 12 | #41⋊C3 | D4 | 1 | 1 | yes | characteristic |
| 13 | GL(2,𝔽3) | — | 4 | 2 | no | — |
| 14 | D24 | — | 4 | 2 | no | — |
| 15 | C3×C16 | — | 4 | 1 | no | — |
| 16 | AGL(1,ℤ/8) | — | 6 | 2 | no | — |
| 17 | C2×D8 | — | 3 | 2 | no | — |
| 18 | #131 | C2×D3 | 1 | 1 | yes | characteristic |
| 19 | D16 | — | 6 | 1 | no | — |
| 20 | D16 | — | 6 | 1 | no | — |
| 21 | M32 | — | 3 | 1 | no | — |
| 22 | C2×C16 | — | 3 | 1 | no | — |
| 23 | #108 | — | 3 | 2 | no | — |
| 24 | #100 | — | 6 | 2 | no | — |
| 25 | D12 | — | 8 | 2 | no | — |
| 26 | SL(2,𝔽3) | D8 | 1 | 1 | yes | characteristic |
| 27 | C3×C8 | — | 4 | 1 | no | — |
| 28 | #41 | D12 | 1 | 1 | yes | characteristic |
| 29 | C2×D4 | — | 6 | 2 | no | — |
| 30 | SD16 | — | 12 | 2 | no | — |
| 31 | D8 | — | 6 | 2 | no | — |
| 32 | D8 | — | 6 | 2 | no | — |
| 33 | D8 | — | 12 | 2 | no | — |
| 34 | M16 | — | 3 | 1 | no | — |
| 35 | M16 | — | 6 | 2 | no | — |
| 36 | C2×C8 | — | 3 | 1 | no | — |
| 37 | C16 | — | 3 | 1 | no | — |
| 38 | C16 | S4 | 1 | 1 | yes | characteristic |
| 39 | C2×D3 | — | 16 | 2 | no | — |
| 40 | C3×C4 | — | 4 | 1 | no | — |
| 41 | C23 | — | 12 | 2 | no | — |
| 42 | Q8 | D24 | 1 | 1 | yes | characteristic |
| 43 | D4 | — | 3 | 1 | no | — |
| 44 | D4 | — | 12 | 2 | no | — |
| 45 | D4 | — | 12 | 2 | no | — |
| 46 | C2×C4 | — | 3 | 1 | no | — |
| 47 | C8 | — | 3 | 1 | no | — |
| 48 | C8 | — | 12 | 2 | no | — |
| 49 | C8 | C2×S4 | 1 | 1 | yes | characteristic |
| 50 | C2×C3 | — | 4 | 1 | no | — |
| 51 | D3 | — | 32 | 2 | no | — |
| 52 | C22 | — | 3 | 1 | no | — |
| 53 | C22 | — | 24 | 2 | no | — |
| 54 | C22 | — | 48 | 2 | no | — |
| 55 | C4 | — | 3 | 1 | no | — |
| 56 | C4 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 57 | C3 | — | 4 | 1 | no | Sylow |
| 58 | C2 | — | 6 | 1 | no | — |
| 59 | C2 | — | 48 | 2 | no | — |
| 60 | C2 | #5402 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #5402 |
| commutator, cosocle | #131⋊C3 | C22 |
| Frattini | C8 | C2×S4 |
| Fitting | #504 | D3 |
| hypercenter | C16 | S4 |
| hypocenter | SL(2,𝔽3) | D8 |
| LCS | G ▷ #131⋊C3 ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5407 | C2 | semidirect |
| 2. #504⋊C3 | C2 | semidirect |
| 8. #504 | D3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5407 | 2 |
| #504⋊C3 | 2 |
| #2221 | 3 |
| D48 | 4 |