Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
100 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Ω+(4,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #5924 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #577⋊C3 | — | 3 | 1 | no | maximal |
| 4 | D3×SL(2,𝔽3) | — | 4 | 1 | no | maximal |
| 5 | C3×GL(2,𝔽3) | — | 4 | 1 | no | maximal |
| 6 | #997 | — | 2 | 1 | no | — |
| 7 | 2−1+4⋊C3 | — | 3 | 1 | no | — |
| 8 | #994 | D3 | 1 | 1 | yes | characteristic, complemented |
| 9 | #994 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 10 | #985 | — | 3 | 1 | no | — |
| 11 | #984 | — | 3 | 1 | no | — |
| 12 | #131⋊C3 | — | 3 | 1 | no | — |
| 13 | C3×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 14 | C3×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 15 | #577 | — | 3 | 1 | no | Sylow |
| 16 | D3×Q8 | — | 4 | 1 | no | — |
| 17 | #41⋊C3 | — | 3 | 1 | no | — |
| 18 | #41⋊C3 | — | 3 | 1 | no | — |
| 19 | C2×SL(2,𝔽3) | — | 6 | 1 | no | — |
| 20 | GL(2,𝔽3) | A4 | 1 | 1 | yes | characteristic, complemented |
| 21 | #226 | — | 3 | 1 | no | — |
| 22 | C3×SD16 | — | 12 | 1 | no | — |
| 23 | C2×C3×D3 | — | 16 | 1 | no | — |
| 24 | 2−1+4 | — | 3 | 1 | no | — |
| 25 | H(5,𝔽2) | C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 26 | #137 | — | 9 | 1 | no | — |
| 27 | AGL(1,ℤ/8) | — | 9 | 1 | no | — |
| 28 | #135 | — | 9 | 1 | no | — |
| 29 | C2×SD16 | — | 9 | 1 | no | — |
| 30 | #131 | — | 3 | 1 | no | — |
| 31 | C2×A4 | — | 24 | 1 | no | — |
| 32 | C3×Q8 | — | 4 | 1 | no | — |
| 33 | C3×Q8 | — | 4 | 1 | no | — |
| 34 | C3×D4 | — | 12 | 1 | no | — |
| 35 | C4×D3 | — | 12 | 1 | no | — |
| 36 | Dic6 | — | 12 | 1 | no | — |
| 37 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 38 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 39 | SL(2,𝔽3) | S4 | 1 | 1 | yes | characteristic |
| 40 | SL(2,𝔽3) | C2×A4 | 1 | 1 | yes | characteristic |
| 41 | C3×C8 | — | 12 | 1 | no | — |
| 42 | C2×C32 | — | 16 | 1 | no | — |
| 43 | C3×D3 | — | 16 | 2 | no | — |
| 44 | #41 | — | 3 | 1 | no | — |
| 45 | #41 | — | 3 | 1 | no | — |
| 46 | #41 | — | 9 | 1 | no | — |
| 47 | #41 | — | 18 | 1 | no | — |
| 48 | C2×Q8 | — | 6 | 1 | no | — |
| 49 | C2×Q8 | — | 9 | 1 | no | — |
| 50 | C2×D4 | — | 9 | 1 | no | — |
| 51 | Q16 | — | 9 | 1 | no | — |
| 52 | SD16 | — | 3 | 1 | no | — |
| 53 | SD16 | — | 9 | 1 | no | — |
| 54 | SD16 | — | 9 | 2 | no | — |
| 55 | D8 | — | 9 | 1 | no | — |
| 56 | M16 | — | 9 | 1 | no | — |
| 57 | C2×C8 | — | 9 | 1 | no | — |
| 58 | C22×C3 | — | 24 | 1 | no | — |
| 59 | C2×D3 | — | 4 | 1 | no | — |
| 60 | A4 | — | 24 | 1 | no | — |
| 61 | C3×C4 | — | 12 | 1 | no | — |
| 62 | C3×C4 | — | 12 | 1 | no | — |
| 63 | Dic3 | — | 12 | 1 | no | — |
| 64 | C32 | — | 16 | 1 | no | Sylow |
| 65 | C23 | — | 6 | 1 | no | — |
| 66 | Q8 | — | 9 | 1 | no | — |
| 67 | Q8 | — | 18 | 1 | no | — |
| 68 | Q8 | C3×S4 | 1 | 1 | yes | characteristic |
| 69 | Q8 | D3×A4 | 1 | 1 | yes | characteristic |
| 70 | D4 | — | 3 | 1 | no | — |
| 71 | D4 | — | 9 | 1 | no | — |
| 72 | D4 | — | 9 | 1 | no | — |
| 73 | D4 | — | 18 | 1 | no | — |
| 74 | C2×C4 | — | 9 | 1 | no | — |
| 75 | C2×C4 | — | 18 | 1 | no | — |
| 76 | C2×C4 | — | 18 | 1 | no | — |
| 77 | C8 | — | 3 | 1 | no | — |
| 78 | C8 | — | 9 | 1 | no | — |
| 79 | C2×C3 | — | 4 | 1 | no | — |
| 80 | C2×C3 | — | 4 | 1 | no | — |
| 81 | C2×C3 | — | 32 | 1 | no | — |
| 82 | C2×C3 | — | 48 | 1 | no | — |
| 83 | D3 | — | 4 | 2 | no | — |
| 84 | C22 | — | 6 | 1 | no | — |
| 85 | C22 | — | 9 | 1 | no | — |
| 86 | C22 | — | 24 | 1 | no | — |
| 87 | C4 | — | 3 | 1 | no | — |
| 88 | C4 | — | 3 | 1 | no | — |
| 89 | C4 | — | 18 | 1 | no | — |
| 90 | C3 | — | 4 | 1 | no | — |
| 91 | C3 | — | 4 | 1 | no | — |
| 92 | C3 | — | 32 | 1 | no | — |
| 93 | C2 | — | 12 | 1 | no | — |
| 94 | C2 | — | 18 | 1 | no | — |
| 95 | C2 | A4×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2 | A4×S4 |
| commutator, cosocle, hypocenter | #994 | C2×C3 |
| Fitting | H(5,𝔽2) | C3×D3 |
| LCS | G ▷ #994 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Ω+(4,𝔽3) | C2 | semidirect |
| 2. #5924 | C3 | semidirect |
| 8. #994 | D3 | semidirect |
| 9. #994 | C2×C3 | semidirect |
| 20. GL(2,𝔽3) | A4 | semidirect |
| 25. H(5,𝔽2) | C3×D3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Ω+(4,𝔽3) | 2 |
| #5924 | 3 |
| #577⋊C3 | 3 |
| D3×SL(2,𝔽3) | 4 |
| C3×GL(2,𝔽3) | 4 |