Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
149 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 149 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C3⋊#994 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #64265 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×GU(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #5428 | — | 3 | 1 | no | maximal |
| 5 | #4825 | — | 3 | 1 | no | maximal |
| 6 | C3×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 7 | C3⋊(#41⋊C3) | — | 2 | 1 | no | — |
| 8 | C3⋊#226 | — | 2 | 1 | no | — |
| 9 | #3719 | — | 4 | 1 | no | maximal |
| 10 | #1009 | D3 | 1 | 1 | yes | characteristic |
| 11 | #994 | — | 3 | 1 | no | — |
| 12 | #984 | — | 3 | 1 | no | — |
| 13 | #984 | — | 6 | 1 | no | — |
| 14 | #951 | — | 3 | 1 | no | — |
| 15 | #909 | — | 3 | 1 | no | — |
| 16 | #875 | — | 3 | 1 | no | — |
| 17 | GU(2,𝔽3) | D3 | 1 | 1 | yes | characteristic, complemented |
| 18 | C3×#104 | — | 3 | 1 | no | — |
| 19 | #823 | — | 3 | 1 | no | — |
| 20 | #805 | — | 3 | 1 | no | — |
| 21 | #634 | — | 4 | 1 | no | — |
| 22 | C3×Dic6 | — | 4 | 1 | no | — |
| 23 | C3×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 24 | C3×C3⋊C8 | — | 4 | 1 | no | — |
| 25 | #455 | — | 9 | 1 | no | Sylow |
| 26 | C3×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 27 | #239 | — | 2 | 1 | no | — |
| 28 | D3×D4 | — | 6 | 1 | no | — |
| 29 | #235 | — | 3 | 1 | no | — |
| 30 | C2×D12 | — | 3 | 1 | no | — |
| 31 | C2×Dic6 | — | 3 | 1 | no | — |
| 32 | #41⋊C3 | — | 2 | 1 | no | — |
| 33 | #41⋊C3 | — | 6 | 1 | no | — |
| 34 | #41⋊C3 | C2×D3 | 1 | 1 | yes | characteristic |
| 35 | #226 | — | 6 | 2 | no | — |
| 36 | #226 | — | 6 | 1 | no | — |
| 37 | C3×M16 | — | 3 | 1 | no | — |
| 38 | C3×C42 | — | 3 | 1 | no | — |
| 39 | #216 | — | 6 | 1 | no | — |
| 40 | #216 | — | 12 | 1 | no | — |
| 41 | #214 | — | 6 | 1 | no | — |
| 42 | (C2×D3)⋊C4 | — | 6 | 2 | no | — |
| 43 | #208 | — | 3 | 1 | no | — |
| 44 | Dic12 | — | 6 | 1 | no | — |
| 45 | #204 | — | 6 | 1 | no | — |
| 46 | C32×C4 | — | 4 | 1 | no | — |
| 47 | C3⋊Dic3 | — | 8 | 1 | no | — |
| 48 | C3×Dic3 | — | 8 | 1 | no | — |
| 49 | H(5,𝔽2) | — | 3 | 1 | no | — |
| 50 | #137 | — | 9 | 1 | no | — |
| 51 | #137 | — | 9 | 1 | no | — |
| 52 | #124 | — | 9 | 1 | no | — |
| 53 | #104 | — | 3 | 1 | no | — |
| 54 | #104 | — | 9 | 1 | no | — |
| 55 | #100 | — | 9 | 1 | no | — |
| 56 | C22×D3 | — | 3 | 2 | no | — |
| 57 | C3×Q8 | — | 12 | 1 | no | — |
| 58 | C3×Q8 | #67 | 1 | 1 | yes | characteristic |
| 59 | C3×D4 | — | 3 | 1 | no | — |
| 60 | C2×C3×C4 | — | 3 | 1 | no | — |
| 61 | C2×C3×C4 | — | 6 | 1 | no | — |
| 62 | #67 | — | 6 | 1 | no | — |
| 63 | C2×Dic3 | — | 6 | 1 | no | — |
| 64 | D12 | — | 3 | 1 | no | — |
| 65 | D12 | — | 6 | 1 | no | — |
| 66 | C4×D3 | — | 6 | 1 | no | — |
| 67 | Dic6 | — | 6 | 1 | no | — |
| 68 | Dic6 | — | 6 | 1 | no | — |
| 69 | Dic6 | — | 12 | 1 | no | — |
| 70 | Dic6 | — | 24 | 1 | no | — |
| 71 | Dic6 | S4 | 1 | 1 | yes | characteristic |
| 72 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 73 | SL(2,𝔽3) | D12 | 1 | 1 | yes | characteristic |
| 74 | C3×C8 | — | 6 | 1 | no | — |
| 75 | C3⋊C8 | — | 4 | 1 | no | — |
| 76 | C3⋊C8 | — | 6 | 1 | no | — |
| 77 | C2×C32 | — | 4 | 1 | no | — |
| 78 | #41 | — | 6 | 1 | no | — |
| 79 | #41 | — | 9 | 1 | no | — |
| 80 | #41 | D32 | 1 | 1 | yes | characteristic |
| 81 | C2×Q8 | — | 9 | 1 | no | — |
| 82 | C2×D4 | — | 9 | 1 | no | — |
| 83 | C2×D4 | — | 18 | 1 | no | — |
| 84 | Q16 | — | 18 | 1 | no | — |
| 85 | Q16 | — | 18 | 1 | no | — |
| 86 | SD16 | — | 18 | 1 | no | — |
| 87 | SD16 | — | 18 | 1 | no | — |
| 88 | M16 | — | 3 | 1 | no | — |
| 89 | M16 | — | 9 | 1 | no | — |
| 90 | C22⋊C4 | — | 18 | 2 | no | — |
| 91 | C42 | — | 3 | 1 | no | — |
| 92 | C22×C3 | — | 3 | 1 | no | — |
| 93 | C2×D3 | — | 6 | 1 | no | — |
| 94 | C2×D3 | — | 12 | 2 | no | — |
| 95 | C3×C4 | — | 3 | 1 | no | — |
| 96 | C3×C4 | — | 4 | 1 | no | — |
| 97 | C3×C4 | — | 8 | 1 | no | — |
| 98 | C3×C4 | — | 12 | 1 | no | — |
| 99 | C3×C4 | — | 24 | 1 | no | — |
| 100 | C3×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 101 | Dic3 | — | 2 | 1 | no | — |
| 102 | Dic3 | — | 12 | 1 | no | — |
| 103 | Dic3 | — | 24 | 1 | no | — |
| 104 | Dic3 | — | 24 | 2 | no | — |
| 105 | C32 | — | 4 | 1 | no | Sylow |
| 106 | C23 | — | 9 | 2 | no | — |
| 107 | Q8 | — | 3 | 1 | no | — |
| 108 | Q8 | — | 18 | 1 | no | — |
| 109 | Q8 | — | 18 | 1 | no | — |
| 110 | Q8 | #626 | 1 | 1 | yes | characteristic |
| 111 | D4 | — | 3 | 1 | no | — |
| 112 | D4 | — | 9 | 1 | no | — |
| 113 | D4 | — | 18 | 1 | no | — |
| 114 | D4 | — | 18 | 1 | no | — |
| 115 | C2×C4 | — | 3 | 1 | no | — |
| 116 | C2×C4 | — | 6 | 1 | no | — |
| 117 | C2×C4 | — | 18 | 1 | no | — |
| 118 | C2×C4 | — | 18 | 1 | no | — |
| 119 | C8 | — | 6 | 1 | no | — |
| 120 | C8 | — | 18 | 1 | no | — |
| 121 | C2×C3 | — | 4 | 1 | no | — |
| 122 | C2×C3 | — | 6 | 1 | no | — |
| 123 | C2×C3 | — | 8 | 1 | no | — |
| 124 | C2×C3 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 125 | D3 | — | 12 | 1 | no | — |
| 126 | C22 | — | 3 | 1 | no | — |
| 127 | C22 | — | 18 | 1 | no | — |
| 128 | C22 | — | 36 | 2 | no | — |
| 129 | C4 | — | 3 | 1 | no | — |
| 130 | C4 | — | 6 | 1 | no | — |
| 131 | C4 | — | 12 | 1 | no | — |
| 132 | C4 | — | 36 | 1 | no | — |
| 133 | C4 | D3×S4 | 1 | 1 | yes | characteristic |
| 134 | C3 | — | 4 | 1 | no | — |
| 135 | C3 | — | 8 | 1 | no | — |
| 136 | C3 | #5428 | 1 | 1 | yes | characteristic, complemented |
| 137 | C2 | — | 6 | 1 | no | — |
| 138 | C2 | — | 36 | 1 | no | — |
| 139 | C2 | #64210 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #64265 | C2 | semidirect |
| 3. C3×GU(2,𝔽3) | C2 | semidirect |
| 17. GU(2,𝔽3) | D3 | semidirect |
| 136. C3 | #5428 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3⋊#994 | 2 |
| #64265 | 2 |
| C3×GU(2,𝔽3) | 2 |
| #5428 | 3 |
| #4825 | 3 |
| #3719 | 4 |