Finite Group DatabasebetaFinite Group DatabasebetaOrder 1944.
225 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 225 conjugacy classes of subgroups — too many to draw. Only its 26 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 | A4×#732 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | A4⋊#272 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C9×C3⋊S4 | — | 3 | 1 | no | maximal |
| 4 | #10619740 | — | 3 | 2 | no | maximal |
| 5 | #10619649 | — | 3 | 1 | no | maximal |
| 6 | C3⋊#4178 | — | 4 | 1 | no | maximal |
| 7 | A4×H(3,𝔽3) | — | 2 | 1 | no | — |
| 8 | A4×H(3,𝔽3) | C2×C3 | 1 | 1 | yes | characteristic |
| 9 | C3×C9×A4 | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | C22×#732 | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | C2×#4178 | — | 3 | 1 | no | — |
| 12 | #66362 | — | 2 | 1 | no | — |
| 13 | #66362 | — | 2 | 2 | no | — |
| 14 | #66358 | D3 | 1 | 2 | yes | complemented |
| 15 | #66325 | — | 3 | 1 | no | — |
| 16 | C3×#732 | — | 4 | 1 | no | Sylow |
| 17 | C3×C3⋊S4 | — | 3 | 1 | no | — |
| 18 | C3×C3⋊S4 | — | 9 | 1 | no | — |
| 19 | #6258 | — | 3 | 2 | no | — |
| 20 | #6258 | — | 9 | 2 | no | — |
| 21 | C9×S4 | — | 9 | 2 | no | — |
| 22 | C9×S4 | — | 9 | 1 | no | — |
| 23 | #6235 | — | 3 | 1 | no | — |
| 24 | C9×#67 | — | 9 | 1 | no | — |
| 25 | C3⋊#272 | — | 4 | 1 | no | — |
| 26 | C9×C3⋊D3 | — | 12 | 1 | no | — |
| 27 | C2×#732 | — | 3 | 1 | no | — |
| 28 | #4178 | — | 6 | 1 | no | — |
| 29 | #4178 | — | 12 | 2 | no | — |
| 30 | C32×A4 | — | 3 | 1 | no | — |
| 31 | C32×A4 | — | 6 | 1 | no | — |
| 32 | C32×A4 | C3×D3 | 1 | 1 | yes | characteristic |
| 33 | C22×H(3,𝔽3) | — | 2 | 1 | no | — |
| 34 | C22×H(3,𝔽3) | C3×D3 | 1 | 1 | yes | characteristic |
| 35 | C22×C3×C9 | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 36 | C2×#272 | — | 3 | 1 | no | — |
| 37 | C2×D3×C9 | — | 9 | 1 | no | — |
| 38 | C22⋊H(3,𝔽3) | — | 2 | 2 | no | — |
| 39 | C22⋊H(3,𝔽3) | — | 3 | 2 | no | — |
| 40 | C22⋊H(3,𝔽3) | — | 6 | 2 | no | — |
| 41 | C22⋊H(3,𝔽3) | C3×D3 | 1 | 2 | yes | — |
| 42 | C3×C22⋊C9 | — | 2 | 1 | no | — |
| 43 | C9×A4 | — | 3 | 1 | no | — |
| 44 | C9×A4 | — | 3 | 2 | no | — |
| 45 | #1084 | — | 3 | 1 | no | — |
| 46 | C9×Dic3 | — | 9 | 1 | no | — |
| 47 | C3×H(3,𝔽3) | — | 4 | 1 | no | — |
| 48 | C3×H(3,𝔽3) | — | 8 | 1 | no | — |
| 49 | C32×C9 | — | 4 | 1 | no | — |
| 50 | #732 | — | 4 | 2 | no | — |
| 51 | #732 | — | 8 | 2 | no | — |
| 52 | #732 | — | 8 | 1 | no | — |
| 53 | #732 | S4 | 1 | 1 | yes | characteristic, complemented |
| 54 | C3⋊S4 | — | 3 | 1 | no | — |
| 55 | C3⋊S4 | — | 9 | 1 | no | — |
| 56 | C3×S4 | — | 9 | 2 | no | — |
| 57 | C3×S4 | — | 9 | 1 | no | — |
| 58 | C3×S4 | — | 27 | 2 | no | — |
| 59 | C3×#67 | — | 9 | 1 | no | — |
| 60 | C3×#67 | — | 27 | 1 | no | — |
| 61 | D4×C9 | — | 27 | 1 | no | — |
| 62 | C3×C3⋊D3 | — | 12 | 1 | no | — |
| 63 | C3×C3⋊D3 | — | 36 | 1 | no | — |
| 64 | C2×H(3,𝔽3) | — | 3 | 1 | no | — |
| 65 | C2×H(3,𝔽3) | — | 6 | 1 | no | — |
| 66 | C2×C3×C9 | — | 3 | 1 | no | — |
| 67 | #272 | — | 6 | 1 | no | — |
| 68 | #272 | — | 12 | 2 | no | — |
| 69 | #272 | — | 36 | 2 | no | — |
| 70 | D3×C9 | — | 18 | 1 | no | — |
| 71 | D3×C9 | — | 36 | 2 | no | — |
| 72 | D3×C9 | — | 36 | 1 | no | — |
| 73 | C22×C32 | — | 3 | 1 | no | — |
| 74 | C22×C32 | — | 6 | 1 | no | — |
| 75 | C22×C32 | C3×C3⋊D3 | 1 | 1 | yes | characteristic |
| 76 | C2×C3×D3 | — | 9 | 1 | no | — |
| 77 | C2×C3×D3 | — | 27 | 1 | no | — |
| 78 | C3×A4 | — | 3 | 1 | no | — |
| 79 | C3×A4 | — | 3 | 2 | no | — |
| 80 | C3×A4 | — | 6 | 1 | no | — |
| 81 | C3×A4 | — | 6 | 2 | no | — |
| 82 | C3×A4 | — | 9 | 1 | no | — |
| 83 | C3×A4 | — | 18 | 1 | no | — |
| 84 | C3×A4 | — | 18 | 2 | no | — |
| 85 | C3×A4 | — | 18 | 1 | no | — |
| 86 | C3×A4 | (C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 87 | C3×A4 | (C3⋊D3)⋊C3 | 1 | 2 | yes | — |
| 88 | C3×Dic3 | — | 9 | 1 | no | — |
| 89 | C3×Dic3 | — | 27 | 1 | no | — |
| 90 | C22×C9 | — | 3 | 1 | no | — |
| 91 | C22×C9 | — | 27 | 1 | no | — |
| 92 | C22⋊C9 | — | 6 | 1 | no | — |
| 93 | C4×C9 | — | 27 | 1 | no | — |
| 94 | C33 | — | 4 | 1 | no | — |
| 95 | C33 | — | 12 | 1 | no | — |
| 96 | C33 | — | 24 | 1 | no | — |
| 97 | H(3,𝔽3) | — | 2 | 1 | no | — |
| 98 | H(3,𝔽3) | — | 4 | 2 | no | — |
| 99 | H(3,𝔽3) | — | 8 | 2 | no | — |
| 100 | H(3,𝔽3) | — | 12 | 2 | no | — |
| 101 | H(3,𝔽3) | — | 24 | 2 | no | — |
| 102 | H(3,𝔽3) | C3×S4 | 1 | 1 | yes | characteristic |
| 103 | C3×C9 | — | 8 | 1 | no | — |
| 104 | C3×C9 | — | 12 | 1 | no | — |
| 105 | C3×C9 | — | 12 | 2 | no | — |
| 106 | C3×C9 | C3⋊S4 | 1 | 1 | yes | characteristic, complemented |
| 107 | S4 | — | 9 | 2 | no | — |
| 108 | S4 | — | 9 | 1 | no | — |
| 109 | S4 | — | 27 | 2 | no | — |
| 110 | C3×D4 | — | 27 | 1 | no | — |
| 111 | #67 | — | 9 | 1 | no | — |
| 112 | #67 | — | 27 | 1 | no | — |
| 113 | C2×C32 | — | 3 | 1 | no | — |
| 114 | C2×C32 | — | 9 | 1 | no | — |
| 115 | C2×C32 | — | 18 | 1 | no | — |
| 116 | C3⋊D3 | — | 12 | 1 | no | — |
| 117 | C3⋊D3 | — | 36 | 1 | no | — |
| 118 | C3×D3 | — | 18 | 1 | no | — |
| 119 | C3×D3 | — | 36 | 2 | no | — |
| 120 | C3×D3 | — | 36 | 1 | no | — |
| 121 | C3×D3 | — | 54 | 1 | no | — |
| 122 | C3×D3 | — | 108 | 2 | no | — |
| 123 | C2×C9 | — | 9 | 1 | no | — |
| 124 | C2×C9 | — | 54 | 1 | no | — |
| 125 | C22×C3 | — | 3 | 1 | no | — |
| 126 | C22×C3 | — | 9 | 1 | no | — |
| 127 | C22×C3 | — | 18 | 1 | no | — |
| 128 | C22×C3 | — | 27 | 1 | no | — |
| 129 | C22×C3 | C3⋊((C3⋊D3)⋊C3) | 1 | 1 | yes | characteristic |
| 130 | C2×D3 | — | 9 | 1 | no | — |
| 131 | C2×D3 | — | 27 | 1 | no | — |
| 132 | A4 | — | 2 | 1 | no | — |
| 133 | A4 | — | 3 | 2 | no | — |
| 134 | A4 | — | 9 | 2 | no | — |
| 135 | A4 | — | 18 | 2 | no | — |
| 136 | A4 | #4178 | 1 | 1 | yes | characteristic, complemented |
| 137 | C3×C4 | — | 27 | 1 | no | — |
| 138 | Dic3 | — | 9 | 1 | no | — |
| 139 | Dic3 | — | 27 | 1 | no | — |
| 140 | C32 | — | 3 | 1 | no | — |
| 141 | C32 | — | 4 | 1 | no | — |
| 142 | C32 | — | 4 | 2 | no | — |
| 143 | C32 | — | 6 | 1 | no | — |
| 144 | C32 | — | 12 | 1 | no | — |
| 145 | C32 | — | 12 | 2 | no | — |
| 146 | C32 | — | 24 | 1 | no | — |
| 147 | C32 | — | 24 | 2 | no | — |
| 148 | C32 | — | 36 | 1 | no | — |
| 149 | C32 | — | 72 | 1 | no | — |
| 150 | C32 | — | 72 | 1 | no | — |
| 151 | C32 | — | 72 | 2 | no | — |
| 152 | C32 | C3×C3⋊S4 | 1 | 1 | yes | characteristic |
| 153 | C9 | — | 3 | 1 | no | — |
| 154 | C9 | — | 24 | 1 | no | — |
| 155 | D4 | — | 27 | 1 | no | Sylow |
| 156 | C2×C3 | — | 3 | 1 | no | — |
| 157 | C2×C3 | — | 9 | 1 | no | — |
| 158 | C2×C3 | — | 27 | 1 | no | — |
| 159 | C2×C3 | — | 54 | 1 | no | — |
| 160 | C2×C3 | — | 54 | 1 | no | — |
| 161 | D3 | — | 18 | 1 | no | — |
| 162 | D3 | — | 36 | 1 | no | — |
| 163 | D3 | — | 36 | 2 | no | — |
| 164 | D3 | — | 54 | 1 | no | — |
| 165 | D3 | — | 108 | 2 | no | — |
| 166 | C22 | — | 27 | 1 | no | — |
| 167 | C22 | C3⋊#4178 | 1 | 1 | yes | characteristic, complemented |
| 168 | C4 | — | 27 | 1 | no | — |
| 169 | C3 | — | 3 | 1 | no | — |
| 170 | C3 | — | 4 | 1 | no | — |
| 171 | C3 | — | 8 | 1 | no | — |
| 172 | C3 | — | 9 | 1 | no | — |
| 173 | C3 | — | 12 | 2 | no | — |
| 174 | C3 | — | 18 | 1 | no | — |
| 175 | C3 | — | 36 | 2 | no | — |
| 176 | C3 | — | 72 | 2 | no | — |
| 177 | C3 | A4⋊((C3⋊D3)⋊C3) | 1 | 1 | yes | characteristic |
| 178 | C2 | — | 3 | 1 | no | — |
| 179 | C2 | — | 54 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | A4⋊((C3⋊D3)⋊C3) |
| commutator, cosocle, hypocenter | A4×H(3,𝔽3) | C2×C3 |
| Frattini | C32 | C3×C3⋊S4 |
| Fitting | C22×#732 | D3 |
| socle | C22×C3 | C3⋊((C3⋊D3)⋊C3) |
| LCS | G ▷ A4×H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. A4×#732 | C2 | semidirect |
| 2. A4⋊#272 | C3 | semidirect |
| 9. C3×C9×A4 | D3 | semidirect |
| 10. C22×#732 | D3 | semidirect |
| 14. #66358 | D3 | semidirect |
| 35. C22×C3×C9 | C3⋊D3 | semidirect |
| 53. #732 | S4 | semidirect |
| 106. C3×C9 | C3⋊S4 | semidirect |
| 136. A4 | #4178 | semidirect |
| 167. C22 | C3⋊#4178 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×#732 | 2 |
| A4⋊#272 | 3 |
| C9×C3⋊S4 | 3 |
| #10619740 | 3 |
| #10619649 | 3 |
| C3⋊#4178 | 4 |