Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
641 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 641 conjugacy classes of subgroups — too many to draw. Only its 114 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 | #10592772 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×#5073 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #10590352 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10590350 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C3⋊C82 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #71276 | — | 3 | 2 | no | maximal |
| 7 | C32×#127 | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×#3753 | — | 2 | 2 | no | — |
| 9 | #64083 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C3×#213 | — | 2 | 4 | no | — |
| 11 | C3×#874 | C22 | 1 | 2 | yes | — |
| 12 | C2×#3714 | — | 2 | 4 | no | — |
| 13 | C2×#3713 | — | 2 | 2 | no | — |
| 14 | C4×C3⋊(C3⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 15 | C3×C4×C3⋊C8 | C22 | 1 | 2 | yes | — |
| 16 | C3×#493 | — | 3 | 2 | no | — |
| 17 | #5073 | — | 3 | 2 | no | — |
| 18 | #5073 | — | 3 | 2 | no | — |
| 19 | #5073 | — | 3 | 2 | no | — |
| 20 | #5073 | — | 6 | 1 | no | — |
| 21 | #5073 | D3 | 1 | 2 | yes | complemented |
| 22 | #4695 | — | 3 | 2 | no | — |
| 23 | C8×C3⋊C8 | — | 3 | 2 | no | — |
| 24 | C2×C32×D4 | — | 2 | 1 | no | — |
| 25 | C2×C32×D4 | — | 2 | 2 | no | — |
| 26 | C2×C4⋊(C3⋊D3) | — | 2 | 1 | no | — |
| 27 | C2×C4⋊(C3⋊D3) | — | 2 | 2 | no | — |
| 28 | C2×C3×D12 | — | 2 | 2 | no | — |
| 29 | C2×C3×D12 | — | 2 | 2 | no | — |
| 30 | C2×C3×D12 | — | 2 | 2 | no | — |
| 31 | C32×C42 | C23 | 1 | 1 | yes | characteristic |
| 32 | #3753 | — | 4 | 4 | no | — |
| 33 | C2×C3⋊(C3⋊C8) | D4 | 1 | 2 | yes | — |
| 34 | C3×#213 | — | 4 | 8 | no | — |
| 35 | C2×C3×C3⋊C8 | D4 | 1 | 4 | yes | — |
| 36 | #3714 | — | 4 | 8 | no | — |
| 37 | #3713 | — | 4 | 4 | no | — |
| 38 | #1712 | — | 9 | 1 | no | Sylow |
| 39 | C2×C3×D8 | — | 6 | 4 | no | — |
| 40 | C3×#127 | — | 2 | 1 | no | — |
| 41 | C3×#127 | — | 3 | 2 | no | — |
| 42 | C3×#127 | C2×D3 | 1 | 2 | yes | — |
| 43 | C2×#213 | — | 2 | 4 | no | — |
| 44 | C2×#213 | — | 6 | 4 | no | — |
| 45 | C2×#213 | — | 6 | 4 | no | — |
| 46 | C2×#213 | — | 6 | 4 | no | — |
| 47 | C2×#213 | — | 6 | 4 | no | — |
| 48 | C2×D24 | — | 6 | 4 | no | — |
| 49 | #874 | — | 3 | 2 | no | — |
| 50 | #874 | — | 6 | 1 | no | — |
| 51 | #874 | C2×D3 | 1 | 2 | yes | — |
| 52 | C3×C4×C8 | — | 3 | 2 | no | — |
| 53 | C4×C3⋊C8 | — | 3 | 2 | no | — |
| 54 | C4×C3⋊C8 | — | 6 | 1 | no | — |
| 55 | C4×C3⋊C8 | C2×D3 | 1 | 2 | yes | — |
| 56 | C23×C32 | — | 4 | 1 | no | — |
| 57 | C22×C3⋊D3 | — | 4 | 1 | no | — |
| 58 | C22×C3×D3 | — | 4 | 2 | no | — |
| 59 | C32×D4 | — | 4 | 2 | no | — |
| 60 | C32×D4 | — | 4 | 4 | no | — |
| 61 | C2×C32×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 62 | C2×C32×C4 | C2×D4 | 1 | 2 | yes | — |
| 63 | C4⋊(C3⋊D3) | — | 4 | 2 | no | — |
| 64 | C4⋊(C3⋊D3) | — | 4 | 4 | no | — |
| 65 | C3×D12 | — | 4 | 4 | no | — |
| 66 | C3×D12 | — | 4 | 4 | no | — |
| 67 | C3×D12 | — | 4 | 4 | no | — |
| 68 | C3⋊(C3⋊C8) | D8 | 1 | 4 | yes | complemented |
| 69 | C3×C3⋊C8 | D8 | 1 | 8 | yes | complemented |
| 70 | #493 | — | 3 | 2 | no | — |
| 71 | #493 | — | 9 | 2 | no | — |
| 72 | #493 | — | 9 | 1 | no | — |
| 73 | #493 | — | 9 | 1 | no | — |
| 74 | C82 | — | 9 | 1 | no | — |
| 75 | C2×C3×D4 | — | 2 | 2 | no | — |
| 76 | C2×C3×D4 | — | 2 | 2 | no | — |
| 77 | C2×C3×D4 | — | 2 | 2 | no | — |
| 78 | C2×C3×D4 | — | 2 | 2 | no | — |
| 79 | C2×C3×D4 | — | 4 | 2 | no | — |
| 80 | C2×C3×D4 | — | 6 | 2 | no | — |
| 81 | C2×C3×D4 | — | 6 | 2 | no | — |
| 82 | C2×C3×D4 | — | 6 | 2 | no | — |
| 83 | C2×D12 | — | 2 | 2 | no | — |
| 84 | C2×D12 | — | 2 | 2 | no | — |
| 85 | C2×D12 | — | 2 | 2 | no | — |
| 86 | C2×D12 | — | 6 | 2 | no | — |
| 87 | C2×D12 | — | 6 | 2 | no | — |
| 88 | C2×D12 | — | 6 | 2 | no | — |
| 89 | C2×D12 | — | 6 | 2 | no | — |
| 90 | C2×D12 | — | 12 | 2 | no | — |
| 91 | C3×D8 | — | 12 | 8 | no | — |
| 92 | C2×C3×C8 | — | 3 | 4 | no | — |
| 93 | C3×C42 | — | 2 | 1 | no | — |
| 94 | C3×C42 | C22×D3 | 1 | 2 | yes | — |
| 95 | #213 | — | 4 | 8 | no | — |
| 96 | #213 | — | 12 | 8 | no | — |
| 97 | #213 | — | 12 | 8 | no | — |
| 98 | #213 | — | 12 | 8 | no | — |
| 99 | #213 | — | 12 | 8 | no | — |
| 100 | C2×C3⋊C8 | — | 3 | 4 | no | — |
| 101 | C2×C3⋊C8 | — | 6 | 2 | no | — |
| 102 | C2×C3⋊C8 | #67 | 1 | 4 | yes | — |
| 103 | D24 | — | 12 | 8 | no | — |
| 104 | C22×C32 | — | 8 | 1 | no | — |
| 105 | C22×C32 | — | 8 | 2 | no | — |
| 106 | C22×C32 | #127 | 1 | 1 | yes | characteristic |
| 107 | C2×C3⋊D3 | — | 8 | 1 | no | — |
| 108 | C2×C3⋊D3 | — | 8 | 2 | no | — |
| 109 | C2×C3×D3 | — | 8 | 2 | no | — |
| 110 | C2×C3×D3 | — | 8 | 2 | no | — |
| 111 | C2×C3×D3 | — | 8 | 2 | no | — |
| 112 | C32×C4 | C2×D8 | 1 | 2 | yes | — |
| 113 | C32×C4 | C2×D8 | 1 | 4 | yes | — |
| 114 | C2×D8 | — | 6 | 4 | no | — |
| 115 | C2×D8 | — | 18 | 4 | no | — |
| 116 | C2×D8 | — | 18 | 2 | no | — |
| 117 | C2×D8 | — | 18 | 2 | no | — |
| 118 | #127 | — | 3 | 2 | no | — |
| 119 | #127 | — | 9 | 1 | no | — |
| 120 | #127 | D32 | 1 | 1 | yes | characteristic |
| 121 | C4×C8 | — | 3 | 2 | no | — |
| 122 | C4×C8 | — | 9 | 1 | no | — |
| 123 | C23×C3 | — | 4 | 2 | no | — |
| 124 | C23×C3 | — | 4 | 2 | no | — |
| 125 | C23×C3 | — | 12 | 2 | no | — |
| 126 | C22×D3 | — | 4 | 2 | no | — |
| 127 | C22×D3 | — | 12 | 2 | no | — |
| 128 | C22×D3 | — | 12 | 2 | no | — |
| 129 | C3×D4 | — | 4 | 4 | no | — |
| 130 | C3×D4 | — | 4 | 4 | no | — |
| 131 | C3×D4 | — | 4 | 4 | no | — |
| 132 | C3×D4 | — | 4 | 4 | no | — |
| 133 | C3×D4 | — | 8 | 4 | no | — |
| 134 | C3×D4 | — | 12 | 4 | no | — |
| 135 | C3×D4 | — | 12 | 4 | no | — |
| 136 | C3×D4 | — | 12 | 4 | no | — |
| 137 | C2×C3×C4 | — | 2 | 2 | no | — |
| 138 | C2×C3×C4 | — | 2 | 1 | no | — |
| 139 | C2×C3×C4 | D3×D4 | 1 | 2 | yes | — |
| 140 | C2×C3×C4 | D3×D4 | 1 | 2 | yes | — |
| 141 | C2×C3×C4 | C2×#67 | 1 | 2 | yes | — |
| 142 | D12 | — | 4 | 4 | no | — |
| 143 | D12 | — | 4 | 4 | no | — |
| 144 | D12 | — | 4 | 4 | no | — |
| 145 | D12 | — | 12 | 4 | no | — |
| 146 | D12 | — | 12 | 4 | no | — |
| 147 | D12 | — | 12 | 4 | no | — |
| 148 | D12 | — | 12 | 4 | no | — |
| 149 | D12 | — | 24 | 4 | no | — |
| 150 | C3×C8 | — | 3 | 8 | no | — |
| 151 | C3⋊C8 | — | 3 | 8 | no | — |
| 152 | C3⋊C8 | — | 6 | 4 | no | — |
| 153 | C3⋊C8 | #213 | 1 | 8 | yes | complemented |
| 154 | C2×C32 | — | 16 | 1 | no | — |
| 155 | C2×C32 | #493 | 1 | 1 | yes | characteristic |
| 156 | C2×C32 | #493 | 1 | 2 | yes | — |
| 157 | C3⋊D3 | — | 16 | 1 | no | — |
| 158 | C3×D3 | — | 16 | 2 | no | — |
| 159 | C2×D4 | — | 2 | 2 | no | — |
| 160 | C2×D4 | — | 2 | 1 | no | — |
| 161 | C2×D4 | — | 6 | 2 | no | — |
| 162 | C2×D4 | — | 6 | 2 | no | — |
| 163 | C2×D4 | — | 6 | 2 | no | — |
| 164 | C2×D4 | — | 18 | 2 | no | — |
| 165 | C2×D4 | — | 18 | 1 | no | — |
| 166 | D8 | — | 12 | 8 | no | — |
| 167 | D8 | — | 36 | 8 | no | — |
| 168 | D8 | — | 36 | 4 | no | — |
| 169 | D8 | — | 36 | 4 | no | — |
| 170 | C2×C8 | — | 3 | 4 | no | — |
| 171 | C2×C8 | — | 9 | 2 | no | — |
| 172 | C42 | C2×D32 | 1 | 1 | yes | characteristic |
| 173 | C22×C3 | — | 2 | 1 | no | — |
| 174 | C22×C3 | — | 8 | 2 | no | — |
| 175 | C22×C3 | — | 8 | 4 | no | — |
| 176 | C22×C3 | — | 8 | 2 | no | — |
| 177 | C22×C3 | — | 8 | 2 | no | — |
| 178 | C22×C3 | — | 8 | 2 | no | — |
| 179 | C22×C3 | — | 24 | 2 | no | — |
| 180 | C22×C3 | — | 24 | 2 | no | — |
| 181 | C22×C3 | — | 24 | 2 | no | — |
| 182 | C22×C3 | #940 | 1 | 2 | yes | — |
| 183 | C2×D3 | — | 8 | 2 | no | — |
| 184 | C2×D3 | — | 8 | 2 | no | — |
| 185 | C2×D3 | — | 8 | 2 | no | — |
| 186 | C2×D3 | — | 24 | 2 | no | — |
| 187 | C2×D3 | — | 24 | 4 | no | — |
| 188 | C2×D3 | — | 24 | 2 | no | — |
| 189 | C2×D3 | — | 24 | 2 | no | — |
| 190 | C2×D3 | — | 24 | 2 | no | — |
| 191 | C3×C4 | — | 2 | 4 | no | — |
| 192 | C3×C4 | — | 2 | 2 | no | — |
| 193 | C3×C4 | D3×D8 | 1 | 4 | yes | — |
| 194 | C3×C4 | D3×D8 | 1 | 4 | yes | — |
| 195 | C3×C4 | C2×#213 | 1 | 4 | yes | — |
| 196 | C32 | #1712 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 197 | C23 | — | 4 | 1 | no | — |
| 198 | C23 | — | 12 | 2 | no | — |
| 199 | C23 | — | 36 | 1 | no | — |
| 200 | D4 | — | 4 | 4 | no | — |
| 201 | D4 | — | 4 | 2 | no | — |
| 202 | D4 | — | 12 | 4 | no | — |
| 203 | D4 | — | 12 | 4 | no | — |
| 204 | D4 | — | 12 | 4 | no | — |
| 205 | D4 | — | 36 | 4 | no | — |
| 206 | D4 | — | 36 | 2 | no | — |
| 207 | C2×C4 | D3×#67 | 1 | 2 | yes | — |
| 208 | C2×C4 | #3811 | 1 | 1 | yes | characteristic |
| 209 | C8 | — | 3 | 8 | no | — |
| 210 | C8 | — | 9 | 4 | no | — |
| 211 | C2×C3 | — | 2 | 2 | no | — |
| 212 | C2×C3 | — | 2 | 1 | no | — |
| 213 | C2×C3 | — | 16 | 2 | no | — |
| 214 | C2×C3 | — | 16 | 2 | no | — |
| 215 | C2×C3 | — | 48 | 2 | no | — |
| 216 | C2×C3 | #5073 | 1 | 2 | yes | — |
| 217 | C2×C3 | #5151 | 1 | 2 | yes | — |
| 218 | C2×C3 | #5151 | 1 | 2 | yes | — |
| 219 | D3 | — | 16 | 2 | no | — |
| 220 | D3 | — | 48 | 2 | no | — |
| 221 | D3 | — | 48 | 2 | no | — |
| 222 | C22 | — | 8 | 2 | no | — |
| 223 | C22 | — | 8 | 1 | no | — |
| 224 | C22 | — | 24 | 2 | no | — |
| 225 | C22 | — | 24 | 2 | no | — |
| 226 | C22 | — | 24 | 2 | no | — |
| 227 | C22 | — | 72 | 2 | no | — |
| 228 | C22 | — | 72 | 1 | no | — |
| 229 | C22 | #63979 | 1 | 1 | yes | characteristic |
| 230 | C4 | D3×#213 | 1 | 4 | yes | — |
| 231 | C4 | #63926 | 1 | 2 | yes | — |
| 232 | C3 | — | 2 | 1 | no | — |
| 233 | C3 | #71276 | 1 | 2 | yes | complemented |
| 234 | C2 | — | 16 | 1 | no | — |
| 235 | C2 | — | 48 | 2 | no | — |
| 236 | C2 | — | 144 | 1 | no | — |
| 237 | C2 | #10591482 | 1 | 2 | yes | — |
| 238 | C2 | #10591484 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #63979 |
| commutator, cosocle | C32×C42 | C23 |
| Frattini | C42 | C2×D32 |
| Fitting | C32×#127 | C22 |
| socle | C22×C32 | #127 |
| hypercenter | #127 | D32 |
| hypocenter | C32 | #1712 |
| LCS | G ▷ C32×C42 ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10592772 | C2 | semidirect |
| 2. C3×#5073 | C2 | semidirect |
| 3. #10590352 | C2 | semidirect |
| 4. #10590350 | C2 | semidirect |
| 5. C3⋊C82 | C2 | semidirect |
| 21. #5073 | D3 | semidirect |
| 68. C3⋊(C3⋊C8) | D8 | semidirect |
| 69. C3×C3⋊C8 | D8 | semidirect |
| 153. C3⋊C8 | #213 | semidirect |
| 196. C32 | #1712 | semidirect |
| 233. C3 | #71276 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10592772 | 2 |
| C3×#5073 | 2 |
| #10590352 | 2 |
| #10590350 | 2 |
| C3⋊C82 | 2 |
| #71276 | 3 |