Finite Group DatabasebetaFinite Group DatabasebetaOrder 1536.
566 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 566 conjugacy classes of subgroups — too many to draw. Only its 125 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 | #10709906 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10709899 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10709898 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10709898 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #10709874 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10681802 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C3×#12509 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #123412 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C3×#2240 | — | 2 | 2 | no | — |
| 10 | C3×#1713 | C22 | 1 | 1 | yes | characteristic |
| 11 | C3×#1712 | C22 | 1 | 1 | yes | characteristic |
| 12 | #71381 | — | 2 | 2 | no | — |
| 13 | #71313 | — | 2 | 2 | no | — |
| 14 | #71290 | — | 2 | 2 | no | — |
| 15 | #71290 | — | 2 | 2 | no | — |
| 16 | #69060 | — | 2 | 2 | no | — |
| 17 | #68062 | C22 | 1 | 1 | yes | characteristic |
| 18 | #68062 | C22 | 1 | 1 | yes | characteristic |
| 19 | C3×C8×C16 | C22 | 1 | 1 | yes | characteristic |
| 20 | C8×C3⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 21 | C8×C3⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 22 | #12517 | — | 3 | 1 | no | — |
| 23 | #12515 | — | 3 | 1 | no | — |
| 24 | #12515 | — | 3 | 1 | no | — |
| 25 | #12515 | — | 3 | 1 | no | — |
| 26 | #12509 | — | 3 | 1 | no | — |
| 27 | #12509 | D3 | 1 | 1 | yes | characteristic, complemented |
| 28 | #7052 | — | 3 | 1 | no | — |
| 29 | C3×#500 | — | 2 | 1 | no | — |
| 30 | C3×#500 | — | 2 | 1 | no | — |
| 31 | C3×#494 | — | 2 | 1 | no | — |
| 32 | C3×#493 | — | 2 | 1 | no | — |
| 33 | C3×#493 | — | 2 | 1 | no | — |
| 34 | C3×#493 | — | 2 | 1 | no | — |
| 35 | C2×#828 | — | 4 | 4 | no | — |
| 36 | C2×#826 | — | 4 | 4 | no | — |
| 37 | C2×#826 | — | 4 | 4 | no | — |
| 38 | C2×D48 | — | 4 | 4 | no | — |
| 39 | #4695 | — | 2 | 1 | no | — |
| 40 | #4695 | — | 2 | 1 | no | — |
| 41 | #4695 | — | 2 | 1 | no | — |
| 42 | #4695 | — | 2 | 1 | no | — |
| 43 | #4695 | — | 2 | 1 | no | — |
| 44 | #4695 | — | 2 | 1 | no | — |
| 45 | C3×#357 | — | 4 | 4 | no | — |
| 46 | C3×C4×C16 | D4 | 1 | 2 | yes | — |
| 47 | C3×C82 | C23 | 1 | 1 | yes | characteristic |
| 48 | #4491 | — | 4 | 4 | no | — |
| 49 | C4×C3⋊C16 | D4 | 1 | 2 | yes | — |
| 50 | C4×C3⋊C16 | D4 | 1 | 2 | yes | — |
| 51 | #2248 | — | 6 | 2 | no | — |
| 52 | #2246 | — | 6 | 2 | no | — |
| 53 | #2246 | — | 6 | 2 | no | — |
| 54 | #2246 | — | 6 | 2 | no | — |
| 55 | #2240 | — | 2 | 2 | no | — |
| 56 | #2240 | — | 6 | 2 | no | — |
| 57 | #1713 | C2×D3 | 1 | 1 | yes | characteristic |
| 58 | #1712 | — | 3 | 1 | no | — |
| 59 | #1712 | — | 3 | 1 | no | — |
| 60 | #1712 | C2×D3 | 1 | 1 | yes | characteristic |
| 61 | C8×C16 | — | 3 | 1 | no | — |
| 62 | C8×C16 | — | 3 | 1 | no | — |
| 63 | C8×C16 | C2×D3 | 1 | 1 | yes | characteristic |
| 64 | C2×C3×Q16 | — | 4 | 2 | no | — |
| 65 | C2×C3×D8 | — | 4 | 2 | no | — |
| 66 | C2×C3×D8 | — | 4 | 2 | no | — |
| 67 | C2×C3×D8 | — | 4 | 2 | no | — |
| 68 | C3×C4⋊Q8 | — | 4 | 1 | no | — |
| 69 | C3×#127 | — | 4 | 1 | no | — |
| 70 | C2×D24 | — | 4 | 2 | no | — |
| 71 | C2×D24 | — | 4 | 2 | no | — |
| 72 | C2×D24 | — | 4 | 2 | no | — |
| 73 | C2×D24 | — | 4 | 2 | no | — |
| 74 | C2×D24 | — | 4 | 2 | no | — |
| 75 | C2×D24 | — | 4 | 2 | no | — |
| 76 | #874 | — | 4 | 1 | no | — |
| 77 | #874 | — | 4 | 1 | no | — |
| 78 | C2×C3×C16 | D8 | 1 | 4 | yes | — |
| 79 | C3×#107 | — | 4 | 2 | no | — |
| 80 | C3×#107 | — | 4 | 2 | no | — |
| 81 | C3×C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 82 | C3×C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | C3×C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | #828 | — | 8 | 8 | no | — |
| 85 | #826 | — | 8 | 8 | no | — |
| 86 | #826 | — | 8 | 8 | no | — |
| 87 | C2×C3⋊C16 | D8 | 1 | 4 | yes | — |
| 88 | C2×C3⋊C16 | D8 | 1 | 4 | yes | — |
| 89 | D48 | — | 8 | 8 | no | — |
| 90 | C2×SD32 | — | 12 | 4 | no | — |
| 91 | C2×D16 | — | 12 | 4 | no | — |
| 92 | C2×D16 | — | 12 | 4 | no | — |
| 93 | C2×D16 | — | 12 | 4 | no | — |
| 94 | #500 | — | 2 | 1 | no | — |
| 95 | #500 | — | 2 | 1 | no | — |
| 96 | #494 | — | 2 | 1 | no | — |
| 97 | #493 | — | 2 | 1 | no | — |
| 98 | #493 | — | 2 | 1 | no | — |
| 99 | #493 | — | 2 | 1 | no | — |
| 100 | #493 | — | 6 | 1 | no | — |
| 101 | #493 | — | 6 | 1 | no | — |
| 102 | #493 | — | 6 | 1 | no | — |
| 103 | #493 | — | 6 | 1 | no | — |
| 104 | #493 | — | 6 | 1 | no | — |
| 105 | #493 | — | 6 | 1 | no | — |
| 106 | #357 | — | 4 | 4 | no | — |
| 107 | #357 | — | 12 | 4 | no | — |
| 108 | C4×C16 | — | 3 | 2 | no | — |
| 109 | C4×C16 | — | 3 | 2 | no | — |
| 110 | C4×C16 | #67 | 1 | 2 | yes | — |
| 111 | C82 | C22×D3 | 1 | 1 | yes | characteristic |
| 112 | C2×C3×Q8 | — | 8 | 1 | no | — |
| 113 | C2×C3×D4 | — | 8 | 1 | no | — |
| 114 | C2×C3×D4 | — | 8 | 1 | no | — |
| 115 | C2×C3×D4 | — | 8 | 1 | no | — |
| 116 | C2×D12 | — | 8 | 1 | no | — |
| 117 | C2×D12 | — | 8 | 1 | no | — |
| 118 | C2×D12 | — | 8 | 1 | no | — |
| 119 | C2×D12 | — | 8 | 1 | no | — |
| 120 | C2×D12 | — | 8 | 1 | no | — |
| 121 | C2×D12 | — | 8 | 1 | no | — |
| 122 | C3×Q16 | — | 8 | 4 | no | — |
| 123 | C3×D8 | — | 8 | 4 | no | — |
| 124 | C3×D8 | — | 8 | 4 | no | — |
| 125 | C3×D8 | — | 8 | 4 | no | — |
| 126 | C2×C3×C8 | C2×D8 | 1 | 2 | yes | — |
| 127 | C2×C3×C8 | C2×D8 | 1 | 2 | yes | — |
| 128 | C2×C3×C8 | C2×D8 | 1 | 2 | yes | — |
| 129 | C3×C4⋊C4 | — | 8 | 1 | no | — |
| 130 | C3×C4⋊C4 | — | 8 | 1 | no | — |
| 131 | C3×C42 | #127 | 1 | 1 | yes | characteristic |
| 132 | D24 | — | 8 | 4 | no | — |
| 133 | D24 | — | 8 | 4 | no | — |
| 134 | D24 | — | 8 | 4 | no | — |
| 135 | D24 | — | 8 | 4 | no | — |
| 136 | D24 | — | 8 | 4 | no | — |
| 137 | D24 | — | 8 | 4 | no | — |
| 138 | C3×C16 | — | 2 | 4 | no | — |
| 139 | C3⋊C16 | — | 2 | 4 | no | — |
| 140 | C3⋊C16 | D16 | 1 | 8 | yes | complemented |
| 141 | C2×Q16 | — | 4 | 2 | no | — |
| 142 | C2×D8 | — | 4 | 2 | no | — |
| 143 | C2×D8 | — | 4 | 2 | no | — |
| 144 | C2×D8 | — | 4 | 2 | no | — |
| 145 | C2×D8 | — | 12 | 2 | no | — |
| 146 | C2×D8 | — | 12 | 2 | no | — |
| 147 | C2×D8 | — | 12 | 2 | no | — |
| 148 | C2×D8 | — | 12 | 2 | no | — |
| 149 | C2×D8 | — | 12 | 2 | no | — |
| 150 | C2×D8 | — | 12 | 2 | no | — |
| 151 | C4⋊Q8 | — | 4 | 1 | no | — |
| 152 | #127 | — | 4 | 1 | no | — |
| 153 | #127 | — | 12 | 1 | no | — |
| 154 | #127 | — | 12 | 1 | no | — |
| 155 | SD32 | — | 24 | 8 | no | — |
| 156 | D16 | — | 24 | 8 | no | — |
| 157 | D16 | — | 24 | 8 | no | — |
| 158 | D16 | — | 24 | 8 | no | — |
| 159 | C2×C16 | — | 3 | 4 | no | — |
| 160 | C2×C16 | — | 3 | 4 | no | — |
| 161 | C2×C16 | #213 | 1 | 4 | yes | — |
| 162 | #107 | — | 4 | 2 | no | — |
| 163 | #107 | — | 4 | 2 | no | — |
| 164 | C4×C8 | D3×D4 | 1 | 1 | yes | characteristic |
| 165 | C4×C8 | D3×D4 | 1 | 1 | yes | characteristic |
| 166 | C4×C8 | C2×#67 | 1 | 1 | yes | characteristic |
| 167 | C23×C3 | — | 16 | 1 | no | — |
| 168 | C22×D3 | — | 16 | 1 | no | — |
| 169 | C22×D3 | — | 16 | 1 | no | — |
| 170 | C3×Q8 | — | 16 | 2 | no | — |
| 171 | C3×D4 | — | 16 | 2 | no | — |
| 172 | C3×D4 | — | 16 | 2 | no | — |
| 173 | C3×D4 | — | 16 | 2 | no | — |
| 174 | C2×C3×C4 | — | 16 | 1 | no | — |
| 175 | C2×C3×C4 | #493 | 1 | 1 | yes | characteristic |
| 176 | C2×C3×C4 | #493 | 1 | 1 | yes | characteristic |
| 177 | C2×C3×C4 | #493 | 1 | 1 | yes | characteristic |
| 178 | D12 | — | 16 | 2 | no | — |
| 179 | D12 | — | 16 | 2 | no | — |
| 180 | D12 | — | 16 | 2 | no | — |
| 181 | D12 | — | 16 | 2 | no | — |
| 182 | D12 | — | 16 | 2 | no | — |
| 183 | D12 | — | 16 | 2 | no | — |
| 184 | C3×C8 | C2×D16 | 1 | 4 | yes | — |
| 185 | C3×C8 | #509 | 1 | 4 | yes | — |
| 186 | C3×C8 | #509 | 1 | 4 | yes | — |
| 187 | C2×Q8 | — | 8 | 1 | no | — |
| 188 | C2×D4 | — | 8 | 1 | no | — |
| 189 | C2×D4 | — | 8 | 1 | no | — |
| 190 | C2×D4 | — | 8 | 1 | no | — |
| 191 | C2×D4 | — | 24 | 1 | no | — |
| 192 | C2×D4 | — | 24 | 1 | no | — |
| 193 | C2×D4 | — | 24 | 1 | no | — |
| 194 | C2×D4 | — | 24 | 1 | no | — |
| 195 | C2×D4 | — | 24 | 1 | no | — |
| 196 | C2×D4 | — | 24 | 1 | no | — |
| 197 | Q16 | — | 8 | 4 | no | — |
| 198 | D8 | — | 8 | 4 | no | — |
| 199 | D8 | — | 8 | 4 | no | — |
| 200 | D8 | — | 8 | 4 | no | — |
| 201 | D8 | — | 24 | 4 | no | — |
| 202 | D8 | — | 24 | 4 | no | — |
| 203 | D8 | — | 24 | 4 | no | — |
| 204 | D8 | — | 24 | 4 | no | — |
| 205 | D8 | — | 24 | 4 | no | — |
| 206 | D8 | — | 24 | 4 | no | — |
| 207 | C2×C8 | D3×D8 | 1 | 2 | yes | — |
| 208 | C2×C8 | D3×D8 | 1 | 2 | yes | — |
| 209 | C2×C8 | C2×#213 | 1 | 2 | yes | — |
| 210 | C4⋊C4 | — | 8 | 1 | no | — |
| 211 | C4⋊C4 | — | 8 | 1 | no | — |
| 212 | C42 | #940 | 1 | 1 | yes | characteristic |
| 213 | C16 | — | 2 | 4 | no | — |
| 214 | C16 | — | 3 | 8 | no | — |
| 215 | C16 | — | 6 | 4 | no | — |
| 216 | C22×C3 | — | 32 | 1 | no | — |
| 217 | C22×C3 | — | 32 | 1 | no | — |
| 218 | C22×C3 | — | 32 | 1 | no | — |
| 219 | C22×C3 | #1712 | 1 | 1 | yes | characteristic |
| 220 | C2×D3 | — | 32 | 1 | no | — |
| 221 | C2×D3 | — | 32 | 1 | no | — |
| 222 | C2×D3 | — | 32 | 1 | no | — |
| 223 | C2×D3 | — | 32 | 1 | no | — |
| 224 | C2×D3 | — | 32 | 1 | no | — |
| 225 | C2×D3 | — | 32 | 1 | no | — |
| 226 | C3×C4 | — | 32 | 1 | no | — |
| 227 | C3×C4 | #2246 | 1 | 2 | yes | — |
| 228 | C3×C4 | #2250 | 1 | 2 | yes | — |
| 229 | C3×C4 | #2250 | 1 | 2 | yes | — |
| 230 | C23 | — | 16 | 1 | no | — |
| 231 | C23 | — | 48 | 1 | no | — |
| 232 | C23 | — | 48 | 1 | no | — |
| 233 | Q8 | — | 16 | 2 | no | — |
| 234 | D4 | — | 16 | 2 | no | — |
| 235 | D4 | — | 16 | 2 | no | — |
| 236 | D4 | — | 16 | 2 | no | — |
| 237 | D4 | — | 48 | 2 | no | — |
| 238 | D4 | — | 48 | 2 | no | — |
| 239 | D4 | — | 48 | 2 | no | — |
| 240 | D4 | — | 48 | 2 | no | — |
| 241 | D4 | — | 48 | 2 | no | — |
| 242 | D4 | — | 48 | 2 | no | — |
| 243 | C2×C4 | — | 16 | 1 | no | — |
| 244 | C2×C4 | #5073 | 1 | 1 | yes | characteristic |
| 245 | C2×C4 | #5151 | 1 | 1 | yes | characteristic |
| 246 | C2×C4 | #5151 | 1 | 1 | yes | characteristic |
| 247 | C8 | D3×D16 | 1 | 4 | yes | — |
| 248 | C8 | #4914 | 1 | 4 | yes | — |
| 249 | C8 | #5193 | 1 | 4 | yes | — |
| 250 | C2×C3 | — | 64 | 1 | no | — |
| 251 | C2×C3 | #12515 | 1 | 1 | yes | characteristic |
| 252 | C2×C3 | #12519 | 1 | 1 | yes | characteristic |
| 253 | C2×C3 | #12519 | 1 | 1 | yes | characteristic |
| 254 | D3 | — | 64 | 1 | no | — |
| 255 | D3 | — | 64 | 1 | no | — |
| 256 | C22 | — | 32 | 1 | no | — |
| 257 | C22 | — | 32 | 1 | no | — |
| 258 | C22 | — | 32 | 1 | no | — |
| 259 | C22 | — | 96 | 1 | no | — |
| 260 | C22 | — | 96 | 1 | no | — |
| 261 | C22 | — | 96 | 1 | no | — |
| 262 | C22 | — | 96 | 1 | no | — |
| 263 | C22 | — | 96 | 1 | no | — |
| 264 | C22 | — | 96 | 1 | no | — |
| 265 | C22 | #71276 | 1 | 1 | yes | characteristic |
| 266 | C4 | — | 32 | 1 | no | — |
| 267 | C4 | #71321 | 1 | 2 | yes | — |
| 268 | C4 | #71464 | 1 | 2 | yes | — |
| 269 | C4 | #71481 | 1 | 2 | yes | — |
| 270 | C3 | #123412 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 271 | C2 | — | 64 | 1 | no | — |
| 272 | C2 | — | 192 | 1 | no | — |
| 273 | C2 | — | 192 | 1 | no | — |
| 274 | C2 | #10709897 | 1 | 1 | yes | characteristic |
| 275 | C2 | #10709914 | 1 | 1 | yes | characteristic |
| 276 | C2 | #10709916 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10709906 | C2 | semidirect |
| 2. #10709899 | C2 | semidirect |
| 3. #10709898 | C2 | semidirect |
| 4. #10709898 | C2 | semidirect |
| 5. #10709874 | C2 | semidirect |
| 6. #10681802 | C2 | semidirect |
| 7. C3×#12509 | C2 | semidirect |
| 27. #12509 | D3 | semidirect |
| 140. C3⋊C16 | D16 | semidirect |
| 270. C3 | #123412 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10709906 | 2 |
| #10709899 | 2 |
| #10709898 | 2 |
| #10709874 | 2 |
| #10681802 | 2 |
| C3×#12509 | 2 |
| #123412 | 3 |