Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
1587 conjugacy classes of subgroups.
The subgroup lattice has 294 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C9×H(3,𝔽4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10589786 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | #10589782 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #10589774 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #10589770 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #10589697 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | #10589688 | C2 | 1 | 6 | yes | maximal, complemented |
| 8 | #87085 | — | 3 | 1 | no | maximal |
| 9 | C23×Dic9 | C22 | 1 | 2 | yes | complemented |
| 10 | C2×#3701 | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | C2×#3699 | C22 | 1 | 3 | yes | — |
| 12 | C9×#124 | C22 | 1 | 3 | yes | — |
| 13 | C9×#124 | C22 | 1 | 6 | yes | — |
| 14 | C9×#120 | C22 | 1 | 6 | yes | — |
| 15 | #63515 | C22 | 1 | 6 | yes | complemented |
| 16 | #63508 | C22 | 1 | 6 | yes | complemented |
| 17 | Q8×Dic9 | C22 | 1 | 2 | yes | complemented |
| 18 | #63506 | C22 | 1 | 3 | yes | complemented |
| 19 | #63502 | C22 | 1 | 3 | yes | — |
| 20 | #63501 | C22 | 1 | 12 | yes | — |
| 21 | #63500 | C22 | 1 | 6 | yes | — |
| 22 | #63497 | C22 | 1 | 6 | yes | — |
| 23 | D4×Dic9 | C22 | 1 | 6 | yes | — |
| 24 | #63448 | C22 | 1 | 6 | yes | complemented |
| 25 | #63446 | C22 | 1 | 12 | yes | complemented |
| 26 | #63445 | C22 | 1 | 12 | yes | complemented |
| 27 | #611⋊C4 | C22 | 1 | 12 | yes | complemented |
| 28 | #63441 | C22 | 1 | 12 | yes | complemented |
| 29 | #63440 | C22 | 1 | 12 | yes | complemented |
| 30 | #63439 | C22 | 1 | 6 | yes | complemented |
| 31 | #63437 | C22 | 1 | 3 | yes | complemented |
| 32 | C42×D9 | C22 | 1 | 3 | yes | complemented |
| 33 | #63432 | C22 | 1 | 6 | yes | complemented |
| 34 | C3×H(3,𝔽4) | D3 | 1 | 1 | yes | characteristic |
| 35 | #5684 | — | 3 | 6 | no | — |
| 36 | #5680 | — | 3 | 6 | no | — |
| 37 | #5672 | — | 3 | 3 | no | — |
| 38 | #5668 | — | 3 | 3 | no | — |
| 39 | #5595 | — | 3 | 6 | no | — |
| 40 | #5586 | — | 3 | 6 | no | — |
| 41 | C24×C9 | C23 | 1 | 2 | yes | — |
| 42 | C2×Q8×C9 | C23 | 1 | 1 | yes | characteristic |
| 43 | C2×Q8×C9 | C23 | 1 | 2 | yes | — |
| 44 | C2×D4×C9 | C23 | 1 | 6 | yes | — |
| 45 | C2×D4×C9 | C23 | 1 | 3 | yes | — |
| 46 | C2×#611 | C23 | 1 | 12 | yes | — |
| 47 | C2×#611 | C23 | 1 | 6 | yes | — |
| 48 | C22×Dic9 | — | 4 | 4 | no | — |
| 49 | C22×Dic9 | C23 | 1 | 6 | yes | — |
| 50 | C22×Dic9 | C23 | 1 | 6 | yes | — |
| 51 | #3701 | — | 4 | 2 | no | — |
| 52 | #3699 | — | 4 | 6 | no | — |
| 53 | C2×D36 | C23 | 1 | 3 | yes | — |
| 54 | C2×C4×D9 | C23 | 1 | 3 | yes | — |
| 55 | C2×C4×D9 | C23 | 1 | 6 | yes | — |
| 56 | C2×Dic18 | C23 | 1 | 3 | yes | — |
| 57 | C9×C22⋊C4 | C23 | 1 | 6 | yes | — |
| 58 | C9×C22⋊C4 | C23 | 1 | 12 | yes | — |
| 59 | C42×C9 | C23 | 1 | 3 | yes | — |
| 60 | #3676 | C23 | 1 | 12 | yes | — |
| 61 | #3676 | C23 | 1 | 6 | yes | — |
| 62 | (C2×D9)⋊C4 | C23 | 1 | 12 | yes | — |
| 63 | C4⋊Dic9 | C23 | 1 | 6 | yes | — |
| 64 | C4⋊Dic9 | C23 | 1 | 6 | yes | — |
| 65 | Dic9⋊C4 | C23 | 1 | 12 | yes | — |
| 66 | Dic9⋊C4 | C23 | 1 | 12 | yes | — |
| 67 | C4×Dic9 | C23 | 1 | 6 | yes | — |
| 68 | C4×Dic9 | C23 | 1 | 3 | yes | — |
| 69 | #3529 | — | 9 | 1 | no | Sylow |
| 70 | C23×Dic3 | — | 3 | 2 | no | — |
| 71 | C2×#239 | — | 3 | 1 | no | — |
| 72 | C2×#237 | — | 3 | 3 | no | — |
| 73 | C3×#124 | C2×D3 | 1 | 3 | yes | — |
| 74 | C3×#124 | C2×D3 | 1 | 6 | yes | — |
| 75 | C3×#120 | C2×D3 | 1 | 6 | yes | — |
| 76 | #953 | — | 3 | 6 | no | — |
| 77 | #946 | — | 3 | 6 | no | — |
| 78 | Q8×Dic3 | — | 3 | 2 | no | — |
| 79 | #944 | — | 3 | 3 | no | — |
| 80 | #940 | — | 3 | 3 | no | — |
| 81 | #939 | — | 3 | 12 | no | — |
| 82 | #938 | — | 3 | 6 | no | — |
| 83 | #935 | — | 3 | 6 | no | — |
| 84 | D4×Dic3 | — | 3 | 6 | no | — |
| 85 | #886 | — | 3 | 6 | no | — |
| 86 | #884 | — | 3 | 12 | no | — |
| 87 | #883 | — | 3 | 12 | no | — |
| 88 | #67⋊C4 | — | 3 | 12 | no | — |
| 89 | #879 | — | 3 | 12 | no | — |
| 90 | #878 | — | 3 | 12 | no | — |
| 91 | #877 | — | 3 | 6 | no | — |
| 92 | #875 | — | 3 | 3 | no | — |
| 93 | C42×D3 | — | 3 | 3 | no | — |
| 94 | #870 | — | 3 | 6 | no | — |
| 95 | C23×C9 | — | 4 | 2 | no | — |
| 96 | C23×C9 | — | 4 | 4 | no | — |
| 97 | C23×C9 | C24 | 1 | 6 | yes | — |
| 98 | C22×D9 | C24 | 1 | 3 | yes | — |
| 99 | Q8×C9 | — | 4 | 1 | no | — |
| 100 | Q8×C9 | — | 4 | 2 | no | — |
| 101 | D4×C9 | — | 4 | 3 | no | — |
| 102 | D4×C9 | — | 4 | 6 | no | — |
| 103 | C2×C4×C9 | C24 | 1 | 3 | yes | — |
| 104 | C2×C4×C9 | C24 | 1 | 6 | yes | — |
| 105 | #611 | — | 4 | 6 | no | — |
| 106 | #611 | — | 4 | 12 | no | — |
| 107 | C2×Dic9 | — | 2 | 12 | no | — |
| 108 | C2×Dic9 | — | 4 | 6 | no | — |
| 109 | C2×Dic9 | C24 | 1 | 6 | yes | — |
| 110 | C2×Dic9 | C24 | 1 | 6 | yes | — |
| 111 | C2×Dic9 | C24 | 1 | 1 | yes | characteristic |
| 112 | D36 | — | 4 | 3 | no | — |
| 113 | C4×D9 | — | 2 | 6 | no | — |
| 114 | C4×D9 | — | 4 | 6 | no | — |
| 115 | Dic18 | — | 4 | 3 | no | — |
| 116 | #563 | — | 9 | 3 | no | — |
| 117 | #562 | — | 9 | 6 | no | — |
| 118 | H(3,𝔽4) | D9 | 1 | 1 | yes | characteristic, complemented |
| 119 | #560 | — | 9 | 6 | no | — |
| 120 | #532 | — | 9 | 3 | no | — |
| 121 | #529 | — | 9 | 6 | no | — |
| 122 | #525 | — | 9 | 6 | no | — |
| 123 | C24×C3 | C22×D3 | 1 | 2 | yes | — |
| 124 | C2×C3×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 125 | C2×C3×Q8 | C22×D3 | 1 | 2 | yes | — |
| 126 | C2×C3×D4 | C22×D3 | 1 | 6 | yes | — |
| 127 | C2×C3×D4 | C22×D3 | 1 | 3 | yes | — |
| 128 | C2×#67 | — | 3 | 12 | no | — |
| 129 | C2×#67 | — | 3 | 6 | no | — |
| 130 | C22×Dic3 | — | 3 | 6 | no | — |
| 131 | C22×Dic3 | — | 3 | 6 | no | — |
| 132 | C22×Dic3 | — | 12 | 4 | no | — |
| 133 | #239 | — | 12 | 2 | no | — |
| 134 | #237 | — | 12 | 6 | no | — |
| 135 | C2×D12 | — | 3 | 3 | no | — |
| 136 | C2×C4×D3 | — | 3 | 3 | no | — |
| 137 | C2×C4×D3 | — | 3 | 6 | no | — |
| 138 | C2×Dic6 | — | 3 | 3 | no | — |
| 139 | C3×C22⋊C4 | C22×D3 | 1 | 6 | yes | — |
| 140 | C3×C22⋊C4 | C22×D3 | 1 | 12 | yes | — |
| 141 | C3×C42 | C22×D3 | 1 | 3 | yes | — |
| 142 | #217 | — | 3 | 6 | no | — |
| 143 | #217 | — | 3 | 12 | no | — |
| 144 | (C2×D3)⋊C4 | — | 3 | 12 | no | — |
| 145 | C4⋊Dic3 | — | 3 | 6 | no | — |
| 146 | C4⋊Dic3 | — | 3 | 6 | no | — |
| 147 | Dic3⋊C4 | — | 3 | 12 | no | — |
| 148 | Dic3⋊C4 | — | 3 | 12 | no | — |
| 149 | C4×Dic3 | — | 3 | 6 | no | — |
| 150 | C4×Dic3 | — | 3 | 3 | no | — |
| 151 | C22×C9 | — | 2 | 6 | no | — |
| 152 | C22×C9 | — | 2 | 12 | no | — |
| 153 | C22×C9 | — | 4 | 8 | no | — |
| 154 | C22×C9 | C25 | 1 | 1 | yes | characteristic |
| 155 | C2×D9 | — | 2 | 3 | no | — |
| 156 | C2×D9 | — | 2 | 6 | no | — |
| 157 | C4×C9 | — | 2 | 3 | no | — |
| 158 | C4×C9 | — | 2 | 6 | no | — |
| 159 | Dic9 | — | 2 | 6 | no | — |
| 160 | Dic9 | — | 2 | 6 | no | — |
| 161 | Dic9 | H(5,𝔽2) | 1 | 2 | yes | — |
| 162 | C2×#41 | — | 9 | 1 | no | — |
| 163 | C2×#41 | — | 9 | 3 | no | — |
| 164 | C23×C4 | — | 9 | 2 | no | — |
| 165 | C4⋊Q8 | — | 9 | 3 | no | — |
| 166 | #127 | — | 9 | 3 | no | — |
| 167 | #126 | — | 9 | 12 | no | — |
| 168 | #125 | — | 9 | 6 | no | — |
| 169 | #124 | — | 9 | 3 | no | — |
| 170 | #124 | C2×D9 | 1 | 6 | yes | — |
| 171 | #124 | C2×D9 | 1 | 3 | yes | — |
| 172 | #123 | — | 9 | 6 | no | — |
| 173 | #123 | — | 9 | 6 | no | — |
| 174 | #123 | — | 9 | 12 | no | — |
| 175 | C22⋊Q8 | — | 9 | 6 | no | — |
| 176 | C22⋊Q8 | — | 9 | 12 | no | — |
| 177 | #121 | — | 9 | 12 | no | — |
| 178 | #121 | — | 9 | 6 | no | — |
| 179 | #121 | — | 9 | 12 | no | — |
| 180 | #120 | — | 9 | 6 | no | — |
| 181 | #120 | C2×D9 | 1 | 6 | yes | — |
| 182 | C4×Q8 | — | 9 | 2 | no | — |
| 183 | C4×D4 | — | 9 | 6 | no | — |
| 184 | C4×D4 | — | 9 | 12 | no | — |
| 185 | #117 | — | 9 | 6 | no | — |
| 186 | C2×C42 | — | 9 | 3 | no | — |
| 187 | C23×C3 | — | 4 | 2 | no | — |
| 188 | C23×C3 | — | 4 | 4 | no | — |
| 189 | C23×C3 | C23×D3 | 1 | 6 | yes | — |
| 190 | C22×D3 | — | 3 | 3 | no | — |
| 191 | C3×Q8 | — | 4 | 1 | no | — |
| 192 | C3×Q8 | — | 4 | 2 | no | — |
| 193 | C3×D4 | — | 4 | 3 | no | — |
| 194 | C3×D4 | — | 4 | 6 | no | — |
| 195 | C2×C3×C4 | C23×D3 | 1 | 3 | yes | — |
| 196 | C2×C3×C4 | C23×D3 | 1 | 6 | yes | — |
| 197 | #67 | — | 12 | 6 | no | — |
| 198 | #67 | — | 12 | 12 | no | — |
| 199 | C2×Dic3 | — | 3 | 1 | no | — |
| 200 | C2×Dic3 | — | 3 | 6 | no | — |
| 201 | C2×Dic3 | — | 3 | 6 | no | — |
| 202 | C2×Dic3 | — | 6 | 12 | no | — |
| 203 | C2×Dic3 | — | 12 | 6 | no | — |
| 204 | D12 | — | 12 | 3 | no | — |
| 205 | C4×D3 | — | 6 | 6 | no | — |
| 206 | C4×D3 | — | 12 | 6 | no | — |
| 207 | Dic6 | — | 12 | 3 | no | — |
| 208 | C2×C9 | — | 4 | 6 | no | — |
| 209 | C2×C9 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 210 | C2×C9 | #585 | 1 | 2 | yes | — |
| 211 | D9 | — | 4 | 3 | no | — |
| 212 | C24 | C22×D9 | 1 | 2 | yes | — |
| 213 | #41 | — | 36 | 6 | no | — |
| 214 | #41 | — | 36 | 2 | no | — |
| 215 | C2×Q8 | — | 9 | 3 | no | — |
| 216 | C2×Q8 | C22×D9 | 1 | 2 | yes | — |
| 217 | C2×Q8 | C22×D9 | 1 | 1 | yes | characteristic |
| 218 | C2×D4 | — | 9 | 12 | no | — |
| 219 | C2×D4 | — | 9 | 6 | no | — |
| 220 | C2×D4 | — | 9 | 3 | no | — |
| 221 | C2×D4 | C22×D9 | 1 | 6 | yes | — |
| 222 | C2×D4 | C22×D9 | 1 | 3 | yes | — |
| 223 | C22×C4 | — | 9 | 6 | no | — |
| 224 | C22×C4 | — | 9 | 6 | no | — |
| 225 | C22×C4 | — | 9 | 3 | no | — |
| 226 | C22×C4 | — | 9 | 6 | no | — |
| 227 | C22×C4 | — | 36 | 4 | no | — |
| 228 | C4⋊C4 | — | 9 | 12 | no | — |
| 229 | C4⋊C4 | — | 9 | 12 | no | — |
| 230 | C4⋊C4 | — | 9 | 6 | no | — |
| 231 | C4⋊C4 | — | 9 | 6 | no | — |
| 232 | C22⋊C4 | — | 9 | 6 | no | — |
| 233 | C22⋊C4 | — | 9 | 12 | no | — |
| 234 | C22⋊C4 | — | 9 | 12 | no | — |
| 235 | C22⋊C4 | C22×D9 | 1 | 6 | yes | — |
| 236 | C22⋊C4 | C22×D9 | 1 | 12 | yes | — |
| 237 | C42 | — | 9 | 6 | no | — |
| 238 | C42 | — | 9 | 3 | no | — |
| 239 | C42 | C22×D9 | 1 | 3 | yes | — |
| 240 | C22×C3 | — | 2 | 6 | no | — |
| 241 | C22×C3 | — | 2 | 12 | no | — |
| 242 | C22×C3 | — | 4 | 8 | no | — |
| 243 | C22×C3 | C24×D3 | 1 | 1 | yes | characteristic |
| 244 | C2×D3 | — | 6 | 3 | no | — |
| 245 | C2×D3 | — | 6 | 6 | no | — |
| 246 | C3×C4 | — | 2 | 3 | no | — |
| 247 | C3×C4 | — | 2 | 6 | no | — |
| 248 | Dic3 | — | 3 | 2 | no | — |
| 249 | Dic3 | — | 6 | 6 | no | — |
| 250 | Dic3 | — | 6 | 6 | no | — |
| 251 | C9 | #3529 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 252 | C23 | — | 4 | 2 | no | — |
| 253 | C23 | — | 4 | 4 | no | — |
| 254 | C23 | — | 9 | 3 | no | — |
| 255 | C23 | C23×D9 | 1 | 6 | yes | — |
| 256 | Q8 | — | 4 | 1 | no | — |
| 257 | Q8 | — | 4 | 2 | no | — |
| 258 | Q8 | — | 36 | 3 | no | — |
| 259 | D4 | — | 4 | 3 | no | — |
| 260 | D4 | — | 4 | 6 | no | — |
| 261 | D4 | — | 36 | 3 | no | — |
| 262 | D4 | — | 36 | 6 | no | — |
| 263 | D4 | — | 36 | 12 | no | — |
| 264 | C2×C4 | — | 9 | 1 | no | — |
| 265 | C2×C4 | — | 9 | 6 | no | — |
| 266 | C2×C4 | — | 9 | 6 | no | — |
| 267 | C2×C4 | — | 18 | 6 | no | — |
| 268 | C2×C4 | — | 18 | 12 | no | — |
| 269 | C2×C4 | — | 36 | 6 | no | — |
| 270 | C2×C4 | — | 36 | 6 | no | — |
| 271 | C2×C4 | C23×D9 | 1 | 6 | yes | — |
| 272 | C2×C4 | C23×D9 | 1 | 3 | yes | — |
| 273 | C2×C3 | — | 4 | 6 | no | — |
| 274 | C2×C3 | D3×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 275 | C2×C3 | #5966 | 1 | 2 | yes | — |
| 276 | D3 | — | 12 | 3 | no | — |
| 277 | C22 | — | 2 | 6 | no | — |
| 278 | C22 | — | 2 | 12 | no | — |
| 279 | C22 | — | 4 | 8 | no | — |
| 280 | C22 | — | 18 | 3 | no | — |
| 281 | C22 | — | 18 | 6 | no | — |
| 282 | C22 | C24×D9 | 1 | 1 | yes | characteristic |
| 283 | C4 | — | 2 | 3 | no | — |
| 284 | C4 | — | 2 | 6 | no | — |
| 285 | C4 | — | 9 | 2 | no | — |
| 286 | C4 | — | 18 | 6 | no | — |
| 287 | C4 | — | 18 | 6 | no | — |
| 288 | C3 | #87085 | 1 | 1 | yes | characteristic |
| 289 | C2 | — | 4 | 6 | no | — |
| 290 | C2 | — | 36 | 3 | no | — |
| 291 | C2 | D9×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 292 | C2 | #10593141 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C24×D9 |
| commutator, cosocle | C22×C9 | C25 |
| Frattini, socle | C22×C3 | C24×D3 |
| Fitting | C9×H(3,𝔽4) | C2 |
| hypercenter | H(3,𝔽4) | D9 |
| hypocenter | C9 | #3529 |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×H(3,𝔽4) | C2 | semidirect |
| 2. #10589786 | C2 | semidirect |
| 3. #10589782 | C2 | semidirect |
| 4. #10589774 | C2 | semidirect |
| 5. #10589770 | C2 | semidirect |
| 6. #10589697 | C2 | semidirect |
| 7. #10589688 | C2 | semidirect |
| 9. C23×Dic9 | C22 | semidirect |
| 10. C2×#3701 | C22 | semidirect |
| 15. #63515 | C22 | semidirect |
| 16. #63508 | C22 | semidirect |
| 17. Q8×Dic9 | C22 | semidirect |
| 18. #63506 | C22 | semidirect |
| 24. #63448 | C22 | semidirect |
| 25. #63446 | C22 | semidirect |
| 26. #63445 | C22 | semidirect |
| 27. #611⋊C4 | C22 | semidirect |
| 28. #63441 | C22 | semidirect |
| 29. #63440 | C22 | semidirect |
| 30. #63439 | C22 | semidirect |
| 31. #63437 | C22 | semidirect |
| 32. C42×D9 | C22 | semidirect |
| 33. #63432 | C22 | semidirect |
| 118. H(3,𝔽4) | D9 | semidirect |
| 251. C9 | #3529 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×H(3,𝔽4) | 2 |
| #10589786 | 2 |
| #10589782 | 2 |
| #10589774 | 2 |
| #10589770 | 2 |
| #10589697 | 2 |
| #10589688 | 2 |
| #87085 | 3 |