Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
493 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 493 conjugacy classes of subgroups — too many to draw. Only its 73 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 | C2×#63691 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C4×#41⋊C9 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10589526 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C4×#3689 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C4×#3688 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C4×#950 | — | 3 | 1 | no | maximal |
| 7 | C2×#41⋊C9 | C22 | 1 | 1 | yes | characteristic |
| 8 | #63691 | C4 | 1 | 4 | yes | direct product factor |
| 9 | C2×#3689 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#3688 | C22 | 1 | 1 | yes | characteristic |
| 11 | C42×D9 | — | 4 | 1 | no | maximal |
| 12 | C4×Q8⋊C9 | C22 | 1 | 2 | yes | — |
| 13 | #63421 | C22 | 1 | 2 | yes | — |
| 14 | C3×C4×#41 | D3 | 1 | 1 | yes | characteristic |
| 15 | C2×#950 | — | 3 | 1 | no | — |
| 16 | C4×#235 | — | 3 | 1 | no | — |
| 17 | #5233 | — | 3 | 1 | no | — |
| 18 | C4×#216 | — | 3 | 2 | no | — |
| 19 | C4×#215 | — | 3 | 2 | no | — |
| 20 | C4×#214 | — | 3 | 2 | no | — |
| 21 | C4×#213 | — | 3 | 2 | no | — |
| 22 | #5002 | — | 3 | 1 | no | — |
| 23 | #4998 | — | 3 | 1 | no | — |
| 24 | C2×C4×C3⋊C8 | — | 3 | 1 | no | — |
| 25 | C2×C4×D9 | — | 4 | 2 | no | — |
| 26 | C2×C4×D9 | — | 4 | 1 | no | — |
| 27 | #41⋊C9 | C2×C4 | 1 | 2 | yes | complemented |
| 28 | C2×Q8⋊C9 | C23 | 1 | 1 | yes | characteristic |
| 29 | #3689 | C2×C4 | 1 | 2 | yes | complemented |
| 30 | #3688 | C2×C4 | 1 | 2 | yes | complemented |
| 31 | C42×C9 | — | 4 | 1 | no | — |
| 32 | C4×Dic9 | — | 4 | 2 | no | — |
| 33 | C4×Dic9 | — | 4 | 1 | no | — |
| 34 | C4×#135 | — | 9 | 1 | no | Sylow |
| 35 | C2×C3×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 36 | C2×#235 | — | 3 | 1 | no | — |
| 37 | C3×C4×Q8 | C2×D3 | 1 | 2 | yes | — |
| 38 | C3×C4×D4 | — | 3 | 2 | no | — |
| 39 | C3×#117 | — | 3 | 1 | no | — |
| 40 | C2×C3×C42 | — | 3 | 1 | no | — |
| 41 | #950 | — | 3 | 4 | no | — |
| 42 | #950 | — | 3 | 4 | no | — |
| 43 | C2×#216 | — | 3 | 1 | no | — |
| 44 | C2×#215 | — | 3 | 1 | no | — |
| 45 | C2×#214 | — | 3 | 1 | no | — |
| 46 | C2×#213 | — | 3 | 1 | no | — |
| 47 | C4×#67 | — | 6 | 2 | no | — |
| 48 | #926 | — | 3 | 1 | no | — |
| 49 | C22×C3⋊C8 | — | 3 | 1 | no | — |
| 50 | C22×C3⋊C8 | — | 3 | 1 | no | — |
| 51 | C4×D12 | — | 3 | 2 | no | — |
| 52 | #872 | — | 6 | 1 | no | — |
| 53 | C42×D3 | — | 6 | 1 | no | — |
| 54 | C4×Dic6 | — | 3 | 2 | no | — |
| 55 | #835 | — | 3 | 2 | no | — |
| 56 | #832 | — | 3 | 2 | no | — |
| 57 | #810 | — | 3 | 2 | no | — |
| 58 | #809 | — | 3 | 2 | no | — |
| 59 | #808 | — | 3 | 2 | no | — |
| 60 | #807 | — | 3 | 2 | no | — |
| 61 | C4×C3⋊C8 | — | 3 | 2 | no | — |
| 62 | C4×C3⋊C8 | — | 3 | 2 | no | — |
| 63 | C22×D9 | — | 4 | 1 | no | — |
| 64 | C2×C4×C9 | — | 4 | 2 | no | — |
| 65 | C2×C4×C9 | — | 4 | 1 | no | — |
| 66 | C2×Dic9 | — | 4 | 2 | no | — |
| 67 | C2×Dic9 | — | 4 | 1 | no | — |
| 68 | C4×D9 | — | 4 | 4 | no | — |
| 69 | C4×D9 | — | 4 | 8 | no | — |
| 70 | Q8⋊C9 | C22×C4 | 1 | 1 | yes | characteristic |
| 71 | C2×#135 | — | 9 | 1 | no | — |
| 72 | C4×#41 | — | 9 | 1 | no | — |
| 73 | C4×#41 | D9 | 1 | 1 | yes | characteristic, complemented |
| 74 | C4×Q16 | — | 9 | 2 | no | — |
| 75 | C4×SD16 | — | 9 | 2 | no | — |
| 76 | C4×SD16 | — | 9 | 2 | no | — |
| 77 | C4×D8 | — | 9 | 2 | no | — |
| 78 | #427 | — | 9 | 1 | no | — |
| 79 | #416 | — | 9 | 1 | no | — |
| 80 | #416 | — | 9 | 1 | no | — |
| 81 | C2×C4×C8 | — | 9 | 1 | no | — |
| 82 | C3×#41 | — | 3 | 2 | no | — |
| 83 | C3×#41 | C4×D3 | 1 | 2 | yes | — |
| 84 | C2×C3×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 85 | C2×C3×D4 | — | 3 | 1 | no | — |
| 86 | C22×C3×C4 | — | 3 | 1 | no | — |
| 87 | C22×C3×C4 | — | 3 | 2 | no | — |
| 88 | C2×#67 | — | 6 | 1 | no | — |
| 89 | #235 | — | 3 | 4 | no | — |
| 90 | #235 | — | 6 | 2 | no | — |
| 91 | C2×D12 | — | 3 | 1 | no | — |
| 92 | C2×C4×D3 | — | 6 | 1 | no | — |
| 93 | C2×C4×D3 | — | 6 | 2 | no | — |
| 94 | C2×Dic6 | — | 3 | 1 | no | — |
| 95 | C3×C4⋊C4 | — | 3 | 2 | no | — |
| 96 | C3×C22⋊C4 | — | 6 | 1 | no | — |
| 97 | C3×C42 | — | 3 | 2 | no | — |
| 98 | C3×C42 | — | 3 | 1 | no | — |
| 99 | C3×C42 | S4 | 1 | 1 | yes | characteristic |
| 100 | #217 | — | 6 | 1 | no | — |
| 101 | #216 | — | 3 | 2 | no | — |
| 102 | #216 | — | 3 | 2 | no | — |
| 103 | #215 | — | 3 | 2 | no | — |
| 104 | #215 | — | 3 | 2 | no | — |
| 105 | #214 | — | 3 | 4 | no | — |
| 106 | #213 | — | 3 | 4 | no | — |
| 107 | (C2×D3)⋊C4 | — | 6 | 2 | no | — |
| 108 | C4⋊Dic3 | — | 3 | 2 | no | — |
| 109 | Dic3⋊C4 | — | 6 | 2 | no | — |
| 110 | C4×Dic3 | — | 6 | 2 | no | — |
| 111 | C4×Dic3 | — | 6 | 1 | no | — |
| 112 | C2×C3⋊C8 | — | 3 | 1 | no | — |
| 113 | C2×C3⋊C8 | — | 3 | 1 | no | — |
| 114 | C2×C3⋊C8 | — | 3 | 4 | no | — |
| 115 | C2×C3⋊C8 | — | 3 | 2 | no | — |
| 116 | C2×C3⋊C8 | — | 6 | 2 | no | — |
| 117 | C22×C9 | — | 4 | 1 | no | — |
| 118 | C2×D9 | — | 4 | 2 | no | — |
| 119 | C2×D9 | — | 4 | 4 | no | — |
| 120 | C4×C9 | — | 4 | 2 | no | — |
| 121 | C4×C9 | — | 4 | 4 | no | — |
| 122 | Dic9 | — | 4 | 2 | no | — |
| 123 | Dic9 | — | 4 | 4 | no | — |
| 124 | C2×#41 | — | 9 | 1 | no | — |
| 125 | C2×#41 | C2×D9 | 1 | 1 | yes | characteristic |
| 126 | #135 | — | 9 | 4 | no | — |
| 127 | #135 | — | 9 | 4 | no | — |
| 128 | C2×Q16 | — | 9 | 1 | no | — |
| 129 | C2×SD16 | — | 9 | 1 | no | — |
| 130 | C2×SD16 | — | 9 | 1 | no | — |
| 131 | C2×D8 | — | 9 | 1 | no | — |
| 132 | C22×C8 | — | 9 | 1 | no | — |
| 133 | C22×C8 | — | 9 | 1 | no | — |
| 134 | C4×Q8 | — | 9 | 2 | no | — |
| 135 | C4×Q8 | C2×D9 | 1 | 2 | yes | — |
| 136 | C4×D4 | — | 3 | 2 | no | — |
| 137 | C4×D4 | — | 9 | 2 | no | — |
| 138 | C4×D4 | — | 18 | 2 | no | — |
| 139 | #117 | — | 3 | 1 | no | — |
| 140 | #117 | — | 9 | 1 | no | — |
| 141 | #117 | — | 18 | 1 | no | — |
| 142 | C2×C42 | — | 3 | 1 | no | — |
| 143 | C2×C42 | — | 18 | 1 | no | — |
| 144 | #107 | — | 9 | 2 | no | — |
| 145 | #106 | — | 9 | 2 | no | — |
| 146 | #103 | — | 9 | 2 | no | — |
| 147 | #103 | — | 9 | 2 | no | — |
| 148 | #102 | — | 9 | 2 | no | — |
| 149 | #102 | — | 9 | 2 | no | — |
| 150 | C4×C8 | — | 9 | 2 | no | — |
| 151 | C4×C8 | — | 9 | 2 | no | — |
| 152 | C23×C3 | — | 3 | 1 | no | — |
| 153 | C22×D3 | — | 6 | 1 | no | — |
| 154 | C3×Q8 | — | 3 | 1 | no | — |
| 155 | C3×Q8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 156 | C3×D4 | — | 3 | 2 | no | — |
| 157 | C3×D4 | — | 6 | 1 | no | — |
| 158 | C2×C3×C4 | — | 3 | 1 | no | — |
| 159 | C2×C3×C4 | — | 3 | 2 | no | — |
| 160 | C2×C3×C4 | — | 3 | 2 | no | — |
| 161 | C2×C3×C4 | — | 3 | 2 | no | — |
| 162 | C2×C3×C4 | — | 6 | 4 | no | — |
| 163 | C2×C3×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 164 | C2×C3×C4 | C2×S4 | 1 | 2 | yes | — |
| 165 | #67 | — | 6 | 4 | no | — |
| 166 | C2×Dic3 | — | 6 | 1 | no | — |
| 167 | C2×Dic3 | — | 6 | 2 | no | — |
| 168 | D12 | — | 3 | 2 | no | — |
| 169 | D12 | — | 6 | 1 | no | — |
| 170 | C4×D3 | — | 6 | 4 | no | — |
| 171 | C4×D3 | — | 12 | 4 | no | — |
| 172 | Dic6 | — | 3 | 2 | no | — |
| 173 | Dic6 | — | 6 | 1 | no | — |
| 174 | C3⋊C8 | — | 3 | 2 | no | — |
| 175 | C3⋊C8 | — | 3 | 2 | no | — |
| 176 | C3⋊C8 | — | 6 | 2 | no | — |
| 177 | C2×C9 | — | 4 | 1 | no | — |
| 178 | C2×C9 | — | 4 | 2 | no | — |
| 179 | D9 | — | 4 | 4 | no | — |
| 180 | #41 | — | 3 | 2 | no | — |
| 181 | #41 | — | 9 | 4 | no | — |
| 182 | #41 | — | 18 | 2 | no | — |
| 183 | #41 | C4×D9 | 1 | 2 | yes | complemented |
| 184 | C2×Q8 | — | 9 | 1 | no | — |
| 185 | C2×Q8 | C22×D9 | 1 | 1 | yes | characteristic |
| 186 | C2×D4 | — | 3 | 1 | no | — |
| 187 | C2×D4 | — | 9 | 1 | no | — |
| 188 | C2×D4 | — | 18 | 1 | no | — |
| 189 | C22×C4 | — | 3 | 1 | no | — |
| 190 | C22×C4 | — | 3 | 2 | no | — |
| 191 | C22×C4 | — | 18 | 1 | no | — |
| 192 | C22×C4 | — | 18 | 2 | no | — |
| 193 | Q16 | — | 9 | 2 | no | — |
| 194 | Q16 | — | 9 | 2 | no | — |
| 195 | SD16 | — | 9 | 4 | no | — |
| 196 | SD16 | — | 9 | 2 | no | — |
| 197 | SD16 | — | 9 | 2 | no | — |
| 198 | D8 | — | 9 | 4 | no | — |
| 199 | C2×C8 | — | 9 | 1 | no | — |
| 200 | C2×C8 | — | 9 | 1 | no | — |
| 201 | C2×C8 | — | 9 | 4 | no | — |
| 202 | C2×C8 | — | 9 | 2 | no | — |
| 203 | C2×C8 | — | 18 | 2 | no | — |
| 204 | C4⋊C4 | — | 3 | 2 | no | — |
| 205 | C4⋊C4 | — | 9 | 2 | no | — |
| 206 | C4⋊C4 | — | 18 | 2 | no | — |
| 207 | C22⋊C4 | — | 6 | 1 | no | — |
| 208 | C22⋊C4 | — | 18 | 2 | no | — |
| 209 | C22⋊C4 | — | 18 | 1 | no | — |
| 210 | C42 | — | 3 | 2 | no | — |
| 211 | C42 | — | 3 | 1 | no | — |
| 212 | C42 | — | 18 | 1 | no | — |
| 213 | C42 | — | 18 | 2 | no | — |
| 214 | C42 | (C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 215 | C22×C3 | — | 3 | 2 | no | — |
| 216 | C22×C3 | — | 6 | 2 | no | — |
| 217 | C22×C3 | C22×S4 | 1 | 1 | yes | characteristic |
| 218 | C2×D3 | — | 6 | 2 | no | — |
| 219 | C2×D3 | — | 12 | 2 | no | — |
| 220 | C3×C4 | — | 3 | 1 | no | — |
| 221 | C3×C4 | — | 3 | 1 | no | — |
| 222 | C3×C4 | — | 6 | 2 | no | — |
| 223 | C3×C4 | C4×S4 | 1 | 2 | yes | — |
| 224 | C3×C4 | #985 | 1 | 4 | yes | — |
| 225 | Dic3 | — | 6 | 2 | no | — |
| 226 | Dic3 | — | 12 | 2 | no | — |
| 227 | C9 | — | 4 | 1 | no | Sylow |
| 228 | C23 | — | 3 | 1 | no | — |
| 229 | C23 | — | 18 | 1 | no | — |
| 230 | Q8 | — | 3 | 1 | no | — |
| 231 | Q8 | — | 9 | 2 | no | — |
| 232 | Q8 | — | 18 | 1 | no | — |
| 233 | Q8 | C2×C4×D9 | 1 | 1 | yes | characteristic |
| 234 | D4 | — | 3 | 2 | no | — |
| 235 | D4 | — | 6 | 1 | no | — |
| 236 | D4 | — | 9 | 2 | no | — |
| 237 | D4 | — | 18 | 1 | no | — |
| 238 | D4 | — | 18 | 4 | no | — |
| 239 | C2×C4 | — | 3 | 1 | no | — |
| 240 | C2×C4 | — | 3 | 2 | no | — |
| 241 | C2×C4 | — | 3 | 2 | no | — |
| 242 | C2×C4 | — | 3 | 2 | no | — |
| 243 | C2×C4 | — | 6 | 4 | no | — |
| 244 | C2×C4 | — | 18 | 1 | no | — |
| 245 | C2×C4 | — | 18 | 4 | no | — |
| 246 | C2×C4 | — | 18 | 2 | no | — |
| 247 | C2×C4 | — | 36 | 4 | no | — |
| 248 | C2×C4 | C2×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 249 | C2×C4 | C2×(C22⋊C9)⋊C2 | 1 | 2 | yes | — |
| 250 | C8 | — | 9 | 2 | no | — |
| 251 | C8 | — | 9 | 2 | no | — |
| 252 | C8 | — | 18 | 2 | no | — |
| 253 | C2×C3 | — | 6 | 2 | no | — |
| 254 | C2×C3 | C2×C4×S4 | 1 | 1 | yes | characteristic |
| 255 | C2×C3 | C2×#985 | 1 | 2 | yes | — |
| 256 | D3 | — | 12 | 2 | no | — |
| 257 | C22 | — | 3 | 2 | no | — |
| 258 | C22 | — | 6 | 2 | no | — |
| 259 | C22 | — | 18 | 2 | no | — |
| 260 | C22 | — | 36 | 2 | no | — |
| 261 | C22 | C22×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 262 | C4 | — | 3 | 1 | no | — |
| 263 | C4 | — | 3 | 1 | no | — |
| 264 | C4 | — | 6 | 2 | no | — |
| 265 | C4 | — | 18 | 2 | no | — |
| 266 | C4 | — | 36 | 2 | no | — |
| 267 | C4 | C4×(C22⋊C9)⋊C2 | 1 | 2 | yes | — |
| 268 | C4 | #63691 | 1 | 4 | yes | direct product factor |
| 269 | C3 | C4×#985 | 1 | 1 | yes | characteristic |
| 270 | C2 | — | 6 | 2 | no | — |
| 271 | C2 | — | 36 | 2 | no | — |
| 272 | C2 | C2×C4×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 273 | C2 | C2×#63691 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C42 | (C22⋊C9)⋊C2 |
| commutator, hypocenter | Q8⋊C9 | C22×C4 |
| Frattini, socle | C22×C3 | C22×S4 |
| Fitting | C3×C4×#41 | D3 |
| prime radical | C2×#63691 | C2 |
| cosocle | C2×Q8⋊C9 | C23 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×#41⋊C9 | C2 | semidirect |
| 3. #10589526 | C2 | semidirect |
| 4. C4×#3689 | C2 | semidirect |
| 5. C4×#3688 | C2 | semidirect |
| 8. #63691 | C4 | direct (N × H) |
| 27. #41⋊C9 | C2×C4 | semidirect |
| 29. #3689 | C2×C4 | semidirect |
| 30. #3688 | C2×C4 | semidirect |
| 73. C4×#41 | D9 | semidirect |
| 183. #41 | C4×D9 | semidirect |
| 268. C4 | #63691 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#63691 | 2 |
| C4×#41⋊C9 | 2 |
| #10589526 | 2 |
| C4×#3689 | 2 |
| C4×#3688 | 2 |
| C4×#950 | 3 |
| C42×D9 | 4 |