Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
592 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 592 conjugacy classes of subgroups — too many to draw. Only its 64 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 | C22×AΓL(1,𝔽9) | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C2×#64231 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×GO+(4,𝔽2)⋊C4 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #10593555 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #10593267 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C22×PSU(3,𝔽2) | C22 | 1 | 1 | yes | characteristic |
| 7 | C22×GO+(4,𝔽2) | — | 2 | 1 | no | — |
| 8 | C22×GO+(4,𝔽2) | C22 | 1 | 2 | yes | — |
| 9 | C22×AGL(1,𝔽9) | C22 | 1 | 2 | yes | — |
| 10 | C2×AΓL(1,𝔽9) | — | 2 | 4 | no | — |
| 11 | C2×AΓL(1,𝔽9) | — | 2 | 4 | no | — |
| 12 | C2×AΓL(1,𝔽9) | — | 2 | 4 | no | — |
| 13 | C2×#3802 | — | 2 | 1 | no | — |
| 14 | C2×C4×#157 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#3777 | C22 | 1 | 1 | yes | characteristic |
| 16 | #64231 | — | 2 | 4 | no | — |
| 17 | GO+(4,𝔽2)⋊C4 | — | 2 | 4 | no | — |
| 18 | #63781 | — | 2 | 1 | no | — |
| 19 | #63747 | — | 2 | 1 | no | — |
| 20 | C22×D32 | — | 2 | 2 | no | — |
| 21 | C22×#157 | — | 2 | 1 | no | — |
| 22 | C22×#157 | — | 2 | 1 | no | — |
| 23 | C22×#157 | D4 | 1 | 1 | yes | characteristic |
| 24 | C22×#157 | C23 | 1 | 1 | yes | characteristic |
| 25 | C2×PSU(3,𝔽2) | — | 2 | 2 | no | — |
| 26 | C2×PSU(3,𝔽2) | — | 2 | 2 | no | — |
| 27 | C2×PSU(3,𝔽2) | — | 2 | 1 | no | — |
| 28 | C2×PSU(3,𝔽2) | D4 | 1 | 2 | yes | complemented |
| 29 | C2×GO+(4,𝔽2) | — | 2 | 2 | no | — |
| 30 | C2×GO+(4,𝔽2) | — | 2 | 2 | no | — |
| 31 | C2×GO+(4,𝔽2) | — | 2 | 2 | no | — |
| 32 | C2×GO+(4,𝔽2) | — | 2 | 2 | no | — |
| 33 | C2×GO+(4,𝔽2) | — | 2 | 4 | no | — |
| 34 | C2×GO+(4,𝔽2) | — | 2 | 2 | no | — |
| 35 | C2×GO+(4,𝔽2) | — | 4 | 2 | no | — |
| 36 | C2×GO+(4,𝔽2) | — | 4 | 2 | no | — |
| 37 | C2×GO+(4,𝔽2) | D4 | 1 | 4 | yes | — |
| 38 | C2×AGL(1,𝔽9) | — | 2 | 2 | no | — |
| 39 | C2×AGL(1,𝔽9) | — | 2 | 2 | no | — |
| 40 | C2×AGL(1,𝔽9) | D4 | 1 | 4 | yes | complemented |
| 41 | AΓL(1,𝔽9) | — | 2 | 16 | no | — |
| 42 | C2×C4×C3⋊D3 | D4 | 1 | 1 | yes | characteristic |
| 43 | C2×C3×D12 | — | 4 | 1 | no | — |
| 44 | C2×#625 | — | 4 | 1 | no | — |
| 45 | #3802 | — | 2 | 4 | no | — |
| 46 | #3802 | — | 4 | 2 | no | — |
| 47 | C4×#157 | — | 2 | 2 | no | — |
| 48 | #3777 | — | 2 | 2 | no | — |
| 49 | #2014 | — | 9 | 1 | no | maximal, Sylow |
| 50 | C2×D3×D4 | — | 12 | 1 | no | — |
| 51 | C22×C3⋊D3 | C2×D4 | 1 | 1 | yes | characteristic |
| 52 | C22×C3×D3 | — | 4 | 2 | no | — |
| 53 | C2×D32 | — | 2 | 4 | no | — |
| 54 | C2×D32 | — | 4 | 2 | no | — |
| 55 | C2×D32 | — | 4 | 2 | no | — |
| 56 | C2×D32 | — | 4 | 2 | no | — |
| 57 | C2×D32 | — | 4 | 2 | no | — |
| 58 | C2×D32 | — | 4 | 2 | no | — |
| 59 | C2×#157 | — | 2 | 1 | no | — |
| 60 | C2×#157 | — | 2 | 1 | no | — |
| 61 | C2×#157 | — | 2 | 2 | no | — |
| 62 | C2×#157 | — | 4 | 1 | no | — |
| 63 | C2×#157 | — | 4 | 1 | no | — |
| 64 | C2×#157 | — | 4 | 1 | no | — |
| 65 | C2×#157 | — | 4 | 1 | no | — |
| 66 | C2×#157 | — | 4 | 1 | no | — |
| 67 | C2×#157 | SD16 | 1 | 2 | yes | complemented |
| 68 | C2×#157 | C2×D4 | 1 | 1 | yes | characteristic |
| 69 | C2×#157 | C2×D4 | 1 | 2 | yes | — |
| 70 | C2×#157 | C2×D4 | 1 | 2 | yes | — |
| 71 | C2×#157 | #41 | 1 | 1 | yes | characteristic |
| 72 | PSU(3,𝔽2) | — | 2 | 4 | no | — |
| 73 | PSU(3,𝔽2) | — | 4 | 2 | no | — |
| 74 | GO+(4,𝔽2) | — | 2 | 4 | no | — |
| 75 | GO+(4,𝔽2) | — | 2 | 4 | no | — |
| 76 | GO+(4,𝔽2) | — | 4 | 8 | no | — |
| 77 | GO+(4,𝔽2) | — | 4 | 2 | no | — |
| 78 | GO+(4,𝔽2) | — | 4 | 2 | no | — |
| 79 | AGL(1,𝔽9) | — | 2 | 4 | no | — |
| 80 | C2×C32×C4 | SD16 | 1 | 1 | yes | characteristic, complemented |
| 81 | C2×C3⋊Dic3 | SD16 | 1 | 1 | yes | characteristic, complemented |
| 82 | C4×C3⋊D3 | — | 2 | 1 | no | — |
| 83 | C4×C3⋊D3 | — | 2 | 1 | no | — |
| 84 | C3×D12 | — | 4 | 4 | no | — |
| 85 | #625 | — | 4 | 4 | no | — |
| 86 | C22×SD16 | — | 9 | 2 | no | — |
| 87 | C2×#127 | — | 9 | 1 | no | — |
| 88 | C2×#102 | — | 9 | 2 | no | — |
| 89 | #391 | — | 9 | 1 | no | — |
| 90 | #336 | — | 9 | 1 | no | — |
| 91 | C23×D3 | — | 12 | 2 | no | — |
| 92 | C2×C3×D4 | — | 12 | 1 | no | — |
| 93 | C2×#67 | — | 12 | 2 | no | — |
| 94 | D3×D4 | — | 12 | 4 | no | — |
| 95 | D3×D4 | — | 12 | 4 | no | — |
| 96 | C2×D12 | — | 4 | 1 | no | — |
| 97 | C2×C4×D3 | — | 12 | 1 | no | — |
| 98 | C22×C32 | C2×SD16 | 1 | 1 | yes | characteristic |
| 99 | C2×C3⋊D3 | #120 | 1 | 1 | yes | characteristic |
| 100 | C2×C3⋊D3 | #121 | 1 | 1 | yes | characteristic |
| 101 | C2×C3⋊D3 | #127 | 1 | 1 | yes | characteristic |
| 102 | C2×C3⋊D3 | C2×SD16 | 1 | 1 | yes | characteristic |
| 103 | C2×C3⋊D3 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 104 | C2×C3⋊D3 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 105 | C2×C3×D3 | — | 4 | 4 | no | — |
| 106 | C2×C3×D3 | — | 8 | 2 | no | — |
| 107 | C2×C3×D3 | — | 8 | 2 | no | — |
| 108 | D32 | — | 4 | 4 | no | — |
| 109 | D32 | — | 4 | 4 | no | — |
| 110 | D32 | — | 8 | 2 | no | — |
| 111 | D32 | — | 8 | 2 | no | — |
| 112 | #157 | — | 2 | 2 | no | — |
| 113 | #157 | — | 4 | 2 | no | — |
| 114 | #157 | — | 8 | 1 | no | — |
| 115 | #157 | #120 | 1 | 2 | yes | — |
| 116 | #157 | #121 | 1 | 2 | yes | — |
| 117 | C32×C4 | — | 2 | 1 | no | — |
| 118 | C3⋊Dic3 | — | 2 | 1 | no | — |
| 119 | C22×Q8 | — | 9 | 1 | no | — |
| 120 | C22×D4 | — | 9 | 2 | no | — |
| 121 | C22×D4 | — | 18 | 1 | no | — |
| 122 | C22×D4 | — | 18 | 1 | no | — |
| 123 | C2×SD16 | — | 18 | 4 | no | — |
| 124 | C2×SD16 | — | 18 | 4 | no | — |
| 125 | C2×SD16 | — | 18 | 4 | no | — |
| 126 | C22×C8 | — | 9 | 2 | no | — |
| 127 | #127 | — | 18 | 4 | no | — |
| 128 | C2×C4⋊C4 | — | 9 | 1 | no | — |
| 129 | C2×C42 | — | 9 | 1 | no | — |
| 130 | #102 | — | 18 | 4 | no | — |
| 131 | #95 | — | 18 | 1 | no | — |
| 132 | #95 | — | 18 | 1 | no | — |
| 133 | C23×C3 | — | 12 | 2 | no | — |
| 134 | C22×D3 | — | 4 | 2 | no | — |
| 135 | C22×D3 | — | 12 | 1 | no | — |
| 136 | C22×D3 | — | 12 | 4 | no | — |
| 137 | C22×D3 | — | 12 | 4 | no | — |
| 138 | C22×D3 | — | 24 | 2 | no | — |
| 139 | C22×D3 | — | 24 | 2 | no | — |
| 140 | C22×D3 | — | 24 | 2 | no | — |
| 141 | C22×D3 | — | 24 | 2 | no | — |
| 142 | C3×D4 | — | 12 | 4 | no | — |
| 143 | C2×C3×C4 | — | 4 | 1 | no | — |
| 144 | #67 | — | 12 | 8 | no | — |
| 145 | C2×Dic3 | — | 12 | 1 | no | — |
| 146 | D12 | — | 4 | 4 | no | — |
| 147 | C4×D3 | — | 12 | 2 | no | — |
| 148 | C4×D3 | — | 12 | 2 | no | — |
| 149 | C2×C32 | #450 | 1 | 1 | yes | characteristic |
| 150 | C2×C32 | #460 | 1 | 1 | yes | characteristic |
| 151 | C2×C32 | #492 | 1 | 1 | yes | characteristic |
| 152 | C3⋊D3 | #392 | 1 | 1 | yes | characteristic |
| 153 | C3⋊D3 | #450 | 1 | 1 | yes | characteristic |
| 154 | C3⋊D3 | #460 | 1 | 1 | yes | characteristic |
| 155 | C3⋊D3 | #496 | 1 | 1 | yes | characteristic |
| 156 | C3×D3 | — | 8 | 4 | no | — |
| 157 | C24 | — | 18 | 2 | no | — |
| 158 | C2×Q8 | — | 9 | 2 | no | — |
| 159 | C2×Q8 | — | 18 | 1 | no | — |
| 160 | C2×Q8 | — | 18 | 2 | no | — |
| 161 | C2×Q8 | — | 18 | 2 | no | — |
| 162 | C2×D4 | — | 9 | 4 | no | — |
| 163 | C2×D4 | — | 12 | 1 | no | — |
| 164 | C2×D4 | — | 18 | 2 | no | — |
| 165 | C2×D4 | — | 18 | 2 | no | — |
| 166 | C2×D4 | — | 18 | 2 | no | — |
| 167 | C2×D4 | — | 18 | 2 | no | — |
| 168 | C2×D4 | — | 18 | 2 | no | — |
| 169 | C2×D4 | — | 18 | 4 | no | — |
| 170 | C2×D4 | — | 18 | 4 | no | — |
| 171 | C2×D4 | — | 36 | 1 | no | — |
| 172 | C2×D4 | — | 36 | 2 | no | — |
| 173 | C2×D4 | — | 36 | 2 | no | — |
| 174 | C2×D4 | — | 36 | 2 | no | — |
| 175 | C22×C4 | — | 9 | 1 | no | — |
| 176 | C22×C4 | — | 9 | 1 | no | — |
| 177 | C22×C4 | — | 9 | 1 | no | — |
| 178 | C22×C4 | — | 18 | 1 | no | — |
| 179 | C22×C4 | — | 18 | 1 | no | — |
| 180 | SD16 | — | 18 | 16 | no | — |
| 181 | C2×C8 | — | 9 | 4 | no | — |
| 182 | C2×C8 | — | 18 | 2 | no | — |
| 183 | C2×C8 | — | 18 | 2 | no | — |
| 184 | C4⋊C4 | — | 18 | 2 | no | — |
| 185 | C42 | — | 18 | 2 | no | — |
| 186 | C22×C3 | — | 4 | 1 | no | — |
| 187 | C22×C3 | — | 12 | 4 | no | — |
| 188 | C22×C3 | — | 24 | 2 | no | — |
| 189 | C22×C3 | — | 24 | 2 | no | — |
| 190 | C2×D3 | — | 4 | 4 | no | — |
| 191 | C2×D3 | — | 8 | 2 | no | — |
| 192 | C2×D3 | — | 8 | 2 | no | — |
| 193 | C2×D3 | — | 12 | 1 | no | — |
| 194 | C2×D3 | — | 12 | 1 | no | — |
| 195 | C2×D3 | — | 12 | 1 | no | — |
| 196 | C2×D3 | — | 12 | 1 | no | — |
| 197 | C2×D3 | — | 12 | 1 | no | — |
| 198 | C2×D3 | — | 12 | 1 | no | — |
| 199 | C2×D3 | — | 24 | 4 | no | — |
| 200 | C2×D3 | — | 24 | 4 | no | — |
| 201 | C2×D3 | — | 24 | 4 | no | — |
| 202 | C2×D3 | — | 24 | 4 | no | — |
| 203 | C3×C4 | — | 4 | 2 | no | — |
| 204 | Dic3 | — | 12 | 2 | no | — |
| 205 | C32 | #2014 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 206 | C23 | — | 9 | 1 | no | — |
| 207 | C23 | — | 12 | 2 | no | — |
| 208 | C23 | — | 18 | 4 | no | — |
| 209 | C23 | — | 36 | 2 | no | — |
| 210 | C23 | — | 36 | 2 | no | — |
| 211 | C23 | — | 36 | 2 | no | — |
| 212 | C23 | — | 36 | 2 | no | — |
| 213 | C23 | — | 36 | 2 | no | — |
| 214 | Q8 | — | 18 | 4 | no | — |
| 215 | Q8 | — | 36 | 2 | no | — |
| 216 | D4 | — | 12 | 4 | no | — |
| 217 | D4 | — | 18 | 4 | no | — |
| 218 | D4 | — | 18 | 4 | no | — |
| 219 | D4 | — | 36 | 2 | no | — |
| 220 | D4 | — | 36 | 2 | no | — |
| 221 | D4 | — | 36 | 4 | no | — |
| 222 | D4 | — | 36 | 8 | no | — |
| 223 | C2×C4 | — | 9 | 2 | no | — |
| 224 | C2×C4 | — | 9 | 2 | no | — |
| 225 | C2×C4 | — | 9 | 1 | no | — |
| 226 | C2×C4 | — | 9 | 1 | no | — |
| 227 | C2×C4 | — | 9 | 2 | no | — |
| 228 | C2×C4 | — | 9 | 1 | no | — |
| 229 | C2×C4 | — | 18 | 1 | no | — |
| 230 | C2×C4 | — | 18 | 1 | no | — |
| 231 | C2×C4 | — | 18 | 1 | no | — |
| 232 | C2×C4 | — | 18 | 1 | no | — |
| 233 | C2×C4 | — | 18 | 2 | no | — |
| 234 | C2×C4 | — | 36 | 1 | no | — |
| 235 | C2×C4 | — | 36 | 1 | no | — |
| 236 | C2×C4 | — | 36 | 1 | no | — |
| 237 | C2×C4 | — | 36 | 1 | no | — |
| 238 | C2×C4 | — | 36 | 1 | no | — |
| 239 | C2×C4 | AΓL(1,𝔽9) | 1 | 1 | yes | characteristic, complemented |
| 240 | C8 | — | 18 | 4 | no | — |
| 241 | C2×C3 | — | 4 | 1 | no | — |
| 242 | C2×C3 | — | 4 | 1 | no | — |
| 243 | C2×C3 | — | 4 | 1 | no | — |
| 244 | C2×C3 | — | 24 | 4 | no | — |
| 245 | D3 | — | 8 | 4 | no | — |
| 246 | D3 | — | 12 | 1 | no | — |
| 247 | D3 | — | 12 | 1 | no | — |
| 248 | D3 | — | 12 | 1 | no | — |
| 249 | D3 | — | 12 | 1 | no | — |
| 250 | C22 | — | 9 | 1 | no | — |
| 251 | C22 | — | 9 | 1 | no | — |
| 252 | C22 | — | 9 | 1 | no | — |
| 253 | C22 | — | 9 | 1 | no | — |
| 254 | C22 | — | 9 | 1 | no | — |
| 255 | C22 | — | 9 | 1 | no | — |
| 256 | C22 | — | 12 | 4 | no | — |
| 257 | C22 | — | 24 | 2 | no | — |
| 258 | C22 | — | 24 | 2 | no | — |
| 259 | C22 | — | 36 | 4 | no | — |
| 260 | C22 | — | 36 | 4 | no | — |
| 261 | C22 | — | 72 | 2 | no | — |
| 262 | C22 | — | 72 | 2 | no | — |
| 263 | C22 | C2×AΓL(1,𝔽9) | 1 | 1 | yes | characteristic |
| 264 | C4 | — | 2 | 1 | no | — |
| 265 | C4 | — | 9 | 2 | no | — |
| 266 | C4 | — | 9 | 2 | no | — |
| 267 | C4 | — | 18 | 2 | no | — |
| 268 | C4 | — | 18 | 1 | no | — |
| 269 | C4 | — | 36 | 2 | no | — |
| 270 | C4 | — | 72 | 1 | no | — |
| 271 | C3 | — | 4 | 1 | no | — |
| 272 | C2 | — | 9 | 1 | no | — |
| 273 | C2 | — | 9 | 1 | no | — |
| 274 | C2 | — | 9 | 1 | no | — |
| 275 | C2 | — | 9 | 1 | no | — |
| 276 | C2 | — | 24 | 4 | no | — |
| 277 | C2 | #10596408 | 1 | 1 | yes | characteristic |
| 278 | C2 | #10596410 | 1 | 1 | yes | characteristic |
| 279 | C2 | #10596417 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×AΓL(1,𝔽9) | C2 | semidirect |
| 3. C2×GO+(4,𝔽2)⋊C4 | C2 | semidirect |
| 4. #10593555 | C2 | semidirect |
| 5. #10593267 | C2 | semidirect |
| 28. C2×PSU(3,𝔽2) | D4 | semidirect |
| 40. C2×AGL(1,𝔽9) | D4 | semidirect |
| 67. C2×#157 | SD16 | semidirect |
| 80. C2×C32×C4 | SD16 | semidirect |
| 81. C2×C3⋊Dic3 | SD16 | semidirect |
| 205. C32 | #2014 | semidirect |
| 239. C2×C4 | AΓL(1,𝔽9) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×AΓL(1,𝔽9) | 2 |
| C2×#64231 | 2 |
| C2×GO+(4,𝔽2)⋊C4 | 2 |
| #10593555 | 2 |
| #10593267 | 2 |
| #2014 | 9 |