Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
2286 conjugacy classes of subgroups; only the 447 normal subgroups are shown.
This group has 2286 conjugacy classes of subgroups — too many to draw. Only its 447 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 | #10596687 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C2×Q8×#157 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10596525 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #10596524 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | Q8×GO+(4,𝔽2) | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #10596513 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | #10596511 | C2 | 1 | 6 | yes | maximal, complemented |
| 8 | C2×Q8×C3⋊D3 | C22 | 1 | 1 | yes | characteristic |
| 9 | C3×#1007 | — | 2 | 2 | no | — |
| 10 | #64319 | C22 | 1 | 8 | yes | — |
| 11 | D32×Q8 | C22 | 1 | 8 | yes | — |
| 12 | #64315 | — | 2 | 8 | no | — |
| 13 | #64306 | C22 | 1 | 12 | yes | — |
| 14 | #64298 | — | 2 | 6 | no | — |
| 15 | Q8×#157 | C22 | 1 | 4 | yes | — |
| 16 | Q8×#157 | C22 | 1 | 4 | yes | — |
| 17 | C2×C4⋊#157 | C22 | 1 | 3 | yes | — |
| 18 | C2×C4×#157 | C22 | 1 | 3 | yes | — |
| 19 | #64242 | C22 | 1 | 4 | yes | — |
| 20 | #64242 | C22 | 1 | 12 | yes | — |
| 21 | #64231 | C22 | 1 | 6 | yes | — |
| 22 | #64230 | C22 | 1 | 24 | yes | — |
| 23 | C4×GO+(4,𝔽2) | C22 | 1 | 12 | yes | — |
| 24 | C4×GO+(4,𝔽2) | C22 | 1 | 12 | yes | — |
| 25 | #64222 | C22 | 1 | 6 | yes | — |
| 26 | #64221 | C22 | 1 | 12 | yes | — |
| 27 | D32⋊Q8 | C22 | 1 | 24 | yes | — |
| 28 | D3×2−1+4 | — | 6 | 2 | no | — |
| 29 | C22×#157 | C23 | 1 | 1 | yes | characteristic |
| 30 | C22×#157 | C23 | 1 | 3 | yes | — |
| 31 | C2×GO+(4,𝔽2) | C23 | 1 | 4 | yes | — |
| 32 | C2×GO+(4,𝔽2) | C23 | 1 | 12 | yes | — |
| 33 | C2×C32×Q8 | D4 | 1 | 1 | yes | characteristic, complemented |
| 34 | Q8×C3⋊D3 | D4 | 1 | 4 | yes | complemented |
| 35 | Q8×C3⋊D3 | C23 | 1 | 4 | yes | — |
| 36 | C2×C4×C3⋊D3 | C23 | 1 | 3 | yes | — |
| 37 | C2×#634 | D4 | 1 | 3 | yes | complemented |
| 38 | C3×#239 | — | 2 | 8 | no | — |
| 39 | C3×D3×Q8 | — | 2 | 8 | no | — |
| 40 | C3×#235 | — | 2 | 12 | no | — |
| 41 | #3811 | C23 | 1 | 8 | yes | — |
| 42 | #3805 | — | 2 | 12 | no | — |
| 43 | #3802 | C23 | 1 | 12 | yes | — |
| 44 | C4×D32 | C23 | 1 | 24 | yes | — |
| 45 | #3798 | — | 2 | 12 | no | — |
| 46 | #3797 | C23 | 1 | 12 | yes | — |
| 47 | #3796 | — | 2 | 12 | no | — |
| 48 | #3795 | — | 2 | 24 | no | — |
| 49 | D3×Dic6 | — | 2 | 24 | no | — |
| 50 | C4⋊#157 | C23 | 1 | 12 | yes | — |
| 51 | C4×#157 | C23 | 1 | 6 | yes | — |
| 52 | C4×#157 | C23 | 1 | 6 | yes | — |
| 53 | #3773 | C23 | 1 | 12 | yes | — |
| 54 | #3773 | C23 | 1 | 4 | yes | — |
| 55 | D32⋊C4 | C23 | 1 | 24 | yes | — |
| 56 | D32⋊C4 | C23 | 1 | 8 | yes | — |
| 57 | #3497 | — | 9 | 1 | no | maximal, Sylow |
| 58 | C3×2−1+4 | — | 6 | 2 | no | — |
| 59 | #1010 | — | 6 | 12 | no | — |
| 60 | #1010 | — | 6 | 8 | no | — |
| 61 | D3×#41 | — | 6 | 12 | no | — |
| 62 | D3×#41 | — | 6 | 8 | no | — |
| 63 | #1007 | — | 2 | 2 | no | — |
| 64 | #1007 | — | 6 | 8 | no | — |
| 65 | C2×D3×Q8 | — | 6 | 8 | no | — |
| 66 | C2×D3×Q8 | — | 6 | 2 | no | — |
| 67 | C22×C3⋊D3 | C24 | 1 | 1 | yes | characteristic |
| 68 | C2×D32 | C24 | 1 | 8 | yes | — |
| 69 | C2×#157 | — | 2 | 2 | no | — |
| 70 | C2×#157 | — | 4 | 3 | no | — |
| 71 | C2×#157 | C24 | 1 | 6 | yes | — |
| 72 | C2×#157 | C24 | 1 | 1 | yes | characteristic |
| 73 | C2×#157 | C24 | 1 | 1 | yes | characteristic |
| 74 | GO+(4,𝔽2) | — | 2 | 8 | no | — |
| 75 | GO+(4,𝔽2) | — | 4 | 12 | no | — |
| 76 | C32×Q8 | C2×D4 | 1 | 4 | yes | — |
| 77 | C2×C32×C4 | C2×D4 | 1 | 3 | yes | — |
| 78 | C2×C3⋊Dic3 | C2×D4 | 1 | 3 | yes | — |
| 79 | C4×C3⋊D3 | C2×D4 | 1 | 6 | yes | — |
| 80 | C4×C3⋊D3 | C24 | 1 | 6 | yes | — |
| 81 | #634 | C2×D4 | 1 | 12 | yes | — |
| 82 | C3×#67 | — | 2 | 8 | no | — |
| 83 | C3×D12 | — | 2 | 12 | no | — |
| 84 | C3×C4×D3 | — | 2 | 24 | no | — |
| 85 | C3×Dic6 | — | 2 | 12 | no | — |
| 86 | #627 | — | 2 | 12 | no | — |
| 87 | #626 | — | 2 | 8 | no | — |
| 88 | #625 | — | 2 | 12 | no | — |
| 89 | #624 | C24 | 1 | 8 | yes | — |
| 90 | D3×Dic3 | — | 2 | 24 | no | — |
| 91 | C2×2−1+4 | — | 9 | 2 | no | — |
| 92 | #550 | — | 9 | 4 | no | — |
| 93 | D4×Q8 | — | 9 | 4 | no | — |
| 94 | #548 | — | 9 | 8 | no | — |
| 95 | #537 | — | 9 | 6 | no | — |
| 96 | #532 | — | 9 | 6 | no | — |
| 97 | C2×C4×Q8 | — | 9 | 1 | no | — |
| 98 | C3×#41 | — | 6 | 12 | no | — |
| 99 | C3×#41 | — | 6 | 8 | no | — |
| 100 | C2×C3×Q8 | — | 2 | 2 | no | — |
| 101 | C2×C3×Q8 | — | 6 | 8 | no | — |
| 102 | #239 | — | 2 | 8 | no | — |
| 103 | #239 | — | 6 | 12 | no | — |
| 104 | D3×Q8 | — | 2 | 8 | no | — |
| 105 | D3×Q8 | — | 6 | 12 | no | — |
| 106 | D3×Q8 | — | 6 | 8 | no | — |
| 107 | D3×Q8 | — | 6 | 8 | no | — |
| 108 | D3×Q8 | — | 6 | 24 | no | — |
| 109 | #237 | — | 6 | 12 | no | — |
| 110 | #237 | — | 6 | 24 | no | — |
| 111 | #237 | — | 6 | 24 | no | — |
| 112 | D3×D4 | — | 6 | 12 | no | — |
| 113 | D3×D4 | — | 6 | 8 | no | — |
| 114 | #235 | — | 2 | 12 | no | — |
| 115 | #235 | — | 6 | 24 | no | — |
| 116 | #235 | — | 6 | 24 | no | — |
| 117 | C2×C4×D3 | — | 6 | 24 | no | — |
| 118 | C2×C4×D3 | — | 6 | 6 | no | — |
| 119 | C2×Dic6 | — | 6 | 6 | no | — |
| 120 | C2×Dic6 | — | 6 | 24 | no | — |
| 121 | C22×C32 | C22×D4 | 1 | 1 | yes | characteristic |
| 122 | C2×C3⋊D3 | — | 2 | 1 | no | — |
| 123 | C2×C3⋊D3 | — | 2 | 1 | no | — |
| 124 | C2×C3⋊D3 | C22×D4 | 1 | 1 | yes | characteristic |
| 125 | C2×C3⋊D3 | C25 | 1 | 1 | yes | characteristic |
| 126 | C2×C3×D3 | — | 2 | 8 | no | — |
| 127 | D32 | — | 2 | 8 | no | — |
| 128 | D32 | — | 2 | 8 | no | — |
| 129 | #157 | — | 2 | 1 | no | — |
| 130 | #157 | — | 2 | 6 | no | — |
| 131 | #157 | 2−1+4 | 1 | 2 | yes | complemented |
| 132 | C32×C4 | C22×D4 | 1 | 6 | yes | — |
| 133 | C3⋊Dic3 | C22×D4 | 1 | 6 | yes | — |
| 134 | C3×Dic3 | — | 2 | 8 | no | — |
| 135 | 2−1+4 | — | 6 | 2 | no | — |
| 136 | 2−1+4 | — | 18 | 8 | no | — |
| 137 | 2−1+4 | — | 18 | 6 | no | — |
| 138 | C2×#41 | — | 9 | 12 | no | — |
| 139 | C2×#41 | — | 9 | 8 | no | — |
| 140 | C22×Q8 | — | 9 | 8 | no | — |
| 141 | C22×Q8 | — | 9 | 1 | no | — |
| 142 | C4⋊Q8 | — | 9 | 6 | no | — |
| 143 | #127 | — | 9 | 6 | no | — |
| 144 | #124 | — | 9 | 12 | no | — |
| 145 | C22⋊Q8 | — | 9 | 24 | no | — |
| 146 | #121 | — | 9 | 4 | no | — |
| 147 | #121 | — | 9 | 12 | no | — |
| 148 | #121 | — | 9 | 24 | no | — |
| 149 | C4×Q8 | — | 9 | 4 | no | — |
| 150 | C4×Q8 | — | 9 | 4 | no | — |
| 151 | C4×D4 | — | 9 | 12 | no | — |
| 152 | C4×D4 | — | 9 | 12 | no | — |
| 153 | C2×C4⋊C4 | — | 9 | 3 | no | — |
| 154 | C2×C42 | — | 9 | 3 | no | — |
| 155 | C22×D3 | — | 6 | 8 | no | — |
| 156 | C22×D3 | — | 6 | 2 | no | — |
| 157 | C3×Q8 | — | 2 | 8 | no | — |
| 158 | C3×Q8 | — | 6 | 12 | no | — |
| 159 | C3×D4 | — | 6 | 12 | no | — |
| 160 | C3×D4 | — | 6 | 8 | no | — |
| 161 | C2×C3×C4 | — | 2 | 6 | no | — |
| 162 | C2×C3×C4 | — | 6 | 24 | no | — |
| 163 | #67 | — | 2 | 8 | no | — |
| 164 | #67 | — | 6 | 8 | no | — |
| 165 | #67 | — | 6 | 24 | no | — |
| 166 | C2×Dic3 | — | 6 | 6 | no | — |
| 167 | C2×Dic3 | — | 6 | 24 | no | — |
| 168 | D12 | — | 2 | 12 | no | — |
| 169 | D12 | — | 6 | 8 | no | — |
| 170 | C4×D3 | — | 2 | 24 | no | — |
| 171 | C4×D3 | — | 6 | 8 | no | — |
| 172 | C4×D3 | — | 6 | 12 | no | — |
| 173 | C4×D3 | — | 6 | 12 | no | — |
| 174 | C4×D3 | — | 6 | 24 | no | — |
| 175 | Dic6 | — | 2 | 12 | no | — |
| 176 | Dic6 | — | 6 | 24 | no | — |
| 177 | Dic6 | — | 6 | 24 | no | — |
| 178 | C2×C32 | — | 2 | 1 | no | — |
| 179 | C2×C32 | C23×D4 | 1 | 1 | yes | characteristic |
| 180 | C3⋊D3 | — | 2 | 1 | no | — |
| 181 | C3⋊D3 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 182 | C3⋊D3 | #585 | 1 | 1 | yes | characteristic |
| 183 | C3×D3 | — | 4 | 8 | no | — |
| 184 | #41 | — | 6 | 8 | no | — |
| 185 | #41 | — | 6 | 12 | no | — |
| 186 | #41 | — | 18 | 12 | no | — |
| 187 | #41 | — | 18 | 24 | no | — |
| 188 | #41 | — | 18 | 12 | no | — |
| 189 | #41 | — | 18 | 12 | no | — |
| 190 | C2×Q8 | — | 6 | 8 | no | — |
| 191 | C2×Q8 | — | 9 | 12 | no | — |
| 192 | C2×Q8 | — | 9 | 4 | no | — |
| 193 | C2×Q8 | — | 9 | 4 | no | — |
| 194 | C2×Q8 | — | 9 | 3 | no | — |
| 195 | C2×Q8 | — | 18 | 24 | no | — |
| 196 | C2×Q8 | GO+(4,𝔽2) | 1 | 1 | yes | characteristic, complemented |
| 197 | C2×D4 | — | 9 | 8 | no | — |
| 198 | C2×D4 | — | 9 | 12 | no | — |
| 199 | C2×D4 | — | 9 | 4 | no | — |
| 200 | C2×D4 | — | 9 | 12 | no | — |
| 201 | C22×C4 | — | 9 | 24 | no | — |
| 202 | C22×C4 | — | 9 | 3 | no | — |
| 203 | C22×C4 | — | 9 | 1 | no | — |
| 204 | C22×C4 | — | 9 | 3 | no | — |
| 205 | C4⋊C4 | — | 9 | 12 | no | — |
| 206 | C4⋊C4 | — | 9 | 4 | no | — |
| 207 | C4⋊C4 | — | 9 | 12 | no | — |
| 208 | C22⋊C4 | — | 9 | 8 | no | — |
| 209 | C22⋊C4 | — | 9 | 24 | no | — |
| 210 | C42 | — | 9 | 6 | no | — |
| 211 | C42 | — | 9 | 6 | no | — |
| 212 | C22×C3 | — | 2 | 2 | no | — |
| 213 | C22×C3 | — | 6 | 8 | no | — |
| 214 | C2×D3 | — | 2 | 8 | no | — |
| 215 | C2×D3 | — | 6 | 2 | no | — |
| 216 | C2×D3 | — | 6 | 2 | no | — |
| 217 | C2×D3 | — | 12 | 8 | no | — |
| 218 | C2×D3 | — | 12 | 8 | no | — |
| 219 | C2×D3 | — | 12 | 2 | no | — |
| 220 | C2×D3 | — | 12 | 2 | no | — |
| 221 | C3×C4 | — | 2 | 12 | no | — |
| 222 | C3×C4 | — | 6 | 8 | no | — |
| 223 | Dic3 | — | 2 | 8 | no | — |
| 224 | Dic3 | — | 6 | 12 | no | — |
| 225 | C32 | #3497 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 226 | C23 | — | 9 | 8 | no | — |
| 227 | C23 | — | 9 | 1 | no | — |
| 228 | Q8 | — | 6 | 12 | no | — |
| 229 | Q8 | — | 9 | 12 | no | — |
| 230 | Q8 | — | 18 | 12 | no | — |
| 231 | Q8 | C2×GO+(4,𝔽2) | 1 | 4 | yes | — |
| 232 | D4 | — | 6 | 8 | no | — |
| 233 | D4 | — | 6 | 12 | no | — |
| 234 | D4 | — | 18 | 8 | no | — |
| 235 | D4 | — | 18 | 12 | no | — |
| 236 | D4 | — | 18 | 8 | no | — |
| 237 | D4 | — | 36 | 12 | no | — |
| 238 | C2×C4 | — | 6 | 24 | no | — |
| 239 | C2×C4 | — | 9 | 8 | no | — |
| 240 | C2×C4 | — | 9 | 1 | no | — |
| 241 | C2×C4 | — | 9 | 3 | no | — |
| 242 | C2×C4 | — | 9 | 6 | no | — |
| 243 | C2×C4 | — | 9 | 6 | no | — |
| 244 | C2×C4 | — | 9 | 1 | no | — |
| 245 | C2×C4 | — | 9 | 6 | no | — |
| 246 | C2×C4 | — | 18 | 24 | no | — |
| 247 | C2×C4 | — | 18 | 2 | no | — |
| 248 | C2×C4 | — | 36 | 3 | no | — |
| 249 | C2×C4 | C2×GO+(4,𝔽2) | 1 | 3 | yes | — |
| 250 | C2×C3 | — | 2 | 2 | no | — |
| 251 | C2×C3 | — | 4 | 2 | no | — |
| 252 | C2×C3 | — | 12 | 8 | no | — |
| 253 | D3 | — | 4 | 8 | no | — |
| 254 | D3 | — | 6 | 2 | no | — |
| 255 | D3 | — | 6 | 2 | no | — |
| 256 | D3 | — | 12 | 2 | no | — |
| 257 | C22 | — | 6 | 8 | no | — |
| 258 | C22 | — | 9 | 1 | no | — |
| 259 | C22 | — | 9 | 1 | no | — |
| 260 | C22 | — | 18 | 1 | no | — |
| 261 | C22 | — | 18 | 1 | no | — |
| 262 | C22 | — | 18 | 8 | no | — |
| 263 | C22 | — | 18 | 8 | no | — |
| 264 | C22 | C22×GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| 265 | C4 | — | 6 | 8 | no | — |
| 266 | C4 | — | 9 | 2 | no | — |
| 267 | C4 | — | 9 | 6 | no | — |
| 268 | C4 | — | 18 | 1 | no | — |
| 269 | C4 | — | 18 | 6 | no | — |
| 270 | C4 | C22×GO+(4,𝔽2) | 1 | 6 | yes | — |
| 271 | C3 | — | 2 | 2 | no | — |
| 272 | C2 | — | 2 | 1 | no | — |
| 273 | C2 | — | 9 | 1 | no | — |
| 274 | C2 | — | 9 | 1 | no | — |
| 275 | C2 | — | 12 | 8 | no | — |
| 276 | C2 | — | 18 | 1 | no | — |
| 277 | C2 | C23×GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×GO+(4,𝔽2) |
| commutator, cosocle | C2×C3⋊D3 | C25 |
| Fitting | C2×C32×Q8 | D4 |
| socle | C2×C32 | C23×D4 |
| hypercenter | C2×Q8 | GO+(4,𝔽2) |
| hypocenter | C32 | #3497 |
| LCS | G ▷ C2×C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10596687 | C2 | semidirect |
| 2. C2×Q8×#157 | C2 | semidirect |
| 3. #10596525 | C2 | semidirect |
| 4. #10596524 | C2 | semidirect |
| 5. Q8×GO+(4,𝔽2) | C2 | semidirect |
| 6. #10596513 | C2 | semidirect |
| 7. #10596511 | C2 | semidirect |
| 33. C2×C32×Q8 | D4 | semidirect |
| 34. Q8×C3⋊D3 | D4 | semidirect |
| 37. C2×#634 | D4 | semidirect |
| 131. #157 | 2−1+4 | semidirect |
| 196. C2×Q8 | GO+(4,𝔽2) | semidirect |
| 225. C32 | #3497 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10596687 | 2 |
| C2×Q8×#157 | 2 |
| #10596525 | 2 |
| #10596524 | 2 |
| Q8×GO+(4,𝔽2) | 2 |
| #10596513 | 2 |
| #10596511 | 2 |
| #3497 | 9 |