Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
390 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 53 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×C3⋊#994 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#568⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #5819⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #83813 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #2888⋊C3 | — | 3 | 1 | no | maximal |
| 6 | C2×C3×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 7 | C3⋊#994 | — | 2 | 1 | no | — |
| 8 | C3⋊#994 | — | 2 | 1 | no | — |
| 9 | C2×C3⋊(#41⋊C3) | C4 | 1 | 2 | yes | — |
| 10 | C3×#131⋊C3 | — | 2 | 1 | no | — |
| 11 | C3⋊(#131⋊C3) | — | 2 | 1 | no | — |
| 12 | C3×#824 | — | 4 | 1 | no | maximal |
| 13 | C2×#1009 | C2×C3 | 1 | 1 | yes | characteristic |
| 14 | C2×#994 | — | 3 | 1 | no | — |
| 15 | C3×#568 | C2×C3 | 1 | 1 | yes | characteristic |
| 16 | #5819 | C2×C3 | 1 | 1 | yes | characteristic |
| 17 | #568⋊C3 | — | 2 | 1 | no | — |
| 18 | #568⋊C3 | — | 3 | 1 | no | — |
| 19 | #568⋊C3 | D3 | 1 | 1 | yes | characteristic, complemented |
| 20 | #5132 | — | 3 | 1 | no | — |
| 21 | #5132 | — | 3 | 1 | no | — |
| 22 | C2×#823 | — | 3 | 1 | no | — |
| 23 | C2×#823 | — | 3 | 1 | no | — |
| 24 | C2×C3×Dic6 | — | 4 | 1 | no | — |
| 25 | C3×#41⋊C3 | D4 | 1 | 1 | yes | characteristic |
| 26 | C3×#41⋊C3 | D4 | 1 | 1 | yes | characteristic |
| 27 | C2×C3×SL(2,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C3⋊(#41⋊C3) | — | 2 | 2 | no | — |
| 29 | C32×M16 | — | 4 | 1 | no | — |
| 30 | C3×#208 | — | 4 | 1 | no | — |
| 31 | #2888 | — | 3 | 1 | no | Sylow |
| 32 | C2×C3×#41 | C22×C3 | 1 | 1 | yes | characteristic |
| 33 | #1009 | — | 2 | 1 | no | — |
| 34 | #1009 | — | 2 | 1 | no | — |
| 35 | #1009 | — | 6 | 1 | no | — |
| 36 | #1009 | — | 6 | 1 | no | — |
| 37 | C2×#239 | C3×C4 | 1 | 2 | yes | — |
| 38 | C2×D3×D4 | — | 3 | 2 | no | — |
| 39 | C2×#235 | — | 3 | 1 | no | — |
| 40 | C22×D12 | — | 3 | 1 | no | — |
| 41 | #994 | — | 6 | 1 | no | — |
| 42 | #994 | — | 6 | 1 | no | — |
| 43 | C2×#41⋊C3 | — | 2 | 1 | no | — |
| 44 | C2×#41⋊C3 | — | 3 | 2 | no | — |
| 45 | C2×#41⋊C3 | C2×D3 | 1 | 1 | yes | characteristic |
| 46 | C3×#131 | — | 2 | 1 | no | — |
| 47 | C3×#131 | — | 6 | 1 | no | — |
| 48 | C2×C3×M16 | — | 3 | 1 | no | — |
| 49 | C2×C3×M16 | — | 3 | 1 | no | — |
| 50 | #948 | — | 2 | 1 | no | — |
| 51 | #948 | — | 6 | 1 | no | — |
| 52 | C2×#208 | — | 3 | 1 | no | — |
| 53 | C2×#208 | — | 3 | 1 | no | — |
| 54 | #131⋊C3 | — | 2 | 2 | no | — |
| 55 | #131⋊C3 | — | 2 | 1 | no | — |
| 56 | #131⋊C3 | — | 6 | 1 | no | — |
| 57 | C3×#101 | — | 12 | 1 | no | — |
| 58 | #824 | — | 3 | 1 | no | — |
| 59 | #824 | A4 | 1 | 1 | yes | characteristic |
| 60 | #823 | — | 3 | 2 | no | — |
| 61 | #823 | — | 3 | 1 | no | — |
| 62 | #823 | — | 3 | 1 | no | — |
| 63 | C2×C32×C4 | — | 4 | 1 | no | — |
| 64 | C2×C3×Dic3 | — | 4 | 2 | no | — |
| 65 | C3×Dic6 | — | 8 | 1 | no | — |
| 66 | C3×Dic6 | — | 8 | 1 | no | — |
| 67 | C3×SL(2,𝔽3) | C22⋊C4 | 1 | 1 | yes | characteristic |
| 68 | C32×C8 | — | 8 | 1 | no | — |
| 69 | C3×C3⋊C8 | — | 8 | 1 | no | — |
| 70 | C2×H(5,𝔽2) | — | 3 | 1 | no | — |
| 71 | #568 | — | 3 | 1 | no | — |
| 72 | #568 | C3×D3 | 1 | 1 | yes | characteristic |
| 73 | #413 | — | 9 | 1 | no | — |
| 74 | #413 | — | 9 | 1 | no | — |
| 75 | C2×#100 | — | 9 | 1 | no | — |
| 76 | C2×#100 | — | 9 | 1 | no | — |
| 77 | C23×D3 | — | 3 | 2 | no | — |
| 78 | C3×#41 | — | 3 | 1 | no | — |
| 79 | C3×#41 | — | 3 | 1 | no | — |
| 80 | C3×#41 | C3×D4 | 1 | 1 | yes | characteristic |
| 81 | C3×#41 | C3×D4 | 1 | 1 | yes | characteristic |
| 82 | C2×C3×Q8 | — | 12 | 1 | no | — |
| 83 | C2×C3×Q8 | C2×C3×C4 | 1 | 1 | yes | characteristic |
| 84 | C2×C3×D4 | — | 3 | 1 | no | — |
| 85 | C22×C3×C4 | — | 3 | 1 | no | — |
| 86 | C2×#67 | — | 3 | 2 | no | — |
| 87 | #239 | — | 2 | 2 | no | — |
| 88 | #239 | — | 6 | 2 | no | — |
| 89 | D3×D4 | — | 6 | 4 | no | — |
| 90 | D3×D4 | — | 6 | 2 | no | — |
| 91 | D3×D4 | — | 6 | 2 | no | — |
| 92 | #235 | — | 6 | 1 | no | — |
| 93 | #235 | — | 6 | 1 | no | — |
| 94 | #235 | — | 6 | 1 | no | — |
| 95 | #235 | — | 6 | 1 | no | — |
| 96 | C2×D12 | — | 3 | 2 | no | — |
| 97 | C2×D12 | — | 3 | 1 | no | — |
| 98 | C2×D12 | — | 6 | 1 | no | — |
| 99 | C2×D12 | — | 6 | 1 | no | — |
| 100 | C2×D12 | — | 6 | 1 | no | — |
| 101 | C2×D12 | — | 6 | 1 | no | — |
| 102 | C2×C4×D3 | — | 3 | 2 | no | — |
| 103 | C2×Dic6 | C2×A4 | 1 | 1 | yes | characteristic |
| 104 | #41⋊C3 | — | 2 | 1 | no | — |
| 105 | #41⋊C3 | — | 2 | 1 | no | — |
| 106 | #41⋊C3 | — | 6 | 2 | no | — |
| 107 | #41⋊C3 | D12 | 1 | 1 | yes | characteristic |
| 108 | #41⋊C3 | #67 | 1 | 1 | yes | characteristic |
| 109 | C2×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 110 | C2×SL(2,𝔽3) | C4×D3 | 1 | 1 | yes | characteristic |
| 111 | C3×M16 | — | 3 | 1 | no | — |
| 112 | C3×M16 | — | 4 | 1 | no | — |
| 113 | C3×M16 | — | 6 | 1 | no | — |
| 114 | C3×M16 | — | 6 | 1 | no | — |
| 115 | C3×M16 | — | 8 | 1 | no | — |
| 116 | C3×M16 | — | 12 | 1 | no | — |
| 117 | C3×M16 | C2×A4 | 1 | 1 | yes | characteristic |
| 118 | C2×C3×C8 | — | 6 | 1 | no | — |
| 119 | C2×C3×C8 | — | 6 | 1 | no | — |
| 120 | #208 | — | 3 | 1 | no | — |
| 121 | #208 | — | 6 | 1 | no | — |
| 122 | #208 | — | 6 | 1 | no | — |
| 123 | #208 | C2×A4 | 1 | 1 | yes | characteristic |
| 124 | C2×C3⋊C8 | — | 6 | 1 | no | — |
| 125 | C2×C3⋊C8 | — | 6 | 1 | no | — |
| 126 | C22×C32 | — | 4 | 1 | no | — |
| 127 | C32×C4 | — | 4 | 1 | no | — |
| 128 | C32×C4 | — | 4 | 1 | no | — |
| 129 | C3×Dic3 | — | 8 | 2 | no | — |
| 130 | H(5,𝔽2) | — | 6 | 1 | no | — |
| 131 | H(5,𝔽2) | — | 6 | 1 | no | — |
| 132 | H(5,𝔽2) | — | 18 | 1 | no | — |
| 133 | H(5,𝔽2) | — | 18 | 1 | no | — |
| 134 | C2×#41 | — | 3 | 2 | no | — |
| 135 | C2×#41 | — | 9 | 1 | no | — |
| 136 | C2×#41 | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 137 | C22×D4 | — | 9 | 1 | no | — |
| 138 | C22×D4 | — | 9 | 2 | no | — |
| 139 | #131 | — | 2 | 1 | no | — |
| 140 | #131 | — | 6 | 1 | no | — |
| 141 | #131 | — | 6 | 1 | no | — |
| 142 | #131 | — | 18 | 1 | no | — |
| 143 | C2×M16 | — | 3 | 1 | no | — |
| 144 | C2×M16 | — | 3 | 1 | no | — |
| 145 | C2×M16 | — | 9 | 1 | no | — |
| 146 | C2×M16 | — | 9 | 1 | no | — |
| 147 | #101 | — | 3 | 1 | no | — |
| 148 | #101 | — | 9 | 1 | no | — |
| 149 | #100 | — | 9 | 1 | no | — |
| 150 | #100 | — | 9 | 2 | no | — |
| 151 | #100 | — | 9 | 1 | no | — |
| 152 | C23×C3 | — | 3 | 1 | no | — |
| 153 | C22×D3 | — | 3 | 2 | no | — |
| 154 | C22×D3 | — | 6 | 2 | no | — |
| 155 | C22×D3 | — | 6 | 2 | no | — |
| 156 | C22×D3 | — | 24 | 2 | no | — |
| 157 | C3×Q8 | — | 3 | 1 | no | — |
| 158 | C3×Q8 | — | 24 | 1 | no | — |
| 159 | C3×Q8 | — | 24 | 1 | no | — |
| 160 | C3×Q8 | C3×C22⋊C4 | 1 | 1 | yes | characteristic |
| 161 | C3×D4 | — | 3 | 1 | no | — |
| 162 | C3×D4 | — | 3 | 1 | no | — |
| 163 | C3×D4 | — | 3 | 2 | no | — |
| 164 | C2×C3×C4 | — | 3 | 1 | no | — |
| 165 | C2×C3×C4 | — | 3 | 1 | no | — |
| 166 | C2×C3×C4 | — | 3 | 1 | no | — |
| 167 | C2×C3×C4 | — | 3 | 1 | no | — |
| 168 | C2×C3×C4 | — | 3 | 1 | no | — |
| 169 | C2×C3×C4 | — | 4 | 1 | no | — |
| 170 | C2×C3×C4 | — | 8 | 1 | no | — |
| 171 | C2×C3×C4 | — | 12 | 2 | no | — |
| 172 | C2×C3×C4 | C22×A4 | 1 | 1 | yes | characteristic |
| 173 | #67 | — | 6 | 2 | no | — |
| 174 | #67 | — | 6 | 2 | no | — |
| 175 | C2×Dic3 | C4×A4 | 1 | 2 | yes | — |
| 176 | D12 | — | 6 | 2 | no | — |
| 177 | D12 | — | 6 | 2 | no | — |
| 178 | D12 | — | 6 | 1 | no | — |
| 179 | D12 | — | 6 | 1 | no | — |
| 180 | C4×D3 | — | 6 | 2 | no | — |
| 181 | C4×D3 | — | 6 | 2 | no | — |
| 182 | Dic6 | — | 2 | 1 | no | — |
| 183 | Dic6 | — | 2 | 1 | no | — |
| 184 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 185 | SL(2,𝔽3) | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 186 | C3×C8 | — | 2 | 1 | no | — |
| 187 | C3×C8 | — | 6 | 1 | no | — |
| 188 | C3×C8 | — | 8 | 1 | no | — |
| 189 | C3×C8 | — | 8 | 2 | no | — |
| 190 | C3×C8 | — | 24 | 1 | no | — |
| 191 | C3⋊C8 | — | 2 | 1 | no | — |
| 192 | C3⋊C8 | — | 6 | 1 | no | — |
| 193 | C2×C32 | — | 4 | 1 | no | — |
| 194 | C2×C32 | — | 8 | 1 | no | — |
| 195 | C24 | — | 9 | 2 | no | — |
| 196 | #41 | — | 3 | 1 | no | — |
| 197 | #41 | — | 3 | 1 | no | — |
| 198 | #41 | — | 6 | 2 | no | — |
| 199 | #41 | — | 18 | 1 | no | — |
| 200 | #41 | — | 18 | 1 | no | — |
| 201 | #41 | — | 18 | 1 | no | — |
| 202 | #41 | — | 18 | 1 | no | — |
| 203 | #41 | — | 18 | 2 | no | — |
| 204 | #41 | C3×D12 | 1 | 1 | yes | characteristic |
| 205 | #41 | C3×#67 | 1 | 1 | yes | characteristic |
| 206 | C2×Q8 | — | 3 | 1 | no | — |
| 207 | C2×Q8 | C3×C4×D3 | 1 | 1 | yes | characteristic |
| 208 | C2×D4 | — | 3 | 1 | no | — |
| 209 | C2×D4 | — | 9 | 2 | no | — |
| 210 | C2×D4 | — | 9 | 2 | no | — |
| 211 | C2×D4 | — | 9 | 1 | no | — |
| 212 | C2×D4 | — | 18 | 1 | no | — |
| 213 | C2×D4 | — | 18 | 1 | no | — |
| 214 | C2×D4 | — | 18 | 1 | no | — |
| 215 | C2×D4 | — | 18 | 1 | no | — |
| 216 | C2×D4 | — | 18 | 4 | no | — |
| 217 | C2×D4 | — | 18 | 2 | no | — |
| 218 | C2×D4 | — | 18 | 2 | no | — |
| 219 | C22×C4 | — | 3 | 1 | no | — |
| 220 | C22×C4 | — | 9 | 2 | no | — |
| 221 | M16 | — | 3 | 1 | no | — |
| 222 | M16 | — | 3 | 1 | no | — |
| 223 | M16 | — | 6 | 1 | no | — |
| 224 | M16 | — | 6 | 1 | no | — |
| 225 | M16 | — | 9 | 1 | no | — |
| 226 | M16 | — | 18 | 1 | no | — |
| 227 | M16 | — | 18 | 1 | no | — |
| 228 | M16 | D3×A4 | 1 | 1 | yes | characteristic |
| 229 | C2×C8 | — | 6 | 1 | no | — |
| 230 | C2×C8 | — | 6 | 1 | no | — |
| 231 | C2×C8 | — | 18 | 1 | no | — |
| 232 | C2×C8 | — | 18 | 1 | no | — |
| 233 | C22×C3 | — | 3 | 1 | no | — |
| 234 | C22×C3 | — | 3 | 1 | no | — |
| 235 | C22×C3 | — | 4 | 1 | no | — |
| 236 | C22×C3 | — | 8 | 1 | no | — |
| 237 | C22×C3 | — | 12 | 1 | no | — |
| 238 | C22×C3 | C2×C4×A4 | 1 | 1 | yes | characteristic |
| 239 | C2×D3 | — | 6 | 2 | no | — |
| 240 | C2×D3 | — | 12 | 2 | no | — |
| 241 | C2×D3 | — | 24 | 2 | no | — |
| 242 | C2×D3 | — | 24 | 2 | no | — |
| 243 | C3×C4 | — | 3 | 1 | no | — |
| 244 | C3×C4 | — | 3 | 1 | no | — |
| 245 | C3×C4 | — | 4 | 1 | no | — |
| 246 | C3×C4 | — | 4 | 1 | no | — |
| 247 | C3×C4 | — | 8 | 1 | no | — |
| 248 | C3×C4 | — | 8 | 1 | no | — |
| 249 | C3×C4 | — | 24 | 2 | no | — |
| 250 | C3×C4 | D4×A4 | 1 | 1 | yes | characteristic |
| 251 | C3×C4 | D4×A4 | 1 | 1 | yes | characteristic |
| 252 | Dic3 | — | 2 | 2 | no | — |
| 253 | C32 | — | 4 | 1 | no | Sylow |
| 254 | C23 | — | 3 | 1 | no | — |
| 255 | C23 | — | 9 | 2 | no | — |
| 256 | C23 | — | 18 | 2 | no | — |
| 257 | C23 | — | 18 | 2 | no | — |
| 258 | C23 | — | 72 | 2 | no | — |
| 259 | Q8 | — | 3 | 1 | no | — |
| 260 | Q8 | — | 6 | 1 | no | — |
| 261 | Q8 | — | 6 | 1 | no | — |
| 262 | Q8 | C3×(C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 263 | D4 | — | 3 | 1 | no | — |
| 264 | D4 | — | 3 | 1 | no | — |
| 265 | D4 | — | 3 | 2 | no | — |
| 266 | D4 | — | 18 | 2 | no | — |
| 267 | D4 | — | 18 | 2 | no | — |
| 268 | D4 | — | 18 | 2 | no | — |
| 269 | D4 | — | 18 | 2 | no | — |
| 270 | D4 | — | 18 | 1 | no | — |
| 271 | D4 | — | 18 | 1 | no | — |
| 272 | C2×C4 | — | 3 | 1 | no | — |
| 273 | C2×C4 | — | 3 | 1 | no | — |
| 274 | C2×C4 | — | 3 | 1 | no | — |
| 275 | C2×C4 | — | 3 | 1 | no | — |
| 276 | C2×C4 | — | 3 | 1 | no | — |
| 277 | C2×C4 | — | 3 | 2 | no | — |
| 278 | C2×C4 | — | 18 | 2 | no | — |
| 279 | C2×C4 | — | 18 | 2 | no | — |
| 280 | C2×C4 | C2×D3×A4 | 1 | 1 | yes | characteristic |
| 281 | C8 | — | 2 | 1 | no | — |
| 282 | C8 | — | 6 | 1 | no | — |
| 283 | C8 | — | 6 | 1 | no | — |
| 284 | C8 | — | 18 | 1 | no | — |
| 285 | C2×C3 | — | 2 | 1 | no | — |
| 286 | C2×C3 | — | 4 | 1 | no | — |
| 287 | C2×C3 | — | 6 | 1 | no | — |
| 288 | C2×C3 | — | 6 | 1 | no | — |
| 289 | C2×C3 | — | 8 | 1 | no | — |
| 290 | C2×C3 | — | 8 | 1 | no | — |
| 291 | C2×C3 | — | 16 | 1 | no | — |
| 292 | C2×C3 | A4×C22⋊C4 | 1 | 1 | yes | characteristic |
| 293 | D3 | — | 12 | 2 | no | — |
| 294 | C22 | — | 3 | 1 | no | — |
| 295 | C22 | — | 3 | 1 | no | — |
| 296 | C22 | — | 12 | 1 | no | — |
| 297 | C22 | — | 18 | 2 | no | — |
| 298 | C22 | — | 36 | 2 | no | — |
| 299 | C22 | — | 72 | 2 | no | — |
| 300 | C22 | — | 72 | 2 | no | — |
| 301 | C22 | C4×D3×A4 | 1 | 1 | yes | characteristic |
| 302 | C4 | — | 3 | 1 | no | — |
| 303 | C4 | — | 3 | 1 | no | — |
| 304 | C4 | — | 6 | 2 | no | — |
| 305 | C4 | A4×D12 | 1 | 1 | yes | characteristic |
| 306 | C4 | A4×#67 | 1 | 1 | yes | characteristic |
| 307 | C3 | — | 4 | 1 | no | — |
| 308 | C3 | — | 8 | 1 | no | — |
| 309 | C3 | #2888⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 310 | C2 | — | 2 | 1 | no | — |
| 311 | C2 | — | 6 | 1 | no | — |
| 312 | C2 | — | 6 | 1 | no | — |
| 313 | C2 | — | 36 | 2 | no | — |
| 314 | C2 | A4×(C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#568⋊C3 | C2 | semidirect |
| 3. #5819⋊C3 | C2 | semidirect |
| 4. #83813 | C3 | semidirect |
| 19. #568⋊C3 | D3 | semidirect |
| 309. C3 | #2888⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3⋊#994 | 2 |
| C3×#568⋊C3 | 2 |
| #5819⋊C3 | 2 |
| #83813 | 3 |
| #2888⋊C3 | 3 |
| C3×#824 | 4 |