Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
1764 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 1764 conjugacy classes of subgroups — too many to draw. Only its 185 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, complemented |
| 2 | #10596587 | C2 | 1 | 8 | yes | maximal, direct product factor |
| 3 | C2×#64266 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C2×#64265 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C2×#5924 | — | 3 | 3 | no | maximal |
| 6 | C2×#5836 | — | 3 | 1 | no | maximal |
| 7 | C2×Q8×C3⋊D3 | — | 4 | 1 | no | maximal |
| 8 | C3×#994 | C22 | 1 | 4 | yes | complemented |
| 9 | C2×C3×#41⋊C3 | C22 | 1 | 3 | yes | — |
| 10 | #64266 | C22 | 1 | 12 | yes | complemented |
| 11 | #64265 | C22 | 1 | 12 | yes | complemented |
| 12 | C2×C3⋊GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 13 | C2×C3⋊#226 | C22 | 1 | 3 | yes | complemented |
| 14 | C2×C3×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 15 | C2×#1010 | — | 3 | 1 | no | — |
| 16 | C2×#994 | D3 | 1 | 3 | yes | complemented |
| 17 | #5924 | — | 3 | 24 | no | — |
| 18 | C2×#985 | — | 3 | 9 | no | — |
| 19 | C2×#984 | — | 3 | 9 | no | — |
| 20 | #5836 | — | 3 | 8 | no | — |
| 21 | #5836 | — | 3 | 8 | no | — |
| 22 | C2×#951 | — | 3 | 3 | no | — |
| 23 | C2×#950 | — | 3 | 3 | no | — |
| 24 | C2×#948 | — | 3 | 1 | no | — |
| 25 | C22×#214 | — | 3 | 3 | no | — |
| 26 | C2×#932 | — | 3 | 3 | no | — |
| 27 | C2×C32×Q8 | — | 4 | 1 | no | — |
| 28 | Q8×C3⋊D3 | — | 4 | 8 | no | — |
| 29 | C2×C4×C3⋊D3 | — | 4 | 3 | no | — |
| 30 | C2×#634 | — | 4 | 3 | no | — |
| 31 | C3×#41⋊C3 | C23 | 1 | 6 | yes | — |
| 32 | C2×C3×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 33 | C3⋊GL(2,𝔽3) | C23 | 1 | 2 | yes | complemented |
| 34 | C3⋊#226 | C23 | 1 | 6 | yes | complemented |
| 35 | C2×#577 | — | 9 | 1 | no | Sylow |
| 36 | C3×H(5,𝔽2) | — | 3 | 4 | no | — |
| 37 | C3×H(5,𝔽2) | C2×D3 | 1 | 4 | yes | complemented |
| 38 | C2×C3×#41 | — | 3 | 1 | no | — |
| 39 | C2×C3×#41 | C2×D3 | 1 | 3 | yes | — |
| 40 | C22×C3×D4 | — | 3 | 3 | no | — |
| 41 | #1010 | — | 3 | 8 | no | — |
| 42 | #1010 | — | 6 | 4 | no | — |
| 43 | C2×D3×Q8 | — | 6 | 1 | no | — |
| 44 | C2×D3×Q8 | — | 12 | 3 | no | — |
| 45 | C2×#237 | — | 6 | 3 | no | — |
| 46 | C2×#235 | — | 3 | 3 | no | — |
| 47 | C22×Dic6 | — | 3 | 3 | no | — |
| 48 | #994 | C2×D3 | 1 | 12 | yes | complemented |
| 49 | C2×#41⋊C3 | C2×D3 | 1 | 9 | yes | — |
| 50 | #985 | — | 3 | 36 | no | — |
| 51 | #984 | — | 3 | 36 | no | — |
| 52 | C2×GL(2,𝔽3) | — | 3 | 3 | no | — |
| 53 | C2×#226 | — | 3 | 9 | no | — |
| 54 | #951 | — | 3 | 12 | no | — |
| 55 | #951 | — | 3 | 12 | no | — |
| 56 | #950 | — | 3 | 12 | no | — |
| 57 | #950 | — | 3 | 12 | no | — |
| 58 | #948 | — | 3 | 8 | no | — |
| 59 | C2×#216 | — | 3 | 3 | no | — |
| 60 | C2×#215 | — | 3 | 1 | no | — |
| 61 | C2×#214 | — | 3 | 6 | no | — |
| 62 | C2×#214 | — | 3 | 3 | no | — |
| 63 | C2×#214 | — | 3 | 24 | no | — |
| 64 | #932 | — | 3 | 24 | no | — |
| 65 | C2×#213 | — | 3 | 3 | no | — |
| 66 | C2×#208 | — | 3 | 3 | no | — |
| 67 | C22×C3⋊C8 | — | 3 | 3 | no | — |
| 68 | C22×C3⋊D3 | — | 4 | 1 | no | — |
| 69 | C32×Q8 | — | 4 | 4 | no | — |
| 70 | C2×C32×C4 | — | 4 | 3 | no | — |
| 71 | C2×C3⋊Dic3 | — | 4 | 3 | no | — |
| 72 | C4×C3⋊D3 | — | 4 | 12 | no | — |
| 73 | #634 | — | 4 | 12 | no | — |
| 74 | C3×SL(2,𝔽3) | C24 | 1 | 1 | yes | characteristic |
| 75 | C2×2−1+4 | — | 9 | 1 | no | — |
| 76 | C2×H(5,𝔽2) | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 77 | #577 | — | 9 | 8 | no | — |
| 78 | #577 | — | 9 | 8 | no | — |
| 79 | C2×#137 | — | 9 | 3 | no | — |
| 80 | C2×AGL(1,ℤ/8) | — | 9 | 3 | no | — |
| 81 | C2×#135 | — | 9 | 3 | no | — |
| 82 | C22×SD16 | — | 9 | 3 | no | — |
| 83 | C2×#131 | — | 9 | 1 | no | — |
| 84 | C24×C3 | — | 6 | 1 | no | — |
| 85 | C3×#41 | — | 3 | 4 | no | — |
| 86 | C3×#41 | — | 3 | 4 | no | — |
| 87 | C3×#41 | — | 3 | 6 | no | — |
| 88 | C3×#41 | C22×D3 | 1 | 6 | yes | — |
| 89 | C2×C3×Q8 | — | 4 | 3 | no | — |
| 90 | C2×C3×Q8 | S4 | 1 | 1 | yes | characteristic |
| 91 | C2×C3×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 92 | C2×C3×D4 | — | 3 | 12 | no | — |
| 93 | C2×C3×D4 | — | 3 | 3 | no | — |
| 94 | C2×C3×D4 | — | 3 | 3 | no | — |
| 95 | C2×C3×D4 | — | 6 | 6 | no | — |
| 96 | C22×C3×C4 | — | 3 | 3 | no | — |
| 97 | C2×#67 | — | 6 | 3 | no | — |
| 98 | C22×Dic3 | — | 6 | 3 | no | — |
| 99 | D3×Q8 | — | 6 | 8 | no | — |
| 100 | D3×Q8 | — | 12 | 24 | no | — |
| 101 | #237 | — | 6 | 24 | no | — |
| 102 | #235 | — | 3 | 12 | no | — |
| 103 | #235 | — | 6 | 6 | no | — |
| 104 | C2×D12 | — | 3 | 1 | no | — |
| 105 | C2×C4×D3 | — | 6 | 3 | no | — |
| 106 | C2×C4×D3 | — | 12 | 9 | no | — |
| 107 | C2×Dic6 | — | 3 | 3 | no | — |
| 108 | C2×Dic6 | — | 3 | 12 | no | — |
| 109 | C2×Dic6 | — | 6 | 6 | no | — |
| 110 | C2×Dic6 | — | 6 | 3 | no | — |
| 111 | C2×Dic6 | — | 12 | 9 | no | — |
| 112 | #41⋊C3 | C22×D3 | 1 | 18 | yes | — |
| 113 | C2×SL(2,𝔽3) | C22×D3 | 1 | 3 | yes | — |
| 114 | GL(2,𝔽3) | — | 3 | 6 | no | — |
| 115 | #226 | — | 3 | 18 | no | — |
| 116 | #216 | — | 3 | 6 | no | — |
| 117 | #216 | — | 3 | 6 | no | — |
| 118 | #215 | — | 3 | 2 | no | — |
| 119 | #215 | — | 3 | 2 | no | — |
| 120 | #214 | — | 3 | 24 | no | — |
| 121 | #214 | — | 3 | 12 | no | — |
| 122 | #213 | — | 3 | 12 | no | — |
| 123 | #208 | — | 3 | 12 | no | — |
| 124 | C2×C3⋊C8 | — | 3 | 1 | no | — |
| 125 | C2×C3⋊C8 | — | 3 | 3 | no | — |
| 126 | C2×C3⋊C8 | — | 3 | 12 | no | — |
| 127 | C22×C32 | — | 4 | 1 | no | — |
| 128 | C2×C3⋊D3 | — | 4 | 2 | no | — |
| 129 | C2×C3⋊D3 | — | 4 | 4 | no | — |
| 130 | C32×C4 | — | 4 | 6 | no | — |
| 131 | C3⋊Dic3 | — | 4 | 6 | no | — |
| 132 | 2−1+4 | — | 9 | 8 | no | — |
| 133 | 2−1+4 | — | 18 | 4 | no | — |
| 134 | H(5,𝔽2) | — | 3 | 4 | no | — |
| 135 | H(5,𝔽2) | C2×C3⋊D3 | 1 | 4 | yes | complemented |
| 136 | C2×#41 | — | 3 | 1 | no | — |
| 137 | C2×#41 | — | 9 | 3 | no | — |
| 138 | C2×#41 | — | 18 | 3 | no | — |
| 139 | C2×#41 | C2×C3⋊D3 | 1 | 3 | yes | — |
| 140 | C22×Q8 | — | 9 | 3 | no | — |
| 141 | C22×Q8 | — | 18 | 1 | no | — |
| 142 | C22×D4 | — | 3 | 3 | no | — |
| 143 | #137 | — | 9 | 12 | no | — |
| 144 | #137 | — | 9 | 12 | no | — |
| 145 | AGL(1,ℤ/8) | — | 9 | 24 | no | — |
| 146 | #135 | — | 9 | 12 | no | — |
| 147 | #135 | — | 9 | 12 | no | — |
| 148 | C2×Q16 | — | 9 | 3 | no | — |
| 149 | C2×SD16 | — | 9 | 3 | no | — |
| 150 | C2×SD16 | — | 9 | 6 | no | — |
| 151 | C2×SD16 | — | 9 | 24 | no | — |
| 152 | C2×SD16 | — | 9 | 1 | no | — |
| 153 | C2×D8 | — | 9 | 3 | no | — |
| 154 | #131 | — | 9 | 8 | no | — |
| 155 | C2×M16 | — | 9 | 3 | no | — |
| 156 | C22×C8 | — | 9 | 3 | no | — |
| 157 | C23×C3 | — | 3 | 3 | no | — |
| 158 | C23×C3 | — | 6 | 4 | no | — |
| 159 | C23×C3 | — | 24 | 2 | no | — |
| 160 | C22×D3 | — | 6 | 1 | no | — |
| 161 | C22×D3 | — | 12 | 3 | no | — |
| 162 | C3×Q8 | — | 3 | 1 | no | — |
| 163 | C3×Q8 | — | 4 | 12 | no | — |
| 164 | C3×Q8 | C2×S4 | 1 | 4 | yes | — |
| 165 | C3×Q8 | C23×D3 | 1 | 1 | yes | characteristic |
| 166 | C3×D4 | — | 3 | 6 | no | — |
| 167 | C3×D4 | — | 3 | 12 | no | — |
| 168 | C3×D4 | — | 6 | 3 | no | — |
| 169 | C2×C3×C4 | — | 3 | 1 | no | — |
| 170 | C2×C3×C4 | — | 3 | 6 | no | — |
| 171 | C2×C3×C4 | — | 3 | 6 | no | — |
| 172 | C2×C3×C4 | — | 4 | 9 | no | — |
| 173 | C2×C3×C4 | C2×S4 | 1 | 3 | yes | — |
| 174 | #67 | — | 6 | 12 | no | — |
| 175 | C2×Dic3 | — | 6 | 12 | no | — |
| 176 | C2×Dic3 | — | 6 | 3 | no | — |
| 177 | C2×Dic3 | — | 12 | 9 | no | — |
| 178 | D12 | — | 3 | 2 | no | — |
| 179 | D12 | — | 6 | 1 | no | — |
| 180 | C4×D3 | — | 6 | 12 | no | — |
| 181 | C4×D3 | — | 12 | 36 | no | — |
| 182 | Dic6 | — | 3 | 6 | no | — |
| 183 | Dic6 | — | 6 | 3 | no | — |
| 184 | Dic6 | — | 6 | 12 | no | — |
| 185 | Dic6 | — | 12 | 36 | no | — |
| 186 | SL(2,𝔽3) | C23×D3 | 1 | 3 | yes | — |
| 187 | C3⋊C8 | — | 3 | 2 | no | — |
| 188 | C3⋊C8 | — | 3 | 6 | no | — |
| 189 | C2×C32 | — | 4 | 2 | no | — |
| 190 | C2×C32 | — | 4 | 1 | no | — |
| 191 | C3⋊D3 | — | 4 | 4 | no | — |
| 192 | C24 | — | 6 | 1 | no | — |
| 193 | #41 | — | 3 | 4 | no | — |
| 194 | #41 | — | 3 | 4 | no | — |
| 195 | #41 | — | 3 | 6 | no | — |
| 196 | #41 | — | 9 | 12 | no | — |
| 197 | #41 | — | 18 | 6 | no | — |
| 198 | #41 | — | 18 | 24 | no | — |
| 199 | #41 | C22×C3⋊D3 | 1 | 6 | yes | — |
| 200 | C2×Q8 | — | 9 | 3 | no | — |
| 201 | C2×Q8 | — | 9 | 12 | no | — |
| 202 | C2×Q8 | — | 18 | 6 | no | — |
| 203 | C2×Q8 | — | 18 | 8 | no | — |
| 204 | C2×Q8 | — | 18 | 3 | no | — |
| 205 | C2×Q8 | C3⋊S4 | 1 | 1 | yes | characteristic |
| 206 | C2×Q8 | C22×C3⋊D3 | 1 | 1 | yes | characteristic |
| 207 | C2×D4 | — | 3 | 12 | no | — |
| 208 | C2×D4 | — | 3 | 3 | no | — |
| 209 | C2×D4 | — | 3 | 3 | no | — |
| 210 | C2×D4 | — | 6 | 6 | no | — |
| 211 | C2×D4 | — | 9 | 1 | no | — |
| 212 | C2×D4 | — | 18 | 3 | no | — |
| 213 | C22×C4 | — | 3 | 3 | no | — |
| 214 | C22×C4 | — | 18 | 3 | no | — |
| 215 | C22×C4 | — | 18 | 3 | no | — |
| 216 | Q16 | — | 9 | 6 | no | — |
| 217 | Q16 | — | 9 | 6 | no | — |
| 218 | SD16 | — | 9 | 12 | no | — |
| 219 | SD16 | — | 9 | 24 | no | — |
| 220 | SD16 | — | 9 | 2 | no | — |
| 221 | SD16 | — | 9 | 2 | no | — |
| 222 | D8 | — | 9 | 12 | no | — |
| 223 | M16 | — | 9 | 12 | no | — |
| 224 | C2×C8 | — | 9 | 1 | no | — |
| 225 | C2×C8 | — | 9 | 3 | no | — |
| 226 | C2×C8 | — | 9 | 12 | no | — |
| 227 | C22×C3 | — | 3 | 6 | no | — |
| 228 | C22×C3 | — | 4 | 3 | no | — |
| 229 | C22×C3 | — | 6 | 6 | no | — |
| 230 | C22×C3 | — | 24 | 4 | no | — |
| 231 | C22×C3 | C22×S4 | 1 | 1 | yes | characteristic |
| 232 | C2×D3 | — | 6 | 2 | no | — |
| 233 | C2×D3 | — | 12 | 6 | no | — |
| 234 | C2×D3 | — | 12 | 12 | no | — |
| 235 | C2×D3 | — | 12 | 2 | no | — |
| 236 | C3×C4 | — | 3 | 1 | no | — |
| 237 | C3×C4 | — | 3 | 1 | no | — |
| 238 | C3×C4 | — | 4 | 18 | no | — |
| 239 | C3×C4 | C22×S4 | 1 | 6 | yes | — |
| 240 | Dic3 | — | 6 | 6 | no | — |
| 241 | Dic3 | — | 12 | 18 | no | — |
| 242 | C32 | — | 4 | 1 | no | Sylow |
| 243 | C23 | — | 3 | 3 | no | — |
| 244 | C23 | — | 6 | 4 | no | — |
| 245 | C23 | — | 18 | 1 | no | — |
| 246 | C23 | — | 24 | 2 | no | — |
| 247 | Q8 | — | 3 | 1 | no | — |
| 248 | Q8 | — | 9 | 6 | no | — |
| 249 | Q8 | — | 18 | 3 | no | — |
| 250 | Q8 | — | 18 | 12 | no | — |
| 251 | Q8 | C2×C3⋊S4 | 1 | 4 | yes | — |
| 252 | Q8 | C23×C3⋊D3 | 1 | 1 | yes | characteristic |
| 253 | D4 | — | 3 | 6 | no | — |
| 254 | D4 | — | 3 | 12 | no | — |
| 255 | D4 | — | 6 | 3 | no | — |
| 256 | D4 | — | 9 | 2 | no | — |
| 257 | D4 | — | 18 | 1 | no | — |
| 258 | D4 | — | 18 | 12 | no | — |
| 259 | C2×C4 | — | 3 | 1 | no | — |
| 260 | C2×C4 | — | 3 | 6 | no | — |
| 261 | C2×C4 | — | 3 | 6 | no | — |
| 262 | C2×C4 | — | 18 | 12 | no | — |
| 263 | C2×C4 | — | 18 | 3 | no | — |
| 264 | C2×C4 | — | 18 | 12 | no | — |
| 265 | C2×C4 | C2×C3⋊S4 | 1 | 3 | yes | — |
| 266 | C8 | — | 9 | 2 | no | — |
| 267 | C8 | — | 9 | 6 | no | — |
| 268 | C2×C3 | — | 4 | 3 | no | — |
| 269 | C2×C3 | — | 4 | 6 | no | — |
| 270 | C2×C3 | — | 6 | 6 | no | — |
| 271 | C2×C3 | #5924 | 1 | 2 | yes | complemented |
| 272 | C2×C3 | C23×S4 | 1 | 1 | yes | characteristic |
| 273 | D3 | — | 12 | 12 | no | — |
| 274 | D3 | — | 12 | 2 | no | — |
| 275 | C22 | — | 3 | 6 | no | — |
| 276 | C22 | — | 6 | 6 | no | — |
| 277 | C22 | — | 18 | 2 | no | — |
| 278 | C22 | — | 24 | 4 | no | — |
| 279 | C22 | — | 36 | 2 | no | — |
| 280 | C22 | C22×C3⋊S4 | 1 | 1 | yes | characteristic |
| 281 | C4 | — | 3 | 1 | no | — |
| 282 | C4 | — | 3 | 1 | no | — |
| 283 | C4 | — | 18 | 6 | no | — |
| 284 | C4 | C22×C3⋊S4 | 1 | 6 | yes | — |
| 285 | C3 | — | 4 | 3 | no | — |
| 286 | C3 | C2×#5924 | 1 | 1 | yes | characteristic, complemented |
| 287 | C2 | — | 6 | 6 | no | — |
| 288 | C2 | — | 36 | 2 | no | — |
| 289 | C2 | #10596587 | 1 | 2 | yes | direct product factor |
| 290 | C2 | C23×C3⋊S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×C3⋊S4 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C24 |
| Frattini | C2 | C23×C3⋊S4 |
| Fitting | C2×C3×H(5,𝔽2) | D3 |
| socle | C22×C3 | C22×S4 |
| hypercenter | C2×Q8 | C3⋊S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×#994 | C2 | semidirect |
| 2. #10596587 | C2 | direct (N × H) |
| 3. C2×#64266 | C2 | semidirect |
| 4. C2×#64265 | C2 | semidirect |
| 8. C3×#994 | C22 | semidirect |
| 10. #64266 | C22 | semidirect |
| 11. #64265 | C22 | semidirect |
| 12. C2×C3⋊GL(2,𝔽3) | C22 | semidirect |
| 13. C2×C3⋊#226 | C22 | semidirect |
| 14. C2×C3×H(5,𝔽2) | D3 | semidirect |
| 16. C2×#994 | D3 | semidirect |
| 33. C3⋊GL(2,𝔽3) | C23 | semidirect |
| 34. C3⋊#226 | C23 | semidirect |
| 37. C3×H(5,𝔽2) | C2×D3 | semidirect |
| 48. #994 | C2×D3 | semidirect |
| 76. C2×H(5,𝔽2) | C3⋊D3 | semidirect |
| 135. H(5,𝔽2) | C2×C3⋊D3 | semidirect |
| 271. C2×C3 | #5924 | semidirect |
| 286. C3 | C2×#5924 | semidirect |
| 289. C2 | #10596587 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#994 | 2 |
| #10596587 | 2 |
| C2×#64266 | 2 |
| C2×#64265 | 2 |
| C2×#5924 | 3 |
| C2×#5836 | 3 |
| C2×Q8×C3⋊D3 | 4 |