Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
810 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 810 conjugacy classes of subgroups — too many to draw. Only its 113 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 | C32×H(3,𝔽4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10592760 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10592731 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #10589407 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #10589402 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #71950 | — | 3 | 4 | no | maximal |
| 7 | #64361 | C22 | 1 | 1 | yes | characteristic |
| 8 | C32×#124 | — | 2 | 1 | no | — |
| 9 | C32×#124 | — | 2 | 2 | no | — |
| 10 | C32×#124 | — | 2 | 1 | no | — |
| 11 | C32×#124 | C22 | 1 | 1 | yes | characteristic |
| 12 | C32×#120 | — | 2 | 2 | no | — |
| 13 | C32×#120 | C22 | 1 | 2 | yes | — |
| 14 | #64162 | — | 2 | 2 | no | — |
| 15 | #64151 | — | 2 | 1 | no | — |
| 16 | #63661 | C22 | 1 | 2 | yes | — |
| 17 | #63660 | C22 | 1 | 1 | yes | characteristic |
| 18 | #63633 | — | 2 | 2 | no | — |
| 19 | #63632 | — | 2 | 1 | no | — |
| 20 | C3×H(3,𝔽4) | D3 | 1 | 4 | yes | complemented |
| 21 | #5061 | — | 3 | 4 | no | — |
| 22 | #5032 | — | 3 | 8 | no | — |
| 23 | #4541 | — | 3 | 4 | no | — |
| 24 | #4536 | — | 3 | 8 | no | — |
| 25 | C24×C32 | D4 | 1 | 2 | yes | complemented |
| 26 | C2×C32×Q8 | — | 2 | 1 | no | — |
| 27 | C2×C32×Q8 | D4 | 1 | 1 | yes | characteristic, complemented |
| 28 | C2×C32×D4 | — | 2 | 1 | no | — |
| 29 | C2×C32×D4 | — | 2 | 1 | no | — |
| 30 | C2×C32×D4 | — | 2 | 2 | no | — |
| 31 | C2×C32×D4 | C23 | 1 | 1 | yes | characteristic |
| 32 | C2×#638 | — | 2 | 2 | no | — |
| 33 | C2×#638 | — | 4 | 2 | no | — |
| 34 | #3830 | — | 4 | 1 | no | — |
| 35 | D4×C3⋊D3 | — | 4 | 1 | no | — |
| 36 | #3828 | — | 2 | 1 | no | — |
| 37 | C32×C22⋊C4 | — | 2 | 2 | no | — |
| 38 | C32×C22⋊C4 | — | 2 | 2 | no | — |
| 39 | C32×C22⋊C4 | — | 2 | 2 | no | — |
| 40 | C32×C22⋊C4 | — | 2 | 2 | no | — |
| 41 | C32×C22⋊C4 | D4 | 1 | 2 | yes | complemented |
| 42 | C32×C42 | — | 2 | 1 | no | — |
| 43 | C32×C42 | D4 | 1 | 1 | yes | characteristic, complemented |
| 44 | #3757 | — | 2 | 2 | no | — |
| 45 | #3757 | — | 4 | 2 | no | — |
| 46 | #3756 | — | 4 | 1 | no | — |
| 47 | #3755 | — | 4 | 1 | no | — |
| 48 | #3748 | — | 2 | 1 | no | — |
| 49 | #2199 | — | 9 | 1 | no | Sylow |
| 50 | #1004 | — | 3 | 4 | no | — |
| 51 | C3×#124 | — | 2 | 4 | no | — |
| 52 | C3×#124 | — | 2 | 8 | no | — |
| 53 | C3×#124 | — | 2 | 4 | no | — |
| 54 | C3×#124 | C2×D3 | 1 | 4 | yes | — |
| 55 | C3×#120 | — | 2 | 8 | no | — |
| 56 | C3×#120 | C2×D3 | 1 | 8 | yes | — |
| 57 | #953 | — | 6 | 8 | no | — |
| 58 | #942 | — | 6 | 4 | no | — |
| 59 | #834 | — | 3 | 8 | no | — |
| 60 | #833 | — | 3 | 4 | no | — |
| 61 | #806 | — | 6 | 8 | no | — |
| 62 | #805 | — | 6 | 4 | no | — |
| 63 | C23×C32 | — | 2 | 2 | no | — |
| 64 | C23×C32 | — | 4 | 2 | no | — |
| 65 | C23×C32 | — | 8 | 2 | no | — |
| 66 | C23×C32 | C2×D4 | 1 | 2 | yes | — |
| 67 | C22×C3⋊D3 | — | 2 | 2 | no | — |
| 68 | C32×Q8 | — | 4 | 1 | no | — |
| 69 | C32×Q8 | — | 8 | 1 | no | — |
| 70 | C32×D4 | — | 4 | 1 | no | — |
| 71 | C32×D4 | — | 8 | 2 | no | — |
| 72 | C32×D4 | — | 8 | 1 | no | — |
| 73 | C32×D4 | — | 8 | 1 | no | — |
| 74 | C2×C32×C4 | — | 2 | 1 | no | — |
| 75 | C2×C32×C4 | — | 2 | 2 | no | — |
| 76 | C2×C32×C4 | — | 2 | 1 | no | — |
| 77 | C2×C32×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 78 | #638 | — | 4 | 2 | no | — |
| 79 | #638 | — | 4 | 1 | no | — |
| 80 | #638 | — | 8 | 4 | no | — |
| 81 | C2×C3⋊Dic3 | — | 2 | 2 | no | — |
| 82 | C2×C3⋊Dic3 | — | 4 | 2 | no | — |
| 83 | C4⋊(C3⋊D3) | — | 2 | 1 | no | — |
| 84 | C4×C3⋊D3 | — | 4 | 1 | no | — |
| 85 | #634 | — | 2 | 1 | no | — |
| 86 | C3⋊(C3⋊C8) | — | 4 | 1 | no | — |
| 87 | H(3,𝔽4) | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 88 | B(4,𝔽2) | — | 9 | 2 | no | — |
| 89 | #455 | — | 9 | 1 | no | — |
| 90 | #354 | — | 9 | 1 | no | — |
| 91 | AΣL(1,𝔽16) | — | 9 | 2 | no | — |
| 92 | C24×C3 | #67 | 1 | 8 | yes | complemented |
| 93 | C2×C3×Q8 | — | 2 | 4 | no | — |
| 94 | C2×C3×Q8 | #67 | 1 | 4 | yes | complemented |
| 95 | C2×C3×D4 | — | 2 | 4 | no | — |
| 96 | C2×C3×D4 | — | 2 | 4 | no | — |
| 97 | C2×C3×D4 | — | 2 | 8 | no | — |
| 98 | C2×C3×D4 | C22×D3 | 1 | 4 | yes | — |
| 99 | C2×#67 | — | 6 | 8 | no | — |
| 100 | C2×#67 | — | 12 | 8 | no | — |
| 101 | #237 | — | 12 | 4 | no | — |
| 102 | D3×D4 | — | 12 | 4 | no | — |
| 103 | #235 | — | 6 | 4 | no | — |
| 104 | C3×C22⋊C4 | — | 2 | 8 | no | — |
| 105 | C3×C22⋊C4 | — | 2 | 8 | no | — |
| 106 | C3×C22⋊C4 | — | 2 | 8 | no | — |
| 107 | C3×C22⋊C4 | — | 2 | 8 | no | — |
| 108 | C3×C22⋊C4 | #67 | 1 | 8 | yes | complemented |
| 109 | C3×C42 | — | 2 | 4 | no | — |
| 110 | C3×C42 | #67 | 1 | 4 | yes | complemented |
| 111 | #217 | — | 6 | 8 | no | — |
| 112 | #217 | — | 12 | 8 | no | — |
| 113 | #216 | — | 12 | 4 | no | — |
| 114 | #215 | — | 12 | 4 | no | — |
| 115 | #208 | — | 6 | 4 | no | — |
| 116 | C22×C32 | — | 2 | 2 | no | — |
| 117 | C22×C32 | — | 4 | 2 | no | — |
| 118 | C22×C32 | — | 4 | 2 | no | — |
| 119 | C22×C32 | — | 4 | 2 | no | — |
| 120 | C22×C32 | — | 4 | 4 | no | — |
| 121 | C22×C32 | — | 4 | 2 | no | — |
| 122 | C22×C32 | — | 8 | 2 | no | — |
| 123 | C22×C32 | #120 | 1 | 1 | yes | characteristic |
| 124 | C2×C3⋊D3 | — | 4 | 1 | no | — |
| 125 | C2×C3⋊D3 | — | 8 | 2 | no | — |
| 126 | C32×C4 | — | 2 | 1 | no | — |
| 127 | C32×C4 | — | 4 | 2 | no | — |
| 128 | C32×C4 | — | 4 | 1 | no | — |
| 129 | C32×C4 | — | 4 | 1 | no | — |
| 130 | C3⋊Dic3 | — | 4 | 1 | no | — |
| 131 | C3⋊Dic3 | — | 8 | 2 | no | — |
| 132 | H(5,𝔽2) | — | 9 | 1 | no | — |
| 133 | #137 | — | 18 | 1 | no | — |
| 134 | #124 | — | 2 | 1 | no | — |
| 135 | #124 | — | 2 | 2 | no | — |
| 136 | #124 | — | 2 | 1 | no | — |
| 137 | #124 | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 138 | #120 | — | 2 | 2 | no | — |
| 139 | #120 | — | 18 | 2 | no | — |
| 140 | #120 | C2×C3⋊D3 | 1 | 2 | yes | — |
| 141 | #104 | — | 18 | 1 | no | — |
| 142 | #100 | — | 9 | 1 | no | — |
| 143 | #99 | — | 9 | 2 | no | — |
| 144 | #99 | — | 18 | 2 | no | — |
| 145 | C23×C3 | — | 2 | 8 | no | — |
| 146 | C23×C3 | — | 4 | 8 | no | — |
| 147 | C23×C3 | — | 8 | 8 | no | — |
| 148 | C23×C3 | C2×#67 | 1 | 8 | yes | — |
| 149 | C22×D3 | — | 6 | 8 | no | — |
| 150 | C3×Q8 | — | 4 | 4 | no | — |
| 151 | C3×Q8 | — | 8 | 4 | no | — |
| 152 | C3×D4 | — | 4 | 4 | no | — |
| 153 | C3×D4 | — | 8 | 8 | no | — |
| 154 | C3×D4 | — | 8 | 4 | no | — |
| 155 | C3×D4 | — | 8 | 4 | no | — |
| 156 | C2×C3×C4 | — | 2 | 4 | no | — |
| 157 | C2×C3×C4 | — | 2 | 8 | no | — |
| 158 | C2×C3×C4 | — | 2 | 4 | no | — |
| 159 | C2×C3×C4 | C2×#67 | 1 | 4 | yes | — |
| 160 | #67 | — | 12 | 8 | no | — |
| 161 | #67 | — | 12 | 4 | no | — |
| 162 | #67 | — | 24 | 16 | no | — |
| 163 | C2×Dic3 | — | 6 | 8 | no | — |
| 164 | C2×Dic3 | — | 12 | 8 | no | — |
| 165 | D12 | — | 6 | 4 | no | — |
| 166 | C4×D3 | — | 12 | 4 | no | — |
| 167 | Dic6 | — | 6 | 4 | no | — |
| 168 | C3⋊C8 | — | 12 | 4 | no | — |
| 169 | C2×C32 | — | 2 | 1 | no | — |
| 170 | C2×C32 | — | 4 | 2 | no | — |
| 171 | C2×C32 | — | 8 | 2 | no | — |
| 172 | C2×C32 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 173 | C3⋊D3 | — | 8 | 1 | no | — |
| 174 | C24 | #638 | 1 | 2 | yes | complemented |
| 175 | #41 | — | 18 | 1 | no | — |
| 176 | #41 | — | 36 | 1 | no | — |
| 177 | C2×Q8 | — | 2 | 1 | no | — |
| 178 | C2×Q8 | #638 | 1 | 1 | yes | characteristic, complemented |
| 179 | C2×D4 | — | 2 | 1 | no | — |
| 180 | C2×D4 | — | 2 | 1 | no | — |
| 181 | C2×D4 | — | 2 | 2 | no | — |
| 182 | C2×D4 | — | 18 | 2 | no | — |
| 183 | C2×D4 | — | 36 | 1 | no | — |
| 184 | C2×D4 | — | 36 | 2 | no | — |
| 185 | C2×D4 | C22×C3⋊D3 | 1 | 1 | yes | characteristic |
| 186 | Q16 | — | 36 | 1 | no | — |
| 187 | SD16 | — | 36 | 1 | no | — |
| 188 | M16 | — | 18 | 1 | no | — |
| 189 | C22⋊C4 | — | 2 | 2 | no | — |
| 190 | C22⋊C4 | — | 2 | 2 | no | — |
| 191 | C22⋊C4 | — | 2 | 2 | no | — |
| 192 | C22⋊C4 | — | 2 | 2 | no | — |
| 193 | C22⋊C4 | — | 18 | 2 | no | — |
| 194 | C22⋊C4 | — | 36 | 2 | no | — |
| 195 | C22⋊C4 | #638 | 1 | 2 | yes | complemented |
| 196 | C42 | — | 2 | 1 | no | — |
| 197 | C42 | #638 | 1 | 1 | yes | characteristic, complemented |
| 198 | C22×C3 | — | 2 | 8 | no | — |
| 199 | C22×C3 | — | 4 | 8 | no | — |
| 200 | C22×C3 | — | 4 | 8 | no | — |
| 201 | C22×C3 | — | 4 | 8 | no | — |
| 202 | C22×C3 | — | 4 | 16 | no | — |
| 203 | C22×C3 | — | 4 | 8 | no | — |
| 204 | C22×C3 | — | 8 | 8 | no | — |
| 205 | C22×C3 | #953 | 1 | 4 | yes | — |
| 206 | C2×D3 | — | 12 | 4 | no | — |
| 207 | C2×D3 | — | 24 | 8 | no | — |
| 208 | C3×C4 | — | 2 | 4 | no | — |
| 209 | C3×C4 | — | 4 | 8 | no | — |
| 210 | C3×C4 | — | 4 | 4 | no | — |
| 211 | C3×C4 | — | 4 | 4 | no | — |
| 212 | Dic3 | — | 12 | 4 | no | — |
| 213 | Dic3 | — | 24 | 8 | no | — |
| 214 | C32 | #2199 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 215 | C23 | — | 2 | 2 | no | — |
| 216 | C23 | — | 4 | 2 | no | — |
| 217 | C23 | — | 8 | 2 | no | — |
| 218 | C23 | — | 18 | 2 | no | — |
| 219 | C23 | C2×#638 | 1 | 2 | yes | — |
| 220 | Q8 | — | 4 | 1 | no | — |
| 221 | Q8 | — | 8 | 1 | no | — |
| 222 | Q8 | — | 18 | 1 | no | — |
| 223 | D4 | — | 4 | 1 | no | — |
| 224 | D4 | — | 8 | 2 | no | — |
| 225 | D4 | — | 8 | 1 | no | — |
| 226 | D4 | — | 8 | 1 | no | — |
| 227 | D4 | — | 18 | 1 | no | — |
| 228 | D4 | — | 36 | 2 | no | — |
| 229 | D4 | — | 36 | 1 | no | — |
| 230 | D4 | — | 72 | 4 | no | — |
| 231 | C2×C4 | — | 2 | 1 | no | — |
| 232 | C2×C4 | — | 2 | 2 | no | — |
| 233 | C2×C4 | — | 2 | 1 | no | — |
| 234 | C2×C4 | — | 18 | 2 | no | — |
| 235 | C2×C4 | — | 36 | 1 | no | — |
| 236 | C2×C4 | — | 36 | 2 | no | — |
| 237 | C2×C4 | C2×#638 | 1 | 1 | yes | characteristic |
| 238 | C8 | — | 36 | 1 | no | — |
| 239 | C2×C3 | — | 2 | 4 | no | — |
| 240 | C2×C3 | — | 4 | 8 | no | — |
| 241 | C2×C3 | — | 8 | 8 | no | — |
| 242 | C2×C3 | #5244 | 1 | 4 | yes | — |
| 243 | D3 | — | 24 | 4 | no | — |
| 244 | C22 | — | 2 | 2 | no | — |
| 245 | C22 | — | 4 | 2 | no | — |
| 246 | C22 | — | 4 | 2 | no | — |
| 247 | C22 | — | 4 | 2 | no | — |
| 248 | C22 | — | 4 | 4 | no | — |
| 249 | C22 | — | 4 | 2 | no | — |
| 250 | C22 | — | 8 | 2 | no | — |
| 251 | C22 | — | 36 | 1 | no | — |
| 252 | C22 | — | 72 | 2 | no | — |
| 253 | C22 | #64162 | 1 | 1 | yes | characteristic |
| 254 | C4 | — | 2 | 1 | no | — |
| 255 | C4 | — | 4 | 2 | no | — |
| 256 | C4 | — | 4 | 1 | no | — |
| 257 | C4 | — | 4 | 1 | no | — |
| 258 | C4 | — | 36 | 1 | no | — |
| 259 | C4 | — | 72 | 2 | no | — |
| 260 | C3 | #71950 | 1 | 4 | yes | complemented |
| 261 | C2 | — | 2 | 1 | no | — |
| 262 | C2 | — | 4 | 2 | no | — |
| 263 | C2 | — | 8 | 2 | no | — |
| 264 | C2 | — | 72 | 1 | no | — |
| 265 | C2 | #10592943 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×H(3,𝔽4) | C2 | semidirect |
| 2. #10592760 | C2 | semidirect |
| 3. #10592731 | C2 | semidirect |
| 4. #10589407 | C2 | semidirect |
| 5. #10589402 | C2 | semidirect |
| 20. C3×H(3,𝔽4) | D3 | semidirect |
| 25. C24×C32 | D4 | semidirect |
| 27. C2×C32×Q8 | D4 | semidirect |
| 41. C32×C22⋊C4 | D4 | semidirect |
| 43. C32×C42 | D4 | semidirect |
| 87. H(3,𝔽4) | C3⋊D3 | semidirect |
| 92. C24×C3 | #67 | semidirect |
| 94. C2×C3×Q8 | #67 | semidirect |
| 108. C3×C22⋊C4 | #67 | semidirect |
| 110. C3×C42 | #67 | semidirect |
| 174. C24 | #638 | semidirect |
| 178. C2×Q8 | #638 | semidirect |
| 195. C22⋊C4 | #638 | semidirect |
| 197. C42 | #638 | semidirect |
| 214. C32 | #2199 | semidirect |
| 260. C3 | #71950 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×H(3,𝔽4) | 2 |
| #10592760 | 2 |
| #10592731 | 2 |
| #10589407 | 2 |
| #10589402 | 2 |
| #71950 | 3 |