Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1032 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 1032 conjugacy classes of subgroups — too many to draw. Only its 127 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 | #11724626 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C3×Q8×#157 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11724344 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #11724343 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C4×#157⋊D3 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #10596525 | — | 3 | 1 | no | maximal |
| 7 | C2×#157⋊D3 | C22 | 1 | 3 | yes | — |
| 8 | C2×#157⋊D3 | C22 | 1 | 6 | yes | — |
| 9 | C3×#3832 | — | 2 | 2 | no | — |
| 10 | C3×Q8×C3⋊D3 | C22 | 1 | 1 | yes | characteristic |
| 11 | C4⋊#1113 | C22 | 1 | 6 | yes | — |
| 12 | C4×#1113 | C22 | 1 | 6 | yes | — |
| 13 | #89041 | — | 2 | 6 | no | — |
| 14 | C3×C4⋊#157 | C22 | 1 | 3 | yes | — |
| 15 | C3×C4×#157 | C22 | 1 | 3 | yes | — |
| 16 | #88940 | C22 | 1 | 1 | yes | characteristic |
| 17 | #88939 | C22 | 1 | 6 | yes | — |
| 18 | #64319 | — | 3 | 2 | no | — |
| 19 | D3×#239 | — | 6 | 2 | no | — |
| 20 | Q8×#157 | D3 | 1 | 1 | yes | characteristic, complemented |
| 21 | #64231 | — | 3 | 3 | no | — |
| 22 | #64230 | — | 3 | 6 | no | — |
| 23 | C4×GO+(4,𝔽2) | — | 3 | 3 | no | — |
| 24 | C2×#1113 | C23 | 1 | 6 | yes | — |
| 25 | C2×C3×#157 | C23 | 1 | 3 | yes | — |
| 26 | C2×C3×#157 | C23 | 1 | 1 | yes | characteristic |
| 27 | #157⋊D3 | — | 2 | 6 | no | — |
| 28 | #157⋊D3 | — | 4 | 6 | no | — |
| 29 | C33×Q8 | D4 | 1 | 1 | yes | characteristic, complemented |
| 30 | C3×C4⋊(C3⋊D3) | — | 2 | 6 | no | — |
| 31 | C3×C4×C3⋊D3 | — | 2 | 6 | no | — |
| 32 | C3×C4×C3⋊D3 | C23 | 1 | 3 | yes | — |
| 33 | C3×#634 | D4 | 1 | 3 | yes | complemented |
| 34 | #6295 | — | 2 | 6 | no | — |
| 35 | #6294 | — | 2 | 6 | no | — |
| 36 | #6294 | C23 | 1 | 2 | yes | — |
| 37 | #5577 | — | 9 | 1 | no | maximal |
| 38 | C2×GO+(4,𝔽2) | — | 3 | 3 | no | — |
| 39 | C2×GO+(4,𝔽2) | — | 3 | 6 | no | — |
| 40 | #3832 | — | 2 | 2 | no | — |
| 41 | Q8×C3⋊D3 | C2×D3 | 1 | 1 | yes | characteristic |
| 42 | C3×#239 | — | 6 | 2 | no | — |
| 43 | C3×#239 | — | 6 | 2 | no | — |
| 44 | C3×#239 | — | 12 | 2 | no | — |
| 45 | C3×D3×Q8 | — | 6 | 2 | no | — |
| 46 | #3802 | — | 3 | 6 | no | — |
| 47 | D3×D12 | — | 6 | 6 | no | — |
| 48 | C4×D32 | — | 3 | 6 | no | — |
| 49 | C4×D32 | — | 6 | 6 | no | — |
| 50 | #3799 | — | 6 | 6 | no | — |
| 51 | #3796 | — | 6 | 6 | no | — |
| 52 | #3795 | — | 6 | 6 | no | — |
| 53 | C4⋊#157 | C2×D3 | 1 | 3 | yes | — |
| 54 | C4×#157 | C2×D3 | 1 | 3 | yes | — |
| 55 | #3773 | — | 3 | 1 | no | — |
| 56 | D32⋊C4 | — | 3 | 6 | no | — |
| 57 | C2×C3×C3⋊D3 | — | 2 | 6 | no | — |
| 58 | C2×C3×C3⋊D3 | C24 | 1 | 1 | yes | characteristic |
| 59 | #1113 | — | 2 | 6 | no | — |
| 60 | #1113 | — | 2 | 6 | no | — |
| 61 | C3×#157 | — | 2 | 3 | no | — |
| 62 | C3×#157 | #41 | 1 | 2 | yes | complemented |
| 63 | C33×C4 | C2×D4 | 1 | 3 | yes | — |
| 64 | C3×C3⋊Dic3 | — | 2 | 2 | no | — |
| 65 | C3×C3⋊Dic3 | C2×D4 | 1 | 3 | yes | — |
| 66 | D3×#41 | — | 18 | 2 | no | — |
| 67 | C2×#239 | — | 9 | 2 | no | — |
| 68 | C3×C4×Q8 | — | 9 | 1 | no | — |
| 69 | #895 | — | 9 | 6 | no | — |
| 70 | #874 | — | 9 | 3 | no | — |
| 71 | C4×D12 | — | 9 | 3 | no | — |
| 72 | C2×D32 | — | 3 | 6 | no | — |
| 73 | C2×D32 | — | 6 | 6 | no | — |
| 74 | C2×#157 | C22×D3 | 1 | 3 | yes | — |
| 75 | C2×#157 | C22×D3 | 1 | 1 | yes | characteristic |
| 76 | GO+(4,𝔽2) | — | 6 | 6 | no | — |
| 77 | GO+(4,𝔽2) | — | 12 | 6 | no | — |
| 78 | C32×Q8 | — | 2 | 2 | no | — |
| 79 | C32×Q8 | — | 4 | 2 | no | — |
| 80 | C32×Q8 | D12 | 1 | 1 | yes | characteristic, complemented |
| 81 | C4⋊(C3⋊D3) | — | 2 | 6 | no | — |
| 82 | C4×C3⋊D3 | — | 2 | 6 | no | — |
| 83 | C4×C3⋊D3 | C22×D3 | 1 | 3 | yes | — |
| 84 | #634 | D12 | 1 | 3 | yes | complemented |
| 85 | C3×D12 | — | 6 | 6 | no | — |
| 86 | C3×D12 | — | 6 | 6 | no | — |
| 87 | C3×D12 | — | 12 | 6 | no | — |
| 88 | C3×C4×D3 | — | 6 | 6 | no | — |
| 89 | C3×C4×D3 | — | 6 | 6 | no | — |
| 90 | C3×C4×D3 | — | 6 | 6 | no | — |
| 91 | C3×C4×D3 | — | 12 | 6 | no | — |
| 92 | C3×Dic6 | — | 6 | 6 | no | — |
| 93 | #626 | — | 6 | 12 | no | — |
| 94 | #625 | — | 6 | 6 | no | — |
| 95 | #624 | — | 3 | 2 | no | — |
| 96 | #624 | — | 6 | 6 | no | — |
| 97 | D3×Dic3 | — | 6 | 2 | no | — |
| 98 | D3×Dic3 | — | 6 | 6 | no | — |
| 99 | D3×Dic3 | — | 6 | 6 | no | — |
| 100 | #550 | — | 27 | 1 | no | Sylow |
| 101 | C2×C33 | C22×D4 | 1 | 1 | yes | characteristic |
| 102 | C3×C3⋊D3 | — | 4 | 6 | no | — |
| 103 | C3×C3⋊D3 | C2×#41 | 1 | 1 | yes | characteristic |
| 104 | C3×C3⋊D3 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 105 | C3×#41 | — | 18 | 2 | no | — |
| 106 | C2×C3×Q8 | — | 9 | 1 | no | — |
| 107 | #239 | — | 6 | 2 | no | — |
| 108 | #239 | — | 6 | 2 | no | — |
| 109 | #239 | — | 12 | 2 | no | — |
| 110 | #239 | — | 18 | 6 | no | — |
| 111 | D3×Q8 | — | 6 | 2 | no | — |
| 112 | #237 | — | 18 | 6 | no | — |
| 113 | D3×D4 | — | 18 | 6 | no | — |
| 114 | #235 | — | 18 | 6 | no | — |
| 115 | C2×D12 | — | 9 | 6 | no | — |
| 116 | C2×D12 | — | 9 | 3 | no | — |
| 117 | C2×D12 | — | 9 | 6 | no | — |
| 118 | C2×C4×D3 | — | 9 | 6 | no | — |
| 119 | C2×C4×D3 | — | 18 | 6 | no | — |
| 120 | C3×C4⋊C4 | — | 9 | 3 | no | — |
| 121 | C3×C42 | — | 9 | 3 | no | — |
| 122 | (C2×D3)⋊C4 | — | 9 | 6 | no | — |
| 123 | C4⋊Dic3 | — | 9 | 1 | no | — |
| 124 | C2×C3⋊D3 | — | 2 | 6 | no | — |
| 125 | C2×C3⋊D3 | C23×D3 | 1 | 1 | yes | characteristic |
| 126 | C2×C3×D3 | — | 6 | 2 | no | — |
| 127 | C2×C3×D3 | — | 6 | 6 | no | — |
| 128 | C2×C3×D3 | — | 6 | 6 | no | — |
| 129 | C2×C3×D3 | — | 12 | 6 | no | — |
| 130 | D32 | — | 6 | 6 | no | — |
| 131 | D32 | — | 6 | 6 | no | — |
| 132 | D32 | — | 12 | 6 | no | — |
| 133 | D32 | — | 12 | 6 | no | — |
| 134 | #157 | — | 2 | 3 | no | — |
| 135 | #157 | #239 | 1 | 2 | yes | complemented |
| 136 | C32×C4 | — | 2 | 6 | no | — |
| 137 | C32×C4 | — | 4 | 6 | no | — |
| 138 | C32×C4 | C2×D12 | 1 | 3 | yes | — |
| 139 | C3⋊Dic3 | — | 2 | 2 | no | — |
| 140 | C3⋊Dic3 | C2×D12 | 1 | 3 | yes | — |
| 141 | C3×Dic3 | — | 6 | 6 | no | — |
| 142 | C3×Dic3 | — | 6 | 2 | no | — |
| 143 | C3×Dic3 | — | 6 | 2 | no | — |
| 144 | C3×Dic3 | — | 12 | 2 | no | — |
| 145 | C2×#41 | — | 27 | 2 | no | — |
| 146 | #127 | — | 27 | 3 | no | — |
| 147 | #121 | — | 27 | 6 | no | — |
| 148 | C4×Q8 | — | 9 | 1 | no | — |
| 149 | C4×D4 | — | 27 | 3 | no | — |
| 150 | C33 | #550 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 151 | C22×D3 | — | 9 | 6 | no | — |
| 152 | C22×D3 | — | 18 | 6 | no | — |
| 153 | C3×Q8 | — | 2 | 2 | no | — |
| 154 | C3×Q8 | — | 4 | 2 | no | — |
| 155 | C3×Q8 | — | 9 | 3 | no | — |
| 156 | C3×Q8 | GO+(4,𝔽2) | 1 | 1 | yes | characteristic, complemented |
| 157 | C3×D4 | — | 18 | 6 | no | — |
| 158 | C2×C3×C4 | — | 9 | 1 | no | — |
| 159 | C2×C3×C4 | — | 9 | 3 | no | — |
| 160 | C2×C3×C4 | — | 9 | 3 | no | — |
| 161 | C2×C3×C4 | — | 18 | 6 | no | — |
| 162 | #67 | — | 18 | 12 | no | — |
| 163 | C2×Dic3 | — | 9 | 2 | no | — |
| 164 | C2×Dic3 | — | 18 | 6 | no | — |
| 165 | D12 | — | 6 | 6 | no | — |
| 166 | D12 | — | 6 | 6 | no | — |
| 167 | D12 | — | 12 | 6 | no | — |
| 168 | D12 | — | 18 | 6 | no | — |
| 169 | D12 | — | 18 | 6 | no | — |
| 170 | D12 | — | 36 | 6 | no | — |
| 171 | C4×D3 | — | 6 | 6 | no | — |
| 172 | C4×D3 | — | 6 | 6 | no | — |
| 173 | C4×D3 | — | 6 | 6 | no | — |
| 174 | C4×D3 | — | 12 | 6 | no | — |
| 175 | C4×D3 | — | 18 | 2 | no | — |
| 176 | C4×D3 | — | 18 | 6 | no | — |
| 177 | C4×D3 | — | 18 | 6 | no | — |
| 178 | Dic6 | — | 6 | 6 | no | — |
| 179 | C2×C32 | — | 2 | 2 | no | — |
| 180 | C2×C32 | — | 4 | 2 | no | — |
| 181 | C2×C32 | C22×D12 | 1 | 1 | yes | characteristic |
| 182 | C3⋊D3 | — | 4 | 6 | no | — |
| 183 | C3⋊D3 | C2×#239 | 1 | 1 | yes | characteristic |
| 184 | C3⋊D3 | #1009 | 1 | 1 | yes | characteristic |
| 185 | C3×D3 | — | 6 | 2 | no | — |
| 186 | C3×D3 | — | 6 | 2 | no | — |
| 187 | C3×D3 | — | 12 | 6 | no | — |
| 188 | C3×D3 | — | 12 | 6 | no | — |
| 189 | C3×D3 | — | 24 | 6 | no | — |
| 190 | #41 | — | 18 | 2 | no | — |
| 191 | #41 | — | 54 | 6 | no | — |
| 192 | C2×Q8 | — | 9 | 1 | no | — |
| 193 | C2×D4 | — | 27 | 6 | no | — |
| 194 | C2×D4 | — | 27 | 3 | no | — |
| 195 | C2×D4 | — | 27 | 6 | no | — |
| 196 | C22×C4 | — | 27 | 6 | no | — |
| 197 | C4⋊C4 | — | 9 | 3 | no | — |
| 198 | C4⋊C4 | — | 27 | 1 | no | — |
| 199 | C22⋊C4 | — | 27 | 6 | no | — |
| 200 | C42 | — | 9 | 3 | no | — |
| 201 | C22×C3 | — | 9 | 1 | no | — |
| 202 | C22×C3 | — | 18 | 6 | no | — |
| 203 | C2×D3 | — | 6 | 2 | no | — |
| 204 | C2×D3 | — | 6 | 6 | no | — |
| 205 | C2×D3 | — | 6 | 6 | no | — |
| 206 | C2×D3 | — | 12 | 6 | no | — |
| 207 | C2×D3 | — | 18 | 6 | no | — |
| 208 | C2×D3 | — | 18 | 6 | no | — |
| 209 | C2×D3 | — | 36 | 6 | no | — |
| 210 | C2×D3 | — | 36 | 6 | no | — |
| 211 | C3×C4 | — | 2 | 6 | no | — |
| 212 | C3×C4 | — | 4 | 6 | no | — |
| 213 | C3×C4 | — | 9 | 2 | no | — |
| 214 | C3×C4 | — | 9 | 3 | no | — |
| 215 | C3×C4 | — | 18 | 2 | no | — |
| 216 | C3×C4 | — | 18 | 3 | no | — |
| 217 | C3×C4 | C2×GO+(4,𝔽2) | 1 | 3 | yes | — |
| 218 | Dic3 | — | 6 | 6 | no | — |
| 219 | Dic3 | — | 6 | 2 | no | — |
| 220 | Dic3 | — | 6 | 2 | no | — |
| 221 | Dic3 | — | 12 | 2 | no | — |
| 222 | C32 | — | 2 | 2 | no | — |
| 223 | C32 | — | 4 | 2 | no | — |
| 224 | C32 | #5577 | 1 | 1 | yes | characteristic, complemented |
| 225 | C23 | — | 27 | 6 | no | — |
| 226 | Q8 | — | 9 | 3 | no | — |
| 227 | Q8 | #157⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 228 | D4 | — | 18 | 6 | no | — |
| 229 | D4 | — | 54 | 6 | no | — |
| 230 | D4 | — | 54 | 6 | no | — |
| 231 | D4 | — | 108 | 6 | no | — |
| 232 | C2×C4 | — | 9 | 1 | no | — |
| 233 | C2×C4 | — | 9 | 3 | no | — |
| 234 | C2×C4 | — | 9 | 3 | no | — |
| 235 | C2×C4 | — | 18 | 6 | no | — |
| 236 | C2×C4 | — | 27 | 2 | no | — |
| 237 | C2×C4 | — | 54 | 6 | no | — |
| 238 | C2×C3 | — | 2 | 2 | no | — |
| 239 | C2×C3 | — | 4 | 2 | no | — |
| 240 | C2×C3 | — | 9 | 1 | no | — |
| 241 | C2×C3 | — | 9 | 1 | no | — |
| 242 | C2×C3 | — | 36 | 6 | no | — |
| 243 | C2×C3 | C22×GO+(4,𝔽2) | 1 | 1 | yes | characteristic |
| 244 | D3 | — | 6 | 2 | no | — |
| 245 | D3 | — | 6 | 2 | no | — |
| 246 | D3 | — | 12 | 6 | no | — |
| 247 | D3 | — | 12 | 6 | no | — |
| 248 | D3 | — | 24 | 6 | no | — |
| 249 | C22 | — | 9 | 1 | no | — |
| 250 | C22 | — | 18 | 6 | no | — |
| 251 | C22 | — | 54 | 6 | no | — |
| 252 | C22 | — | 54 | 6 | no | — |
| 253 | C4 | — | 9 | 2 | no | — |
| 254 | C4 | — | 9 | 3 | no | — |
| 255 | C4 | — | 18 | 3 | no | — |
| 256 | C4 | — | 18 | 2 | no | — |
| 257 | C4 | C2×#157⋊D3 | 1 | 3 | yes | — |
| 258 | C3 | — | 2 | 2 | no | — |
| 259 | C3 | — | 4 | 2 | no | — |
| 260 | C3 | #10596525 | 1 | 1 | yes | characteristic, complemented |
| 261 | C2 | — | 9 | 1 | no | — |
| 262 | C2 | — | 9 | 1 | no | — |
| 263 | C2 | — | 36 | 6 | no | — |
| 264 | C2 | C22×#157⋊D3 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×#157⋊D3 |
| commutator, cosocle | C2×C3×C3⋊D3 | C24 |
| Fitting | C33×Q8 | D4 |
| socle | C2×C33 | C22×D4 |
| hypercenter | Q8 | #157⋊D3 |
| hypocenter | C33 | #550 |
| LCS | G ▷ C2×C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11724626 | C2 | semidirect |
| 2. C3×Q8×#157 | C2 | semidirect |
| 3. #11724344 | C2 | semidirect |
| 4. #11724343 | C2 | semidirect |
| 5. C4×#157⋊D3 | C2 | semidirect |
| 20. Q8×#157 | D3 | semidirect |
| 29. C33×Q8 | D4 | semidirect |
| 33. C3×#634 | D4 | semidirect |
| 62. C3×#157 | #41 | semidirect |
| 80. C32×Q8 | D12 | semidirect |
| 84. #634 | D12 | semidirect |
| 135. #157 | #239 | semidirect |
| 150. C33 | #550 | semidirect |
| 156. C3×Q8 | GO+(4,𝔽2) | semidirect |
| 224. C32 | #5577 | semidirect |
| 227. Q8 | #157⋊D3 | semidirect |
| 260. C3 | #10596525 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11724626 | 2 |
| C3×Q8×#157 | 2 |
| #11724344 | 2 |
| #11724343 | 2 |
| C4×#157⋊D3 | 2 |
| #10596525 | 3 |
| #5577 | 9 |