Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
930 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 930 conjugacy classes of subgroups — too many to draw. Only its 191 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 | #78579 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #78553 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C3×#1747 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #68727 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #68721 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #68715 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #68703 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #23794 | — | 3 | 1 | no | maximal, Sylow |
| 9 | #5748 | C22 | 1 | 3 | yes | — |
| 10 | C2×#901 | C22 | 1 | 3 | yes | — |
| 11 | C22×D3×C8 | C22 | 1 | 1 | yes | characteristic |
| 12 | #5524 | C22 | 1 | 3 | yes | — |
| 13 | C3×#405 | C22 | 1 | 3 | yes | — |
| 14 | #5129 | C22 | 1 | 3 | yes | — |
| 15 | #5126 | C22 | 1 | 3 | yes | — |
| 16 | #5125 | C22 | 1 | 3 | yes | — |
| 17 | #5005 | C22 | 1 | 6 | yes | — |
| 18 | #4999 | C22 | 1 | 3 | yes | — |
| 19 | #4717 | — | 2 | 6 | no | — |
| 20 | #4706 | — | 2 | 6 | no | — |
| 21 | #4700 | — | 2 | 6 | no | — |
| 22 | C8×D12 | — | 2 | 6 | no | — |
| 23 | C3×#338 | C22 | 1 | 1 | yes | characteristic |
| 24 | C3×H(3,ℤ/4) | C22 | 1 | 3 | yes | — |
| 25 | #2955 | — | 3 | 3 | no | — |
| 26 | #2929 | — | 3 | 3 | no | — |
| 27 | #2098 | — | 3 | 3 | no | — |
| 28 | #2091 | — | 3 | 1 | no | — |
| 29 | #2017 | — | 3 | 1 | no | — |
| 30 | #2010 | — | 3 | 3 | no | — |
| 31 | #1747 | D3 | 1 | 1 | yes | characteristic, complemented |
| 32 | C2×#235 | C23 | 1 | 3 | yes | — |
| 33 | C22×C4×D3 | C23 | 1 | 1 | yes | characteristic |
| 34 | C2×C3×M16 | C23 | 1 | 3 | yes | — |
| 35 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 36 | C3×#117 | C23 | 1 | 3 | yes | — |
| 37 | C2×#208 | C23 | 1 | 3 | yes | — |
| 38 | C22×C3⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 39 | #909 | — | 2 | 6 | no | — |
| 40 | #908 | — | 2 | 6 | no | — |
| 41 | #904 | — | 2 | 6 | no | — |
| 42 | #901 | — | 2 | 6 | no | — |
| 43 | #901 | — | 2 | 6 | no | — |
| 44 | C2×ΓL(1,𝔽25) | — | 2 | 3 | no | — |
| 45 | C2×D3×C8 | — | 2 | 3 | no | — |
| 46 | C2×D3×C8 | — | 2 | 3 | no | — |
| 47 | C2×D3×C8 | — | 2 | 3 | no | — |
| 48 | #897 | — | 2 | 3 | no | — |
| 49 | #895 | — | 2 | 3 | no | — |
| 50 | #886 | — | 2 | 3 | no | — |
| 51 | #882 | — | 2 | 3 | no | — |
| 52 | C4×D12 | — | 2 | 6 | no | — |
| 53 | C3×#97 | D4 | 1 | 6 | yes | complemented |
| 54 | C3×C4×C8 | D4 | 1 | 6 | yes | complemented |
| 55 | (C2×D3)⋊C8 | — | 2 | 6 | no | — |
| 56 | (C2×D3)⋊C8 | — | 2 | 6 | no | — |
| 57 | #819 | — | 2 | 6 | no | — |
| 58 | #805 | — | 2 | 12 | no | — |
| 59 | C4⋊(C3⋊C8) | — | 2 | 6 | no | — |
| 60 | #575 | — | 3 | 3 | no | — |
| 61 | C2×#131 | — | 3 | 3 | no | — |
| 62 | C23×C8 | — | 3 | 1 | no | — |
| 63 | #525 | — | 3 | 3 | no | — |
| 64 | #444 | — | 6 | 6 | no | — |
| 65 | #443 | — | 6 | 6 | no | — |
| 66 | #435 | — | 6 | 6 | no | — |
| 67 | C8×D4 | — | 6 | 6 | no | — |
| 68 | #430 | — | 3 | 3 | no | — |
| 69 | #424 | — | 3 | 3 | no | — |
| 70 | #421 | — | 3 | 6 | no | — |
| 71 | #408 | — | 3 | 3 | no | — |
| 72 | #407 | — | 3 | 3 | no | — |
| 73 | #405 | C2×D3 | 1 | 3 | yes | — |
| 74 | #338 | C2×D3 | 1 | 1 | yes | characteristic |
| 75 | H(3,ℤ/4) | C2×D3 | 1 | 3 | yes | — |
| 76 | C23×D3 | #41 | 1 | 1 | yes | characteristic |
| 77 | C22×C3×C4 | C24 | 1 | 1 | yes | characteristic |
| 78 | C2×#67 | — | 2 | 3 | no | — |
| 79 | C22×Dic3 | #41 | 1 | 1 | yes | characteristic |
| 80 | #235 | — | 2 | 6 | no | — |
| 81 | #235 | — | 2 | 6 | no | — |
| 82 | C2×D12 | — | 2 | 6 | no | — |
| 83 | C2×D12 | #41 | 1 | 3 | yes | — |
| 84 | C2×C4×D3 | — | 2 | 3 | no | — |
| 85 | C2×C4×D3 | — | 2 | 3 | no | — |
| 86 | C2×C4×D3 | — | 2 | 3 | no | — |
| 87 | C2×Dic6 | #41 | 1 | 3 | yes | — |
| 88 | C3×M16 | — | 2 | 6 | no | — |
| 89 | C2×C3×C8 | C2×D4 | 1 | 6 | yes | — |
| 90 | C2×C3×C8 | C2×D4 | 1 | 3 | yes | — |
| 91 | C2×C3×C8 | C2×D4 | 1 | 3 | yes | — |
| 92 | C3×C4⋊C4 | — | 2 | 3 | no | — |
| 93 | C3×C22⋊C4 | — | 2 | 3 | no | — |
| 94 | C3×C42 | C2×D4 | 1 | 6 | yes | — |
| 95 | (C2×D3)⋊C4 | — | 2 | 6 | no | — |
| 96 | (C2×D3)⋊C4 | — | 2 | 6 | no | — |
| 97 | C4⋊Dic3 | — | 2 | 6 | no | — |
| 98 | #208 | — | 2 | 6 | no | — |
| 99 | C2×C3⋊C8 | — | 2 | 3 | no | — |
| 100 | C2×C3⋊C8 | — | 2 | 3 | no | — |
| 101 | Dic12 | — | 2 | 6 | no | — |
| 102 | D24 | — | 2 | 6 | no | — |
| 103 | #204 | — | 2 | 12 | no | — |
| 104 | ΓL(1,𝔽25) | — | 4 | 6 | no | — |
| 105 | D3×C8 | — | 4 | 3 | no | — |
| 106 | D3×C8 | — | 4 | 1 | no | — |
| 107 | D3×C8 | — | 4 | 3 | no | — |
| 108 | D3×C8 | — | 4 | 3 | no | — |
| 109 | C2×#41 | — | 3 | 3 | no | — |
| 110 | C23×C4 | — | 3 | 1 | no | — |
| 111 | #137 | — | 6 | 6 | no | — |
| 112 | AGL(1,ℤ/8) | — | 6 | 6 | no | — |
| 113 | #135 | — | 6 | 6 | no | — |
| 114 | #131 | — | 6 | 6 | no | — |
| 115 | #131 | — | 6 | 6 | no | — |
| 116 | C2×M16 | — | 3 | 3 | no | — |
| 117 | C2×M16 | — | 6 | 3 | no | — |
| 118 | C2×M16 | C22×D3 | 1 | 3 | yes | — |
| 119 | C22×C8 | — | 3 | 1 | no | — |
| 120 | C22×C8 | — | 6 | 3 | no | — |
| 121 | C22×C8 | — | 6 | 3 | no | — |
| 122 | C22×C8 | — | 6 | 3 | no | — |
| 123 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 124 | #123 | — | 6 | 3 | no | — |
| 125 | C22⋊Q8 | — | 6 | 3 | no | — |
| 126 | #121 | — | 6 | 3 | no | — |
| 127 | #120 | — | 6 | 3 | no | — |
| 128 | C4×D4 | — | 6 | 6 | no | — |
| 129 | #117 | C22×D3 | 1 | 3 | yes | — |
| 130 | #108 | — | 6 | 6 | no | — |
| 131 | C4⋊C8 | — | 6 | 6 | no | — |
| 132 | #104 | — | 6 | 12 | no | — |
| 133 | C22⋊C8 | — | 6 | 6 | no | — |
| 134 | C22⋊C8 | — | 6 | 6 | no | — |
| 135 | #97 | D12 | 1 | 6 | yes | complemented |
| 136 | C4×C8 | D12 | 1 | 6 | yes | complemented |
| 137 | C23×C3 | C2×#41 | 1 | 1 | yes | characteristic |
| 138 | C22×D3 | — | 2 | 3 | no | — |
| 139 | C22×D3 | — | 2 | 3 | no | — |
| 140 | C22×D3 | — | 4 | 2 | no | — |
| 141 | C2×C3×C4 | — | 2 | 6 | no | — |
| 142 | C2×C3×C4 | C22×D4 | 1 | 3 | yes | — |
| 143 | C2×C3×C4 | C2×#41 | 1 | 3 | yes | — |
| 144 | #67 | — | 4 | 6 | no | — |
| 145 | C2×Dic3 | — | 2 | 3 | no | — |
| 146 | C2×Dic3 | — | 2 | 3 | no | — |
| 147 | D12 | — | 2 | 6 | no | — |
| 148 | D12 | — | 4 | 12 | no | — |
| 149 | C4×D3 | — | 4 | 3 | no | — |
| 150 | C4×D3 | — | 4 | 3 | no | — |
| 151 | C4×D3 | — | 4 | 1 | no | — |
| 152 | C4×D3 | — | 4 | 3 | no | — |
| 153 | Dic6 | — | 2 | 6 | no | — |
| 154 | C3×C8 | — | 2 | 6 | no | — |
| 155 | C3×C8 | #127 | 1 | 3 | yes | — |
| 156 | C3×C8 | #127 | 1 | 1 | yes | characteristic |
| 157 | C3⋊C8 | — | 4 | 1 | no | — |
| 158 | C3⋊C8 | — | 4 | 3 | no | — |
| 159 | C24 | — | 3 | 1 | no | — |
| 160 | #41 | — | 6 | 6 | no | — |
| 161 | #41 | — | 6 | 6 | no | — |
| 162 | C2×Q8 | — | 3 | 3 | no | — |
| 163 | C2×D4 | — | 3 | 3 | no | — |
| 164 | C2×D4 | — | 6 | 6 | no | — |
| 165 | C2×D4 | — | 6 | 3 | no | — |
| 166 | C22×C4 | — | 3 | 1 | no | — |
| 167 | C22×C4 | — | 6 | 3 | no | — |
| 168 | C22×C4 | — | 6 | 3 | no | — |
| 169 | C22×C4 | — | 6 | 3 | no | — |
| 170 | C22×C4 | C23×D3 | 1 | 1 | yes | characteristic |
| 171 | Q16 | — | 6 | 6 | no | — |
| 172 | SD16 | — | 6 | 12 | no | — |
| 173 | D8 | — | 6 | 6 | no | — |
| 174 | M16 | — | 2 | 6 | no | — |
| 175 | M16 | — | 6 | 6 | no | — |
| 176 | M16 | — | 12 | 6 | no | — |
| 177 | C2×C8 | — | 6 | 3 | no | — |
| 178 | C2×C8 | — | 6 | 3 | no | — |
| 179 | C2×C8 | — | 12 | 3 | no | — |
| 180 | C2×C8 | — | 12 | 3 | no | — |
| 181 | C2×C8 | — | 12 | 1 | no | — |
| 182 | C2×C8 | — | 12 | 3 | no | — |
| 183 | C2×C8 | C2×D12 | 1 | 6 | yes | — |
| 184 | C2×C8 | C2×D12 | 1 | 3 | yes | — |
| 185 | C2×C8 | C2×D12 | 1 | 3 | yes | — |
| 186 | C4⋊C4 | — | 2 | 3 | no | — |
| 187 | C4⋊C4 | — | 6 | 6 | no | — |
| 188 | C22⋊C4 | — | 2 | 3 | no | — |
| 189 | C22⋊C4 | — | 6 | 6 | no | — |
| 190 | C22⋊C4 | — | 6 | 6 | no | — |
| 191 | C42 | C2×D12 | 1 | 6 | yes | — |
| 192 | C22×C3 | — | 4 | 1 | no | — |
| 193 | C22×C3 | #525 | 1 | 3 | yes | — |
| 194 | C2×D3 | — | 2 | 6 | no | — |
| 195 | C2×D3 | — | 4 | 3 | no | — |
| 196 | C2×D3 | — | 4 | 1 | no | — |
| 197 | C2×D3 | — | 4 | 6 | no | — |
| 198 | C2×D3 | — | 4 | 3 | no | — |
| 199 | C3×C4 | — | 4 | 6 | no | — |
| 200 | C3×C4 | #525 | 1 | 3 | yes | — |
| 201 | C3×C4 | C2×#127 | 1 | 1 | yes | characteristic |
| 202 | Dic3 | — | 4 | 3 | no | — |
| 203 | Dic3 | — | 4 | 1 | no | — |
| 204 | C23 | — | 6 | 3 | no | — |
| 205 | C23 | — | 6 | 3 | no | — |
| 206 | C23 | — | 12 | 2 | no | — |
| 207 | C23 | D3×#41 | 1 | 1 | yes | characteristic |
| 208 | Q8 | — | 6 | 6 | no | — |
| 209 | D4 | — | 6 | 6 | no | — |
| 210 | D4 | — | 12 | 12 | no | — |
| 211 | D4 | — | 12 | 6 | no | — |
| 212 | C2×C4 | — | 2 | 6 | no | — |
| 213 | C2×C4 | — | 6 | 3 | no | — |
| 214 | C2×C4 | — | 6 | 3 | no | — |
| 215 | C2×C4 | — | 12 | 1 | no | — |
| 216 | C2×C4 | — | 12 | 3 | no | — |
| 217 | C2×C4 | — | 12 | 3 | no | — |
| 218 | C2×C4 | — | 12 | 3 | no | — |
| 219 | C2×C4 | C22×D12 | 1 | 3 | yes | — |
| 220 | C2×C4 | D3×#41 | 1 | 3 | yes | — |
| 221 | C8 | — | 2 | 6 | no | — |
| 222 | C8 | — | 12 | 1 | no | — |
| 223 | C8 | — | 12 | 3 | no | — |
| 224 | C8 | #874 | 1 | 3 | yes | — |
| 225 | C8 | #874 | 1 | 1 | yes | characteristic |
| 226 | C2×C3 | — | 2 | 3 | no | — |
| 227 | C2×C3 | #2397 | 1 | 1 | yes | characteristic |
| 228 | D3 | — | 4 | 2 | no | — |
| 229 | D3 | — | 8 | 3 | no | — |
| 230 | C22 | — | 4 | 1 | no | — |
| 231 | C22 | — | 6 | 6 | no | — |
| 232 | C22 | — | 12 | 3 | no | — |
| 233 | C22 | — | 12 | 6 | no | — |
| 234 | C22 | — | 12 | 1 | no | — |
| 235 | C22 | — | 12 | 3 | no | — |
| 236 | C22 | #5524 | 1 | 3 | yes | — |
| 237 | C4 | — | 4 | 6 | no | — |
| 238 | C4 | — | 12 | 1 | no | — |
| 239 | C4 | — | 12 | 3 | no | — |
| 240 | C4 | C2×#874 | 1 | 1 | yes | characteristic |
| 241 | C4 | #5524 | 1 | 3 | yes | — |
| 242 | C3 | #23794 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 243 | C2 | — | 2 | 3 | no | — |
| 244 | C2 | — | 12 | 2 | no | — |
| 245 | C2 | — | 24 | 3 | no | — |
| 246 | C2 | #73418 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C8 | #874 |
| commutator, cosocle | C22×C3×C4 | C24 |
| Frattini | C22×C4 | C23×D3 |
| Fitting | C3×#1747 | C2 |
| socle | C2×C3 | #2397 |
| hypercenter | #1747 | D3 |
| hypocenter | C3 | #23794 |
| LCS | G ▷ C22×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ #1747 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #78579 | C2 | semidirect |
| 2. #78553 | C2 | semidirect |
| 3. C3×#1747 | C2 | semidirect |
| 4. #68727 | C2 | semidirect |
| 5. #68721 | C2 | semidirect |
| 6. #68715 | C2 | semidirect |
| 7. #68703 | C2 | semidirect |
| 31. #1747 | D3 | semidirect |
| 53. C3×#97 | D4 | semidirect |
| 54. C3×C4×C8 | D4 | semidirect |
| 135. #97 | D12 | semidirect |
| 136. C4×C8 | D12 | semidirect |
| 242. C3 | #23794 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #78579 | 2 |
| #78553 | 2 |
| C3×#1747 | 2 |
| #68727 | 2 |
| #68721 | 2 |
| #68715 | 2 |
| #68703 | 2 |
| #23794 | 3 |