Finite Group DatabasebetaFinite Group DatabasebetaOrder 7808.
716 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 716 conjugacy classes of subgroups — too many to draw. Only its 133 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 | #16840597 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #16840334 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #16839953 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #16839952 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #16839951 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #16839734 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C61×#458 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #14519824 | — | 2 | 3 | no | — |
| 9 | #14519822 | — | 2 | 3 | no | — |
| 10 | #14519822 | C22 | 1 | 3 | yes | — |
| 11 | C2×#11758224 | — | 2 | 3 | no | — |
| 12 | C2×#11758224 | C22 | 1 | 3 | yes | — |
| 13 | #41×D61 | — | 2 | 3 | no | — |
| 14 | C22×Dic122 | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | #14519776 | — | 2 | 3 | no | — |
| 16 | #14519763 | — | 2 | 3 | no | — |
| 17 | #14519761 | C22 | 1 | 3 | yes | complemented |
| 18 | #14519760 | C22 | 1 | 3 | yes | complemented |
| 19 | #14519758 | C22 | 1 | 3 | yes | complemented |
| 20 | #14519755 | — | 2 | 3 | no | — |
| 21 | #14519754 | — | 2 | 3 | no | — |
| 22 | #14519733 | C22 | 1 | 3 | yes | complemented |
| 23 | #14519692 | C22 | 1 | 3 | yes | complemented |
| 24 | #14519691 | C22 | 1 | 6 | yes | complemented |
| 25 | H(5,𝔽2)×C61 | C22 | 1 | 1 | yes | characteristic |
| 26 | #123×C61 | C22 | 1 | 3 | yes | complemented |
| 27 | #99×C61 | C22 | 1 | 3 | yes | complemented |
| 28 | #41×C61 | — | 2 | 3 | no | — |
| 29 | C2×D4×C61 | — | 2 | 3 | no | — |
| 30 | C2×D4×C61 | C23 | 1 | 3 | yes | — |
| 31 | C22×C4×C61 | C23 | 1 | 1 | yes | characteristic |
| 32 | C2×C22⋊D61 | — | 2 | 3 | no | — |
| 33 | C2×C22⋊D61 | — | 2 | 3 | no | — |
| 34 | C2×C22⋊D61 | D4 | 1 | 3 | yes | — |
| 35 | C2×C22⋊D61 | C23 | 1 | 3 | yes | — |
| 36 | C22×Dic61 | — | 2 | 1 | no | — |
| 37 | C22×Dic61 | D4 | 1 | 3 | yes | — |
| 38 | C22×Dic61 | C23 | 1 | 1 | yes | characteristic |
| 39 | C22×Dic61 | C23 | 1 | 1 | yes | characteristic |
| 40 | #11758226 | — | 2 | 1 | no | — |
| 41 | Q8×D61 | — | 2 | 3 | no | — |
| 42 | #11758224 | — | 2 | 3 | no | — |
| 43 | #11758224 | — | 2 | 6 | no | — |
| 44 | #11758224 | — | 2 | 6 | no | — |
| 45 | #11758224 | — | 2 | 6 | no | — |
| 46 | #11758224 | — | 2 | 6 | no | — |
| 47 | D4×D61 | — | 2 | 3 | no | — |
| 48 | D4×D61 | — | 2 | 6 | no | — |
| 49 | #11758222 | — | 2 | 6 | no | — |
| 50 | #11758222 | — | 2 | 3 | no | — |
| 51 | C2×C4×D61 | — | 2 | 3 | no | — |
| 52 | C2×C4×D61 | D4 | 1 | 3 | yes | — |
| 53 | C2×Dic122 | — | 2 | 3 | no | — |
| 54 | C2×Dic122 | — | 2 | 3 | no | — |
| 55 | C2×Dic122 | — | 2 | 3 | no | — |
| 56 | C2×Dic122 | D4 | 1 | 3 | yes | — |
| 57 | C4⋊C4×C61 | — | 2 | 3 | no | — |
| 58 | C22⋊C4×C61 | — | 2 | 3 | no | — |
| 59 | C22⋊C4×C61 | C23 | 1 | 3 | yes | — |
| 60 | C22⋊Dic61 | — | 2 | 3 | no | — |
| 61 | C22⋊Dic61 | C23 | 1 | 3 | yes | — |
| 62 | #11758198 | — | 2 | 3 | no | — |
| 63 | #11758198 | — | 2 | 3 | no | — |
| 64 | C4⋊Dic61 | — | 2 | 3 | no | — |
| 65 | #11758196 | — | 2 | 3 | no | — |
| 66 | #11758196 | — | 2 | 3 | no | — |
| 67 | #11758196 | — | 2 | 3 | no | — |
| 68 | C4×Dic61 | — | 2 | 3 | no | — |
| 69 | C23×C61 | — | 2 | 1 | no | — |
| 70 | C23×C61 | C2×D4 | 1 | 3 | yes | — |
| 71 | C23×C61 | C24 | 1 | 1 | yes | characteristic |
| 72 | C22×D61 | — | 2 | 3 | no | — |
| 73 | C22×D61 | C2×D4 | 1 | 3 | yes | — |
| 74 | Q8×C61 | — | 2 | 1 | no | — |
| 75 | D4×C61 | — | 2 | 3 | no | — |
| 76 | D4×C61 | — | 2 | 6 | no | — |
| 77 | C2×C4×C61 | — | 2 | 3 | no | — |
| 78 | C2×C4×C61 | — | 2 | 3 | no | — |
| 79 | C2×C4×C61 | — | 2 | 3 | no | — |
| 80 | C2×C4×C61 | C2×D4 | 1 | 3 | yes | — |
| 81 | C22⋊D61 | — | 2 | 6 | no | — |
| 82 | C22⋊D61 | — | 2 | 6 | no | — |
| 83 | C22⋊D61 | — | 2 | 6 | no | — |
| 84 | C2×Dic61 | — | 2 | 3 | no | — |
| 85 | C2×Dic61 | — | 2 | 3 | no | — |
| 86 | C2×Dic61 | — | 2 | 3 | no | — |
| 87 | C2×Dic61 | — | 2 | 3 | no | — |
| 88 | C2×Dic61 | — | 2 | 3 | no | — |
| 89 | C2×Dic61 | C2×D4 | 1 | 3 | yes | — |
| 90 | C2×Dic61 | C2×D4 | 1 | 3 | yes | — |
| 91 | C2×Dic61 | C2×D4 | 1 | 3 | yes | — |
| 92 | D244 | — | 2 | 3 | no | — |
| 93 | C4×D61 | — | 2 | 6 | no | — |
| 94 | C4×D61 | — | 2 | 3 | no | — |
| 95 | C4×D61 | — | 2 | 3 | no | — |
| 96 | Dic122 | — | 2 | 3 | no | — |
| 97 | Dic122 | — | 2 | 6 | no | — |
| 98 | Dic122 | — | 4 | 3 | no | — |
| 99 | Dic122 | — | 4 | 1 | no | — |
| 100 | C22×C61 | — | 2 | 3 | no | — |
| 101 | C22×C61 | — | 4 | 1 | no | — |
| 102 | C22×C61 | — | 4 | 3 | no | — |
| 103 | C22×C61 | — | 8 | 1 | no | — |
| 104 | C22×C61 | C22×D4 | 1 | 3 | yes | — |
| 105 | C2×D61 | — | 2 | 3 | no | — |
| 106 | C2×D61 | — | 4 | 3 | no | — |
| 107 | C2×D61 | — | 8 | 3 | no | — |
| 108 | C4×C61 | — | 2 | 3 | no | — |
| 109 | C4×C61 | — | 4 | 3 | no | — |
| 110 | C4×C61 | — | 4 | 1 | no | — |
| 111 | Dic61 | — | 2 | 3 | no | — |
| 112 | Dic61 | — | 4 | 3 | no | — |
| 113 | Dic61 | — | 4 | 1 | no | — |
| 114 | Dic61 | #120 | 1 | 3 | yes | — |
| 115 | Dic61 | #120 | 1 | 1 | yes | characteristic |
| 116 | #3028 | — | 61 | 1 | no | maximal, Sylow |
| 117 | C2×C61 | — | 2 | 3 | no | — |
| 118 | C2×C61 | — | 4 | 3 | no | — |
| 119 | C2×C61 | C2×#120 | 1 | 1 | yes | characteristic |
| 120 | D61 | — | 4 | 3 | no | — |
| 121 | #585 | — | 61 | 1 | no | — |
| 122 | #536 | — | 61 | 3 | no | — |
| 123 | #458 | — | 61 | 1 | no | — |
| 124 | #458 | — | 61 | 3 | no | — |
| 125 | #458 | — | 61 | 3 | no | — |
| 126 | #458 | D61 | 1 | 1 | yes | characteristic, complemented |
| 127 | #410 | — | 61 | 3 | no | — |
| 128 | C61 | #3028 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 129 | 2−1+4 | — | 122 | 3 | no | — |
| 130 | H(5,𝔽2) | — | 61 | 3 | no | — |
| 131 | H(5,𝔽2) | — | 122 | 3 | no | — |
| 132 | H(5,𝔽2) | C2×D61 | 1 | 1 | yes | characteristic |
| 133 | C2×#41 | — | 61 | 3 | no | — |
| 134 | C2×#41 | — | 122 | 3 | no | — |
| 135 | C2×#41 | — | 122 | 3 | no | — |
| 136 | C22×Q8 | — | 61 | 1 | no | — |
| 137 | C4⋊Q8 | — | 122 | 3 | no | — |
| 138 | #124 | — | 122 | 3 | no | — |
| 139 | #123 | — | 61 | 3 | no | — |
| 140 | #123 | — | 61 | 3 | no | — |
| 141 | #123 | — | 61 | 3 | no | — |
| 142 | #123 | C2×D61 | 1 | 3 | yes | complemented |
| 143 | C22⋊Q8 | — | 122 | 3 | no | — |
| 144 | C22⋊Q8 | — | 122 | 3 | no | — |
| 145 | #117 | — | 61 | 3 | no | — |
| 146 | #99 | — | 61 | 3 | no | — |
| 147 | #99 | — | 61 | 6 | no | — |
| 148 | #99 | C2×D61 | 1 | 3 | yes | complemented |
| 149 | #41 | — | 2 | 3 | no | — |
| 150 | #41 | — | 122 | 6 | no | — |
| 151 | #41 | — | 122 | 6 | no | — |
| 152 | #41 | — | 122 | 6 | no | — |
| 153 | #41 | — | 122 | 6 | no | — |
| 154 | #41 | — | 122 | 3 | no | — |
| 155 | #41 | — | 122 | 6 | no | — |
| 156 | #41 | — | 122 | 1 | no | — |
| 157 | #41 | — | 122 | 3 | no | — |
| 158 | C2×Q8 | — | 61 | 3 | no | — |
| 159 | C2×Q8 | — | 122 | 3 | no | — |
| 160 | C2×Q8 | — | 122 | 3 | no | — |
| 161 | C2×Q8 | — | 122 | 3 | no | — |
| 162 | C2×Q8 | — | 122 | 3 | no | — |
| 163 | C2×D4 | — | 2 | 3 | no | — |
| 164 | C2×D4 | — | 61 | 3 | no | — |
| 165 | C2×D4 | — | 61 | 3 | no | — |
| 166 | C2×D4 | — | 122 | 6 | no | — |
| 167 | C2×D4 | — | 122 | 3 | no | — |
| 168 | C2×D4 | — | 122 | 3 | no | — |
| 169 | C2×D4 | — | 122 | 3 | no | — |
| 170 | C2×D4 | C22×D61 | 1 | 3 | yes | — |
| 171 | C22×C4 | — | 61 | 1 | no | — |
| 172 | C22×C4 | — | 61 | 3 | no | — |
| 173 | C22×C4 | — | 61 | 3 | no | — |
| 174 | C22×C4 | — | 61 | 1 | no | — |
| 175 | C22×C4 | — | 122 | 3 | no | — |
| 176 | C22×C4 | — | 122 | 1 | no | — |
| 177 | C22×C4 | C22×D61 | 1 | 1 | yes | characteristic |
| 178 | C4⋊C4 | — | 2 | 3 | no | — |
| 179 | C4⋊C4 | — | 122 | 3 | no | — |
| 180 | C4⋊C4 | — | 122 | 3 | no | — |
| 181 | C4⋊C4 | — | 122 | 3 | no | — |
| 182 | C4⋊C4 | — | 122 | 3 | no | — |
| 183 | C22⋊C4 | — | 2 | 3 | no | — |
| 184 | C22⋊C4 | — | 61 | 3 | no | — |
| 185 | C22⋊C4 | — | 122 | 3 | no | — |
| 186 | C22⋊C4 | — | 122 | 3 | no | — |
| 187 | C22⋊C4 | — | 122 | 3 | no | — |
| 188 | C22⋊C4 | C22×D61 | 1 | 3 | yes | — |
| 189 | C42 | — | 122 | 3 | no | — |
| 190 | C23 | — | 2 | 1 | no | — |
| 191 | C23 | — | 61 | 3 | no | — |
| 192 | C23 | — | 122 | 3 | no | — |
| 193 | C23 | D4×D61 | 1 | 3 | yes | — |
| 194 | C23 | C23×D61 | 1 | 1 | yes | characteristic |
| 195 | Q8 | — | 2 | 1 | no | — |
| 196 | Q8 | — | 122 | 6 | no | — |
| 197 | Q8 | — | 122 | 3 | no | — |
| 198 | Q8 | — | 244 | 1 | no | — |
| 199 | Q8 | — | 244 | 3 | no | — |
| 200 | D4 | — | 2 | 3 | no | — |
| 201 | D4 | — | 2 | 6 | no | — |
| 202 | D4 | — | 122 | 3 | no | — |
| 203 | D4 | — | 122 | 6 | no | — |
| 204 | D4 | — | 122 | 6 | no | — |
| 205 | D4 | — | 122 | 6 | no | — |
| 206 | C2×C4 | — | 2 | 3 | no | — |
| 207 | C2×C4 | — | 2 | 3 | no | — |
| 208 | C2×C4 | — | 2 | 3 | no | — |
| 209 | C2×C4 | — | 61 | 3 | no | — |
| 210 | C2×C4 | — | 61 | 3 | no | — |
| 211 | C2×C4 | — | 61 | 3 | no | — |
| 212 | C2×C4 | — | 122 | 3 | no | — |
| 213 | C2×C4 | — | 122 | 3 | no | — |
| 214 | C2×C4 | — | 122 | 6 | no | — |
| 215 | C2×C4 | — | 122 | 3 | no | — |
| 216 | C2×C4 | — | 122 | 3 | no | — |
| 217 | C2×C4 | — | 122 | 3 | no | — |
| 218 | C2×C4 | — | 122 | 3 | no | — |
| 219 | C2×C4 | — | 122 | 3 | no | — |
| 220 | C2×C4 | D4×D61 | 1 | 3 | yes | — |
| 221 | C22 | — | 2 | 3 | no | — |
| 222 | C22 | — | 4 | 1 | no | — |
| 223 | C22 | — | 4 | 3 | no | — |
| 224 | C22 | — | 8 | 1 | no | — |
| 225 | C22 | — | 122 | 3 | no | — |
| 226 | C22 | — | 488 | 3 | no | — |
| 227 | C22 | — | 244 | 3 | no | — |
| 228 | C22 | C2×D4×D61 | 1 | 3 | yes | — |
| 229 | C4 | — | 2 | 3 | no | — |
| 230 | C4 | — | 4 | 3 | no | — |
| 231 | C4 | — | 4 | 1 | no | — |
| 232 | C4 | — | 61 | 1 | no | — |
| 233 | C4 | — | 61 | 3 | no | — |
| 234 | C4 | — | 122 | 3 | no | — |
| 235 | C4 | — | 244 | 3 | no | — |
| 236 | C4 | — | 244 | 1 | no | — |
| 237 | C2 | — | 2 | 3 | no | — |
| 238 | C2 | — | 4 | 3 | no | — |
| 239 | C2 | — | 244 | 3 | no | — |
| 240 | C2 | #120×D61 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #120×D61 |
| commutator | C23×C61 | C24 |
| Frattini | C23 | C23×D61 |
| Fitting | C61×#458 | C2 |
| socle | C2×C61 | C2×#120 |
| hypercenter | #458 | D61 |
| hypocenter | C61 | #3028 |
| LCS | G ▷ C23×C61 ▷ C2×C61 ▷ C61 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #458 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #16840334 | C2 | semidirect |
| 3. #16839953 | C2 | semidirect |
| 4. #16839952 | C2 | semidirect |
| 5. #16839951 | C2 | semidirect |
| 6. #16839734 | C2 | semidirect |
| 7. C61×#458 | C2 | semidirect |
| 14. C22×Dic122 | C22 | semidirect |
| 17. #14519761 | C22 | semidirect |
| 18. #14519760 | C22 | semidirect |
| 19. #14519758 | C22 | semidirect |
| 22. #14519733 | C22 | semidirect |
| 23. #14519692 | C22 | semidirect |
| 24. #14519691 | C22 | semidirect |
| 26. #123×C61 | C22 | semidirect |
| 27. #99×C61 | C22 | semidirect |
| 126. #458 | D61 | semidirect |
| 128. C61 | #3028 | semidirect |
| 142. #123 | C2×D61 | semidirect |
| 148. #99 | C2×D61 | semidirect |