Finite Group DatabasebetaFinite Group DatabasebetaOrder 3968.
354 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 41 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 | #14521684 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #14521237 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #14521216 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #14520835 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C31×#353 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #11758561 | C22 | 1 | 1 | yes | characteristic, complemented |
| 7 | C2×D4×D31 | — | 2 | 2 | no | — |
| 8 | C31×#127 | C22 | 1 | 1 | yes | characteristic |
| 9 | #11758515 | — | 2 | 2 | no | — |
| 10 | #11758514 | — | 2 | 1 | no | — |
| 11 | #11758514 | C22 | 1 | 2 | yes | — |
| 12 | #11758509 | — | 2 | 1 | no | — |
| 13 | #11758459 | — | 2 | 2 | no | — |
| 14 | C4×D124 | — | 2 | 1 | no | — |
| 15 | C31×#99 | C22 | 1 | 2 | yes | complemented |
| 16 | #11758421 | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | #11758394 | — | 2 | 2 | no | — |
| 18 | #11758393 | — | 2 | 1 | no | — |
| 19 | C23×D31 | — | 2 | 1 | no | — |
| 20 | C23×D31 | D4 | 1 | 2 | yes | complemented |
| 21 | C2×D4×C31 | — | 2 | 2 | no | — |
| 22 | C2×D4×C31 | D4 | 1 | 1 | yes | characteristic, complemented |
| 23 | C2×D4×C31 | C23 | 1 | 1 | yes | characteristic |
| 24 | C2×C22⋊D31 | — | 2 | 2 | no | — |
| 25 | C2×C22⋊D31 | — | 2 | 2 | no | — |
| 26 | C2×C22⋊D31 | — | 2 | 2 | no | — |
| 27 | #92497 | — | 4 | 1 | no | — |
| 28 | D4×D31 | — | 4 | 1 | no | — |
| 29 | D4×D31 | — | 4 | 8 | no | — |
| 30 | #92495 | — | 2 | 1 | no | — |
| 31 | C2×D124 | — | 2 | 1 | no | — |
| 32 | C2×D124 | — | 4 | 2 | no | — |
| 33 | C2×C4×D31 | — | 2 | 2 | no | — |
| 34 | C22⋊C4×C31 | — | 2 | 2 | no | — |
| 35 | C42×C31 | D4 | 1 | 1 | yes | characteristic, complemented |
| 36 | C22⋊Dic31 | — | 2 | 1 | no | — |
| 37 | C22⋊Dic31 | D4 | 1 | 2 | yes | complemented |
| 38 | #92480 | — | 4 | 1 | no | — |
| 39 | #92479 | — | 4 | 1 | no | — |
| 40 | (C2×D31)⋊C4 | — | 2 | 2 | no | — |
| 41 | (C2×D31)⋊C4 | — | 4 | 2 | no | — |
| 42 | C4⋊Dic31 | — | 2 | 1 | no | — |
| 43 | #92474 | — | 2 | 1 | no | — |
| 44 | C23×C31 | — | 2 | 1 | no | — |
| 45 | C23×C31 | C2×D4 | 1 | 2 | yes | — |
| 46 | C22×D31 | — | 2 | 2 | no | — |
| 47 | C22×D31 | — | 2 | 2 | no | — |
| 48 | C22×D31 | — | 4 | 4 | no | — |
| 49 | C22×D31 | — | 4 | 4 | no | — |
| 50 | C22×D31 | — | 4 | 2 | no | — |
| 51 | C22×D31 | — | 8 | 1 | no | — |
| 52 | D4×C31 | — | 4 | 1 | no | — |
| 53 | D4×C31 | — | 4 | 1 | no | — |
| 54 | D4×C31 | — | 4 | 4 | no | — |
| 55 | C2×C4×C31 | — | 2 | 1 | no | — |
| 56 | C2×C4×C31 | — | 4 | 2 | no | — |
| 57 | C2×C4×C31 | C2×D4 | 1 | 1 | yes | characteristic |
| 58 | C22⋊D31 | — | 4 | 4 | no | — |
| 59 | C22⋊D31 | — | 4 | 4 | no | — |
| 60 | C22⋊D31 | — | 4 | 2 | no | — |
| 61 | C22⋊D31 | — | 4 | 1 | no | — |
| 62 | C2×Dic31 | — | 2 | 2 | no | — |
| 63 | C2×Dic31 | — | 2 | 2 | no | — |
| 64 | D124 | — | 2 | 4 | no | — |
| 65 | D124 | — | 2 | 1 | no | — |
| 66 | D124 | — | 8 | 4 | no | — |
| 67 | C4×D31 | — | 4 | 4 | no | — |
| 68 | C4×D31 | — | 4 | 1 | no | — |
| 69 | Dic62 | — | 2 | 1 | no | — |
| 70 | C31⋊C8 | — | 4 | 1 | no | — |
| 71 | #2196 | — | 31 | 1 | no | maximal, Sylow |
| 72 | C22×C31 | — | 2 | 2 | no | — |
| 73 | C22×C31 | — | 4 | 2 | no | — |
| 74 | C22×C31 | — | 4 | 2 | no | — |
| 75 | C22×C31 | — | 4 | 1 | no | — |
| 76 | C22×C31 | #120 | 1 | 1 | yes | characteristic |
| 77 | C2×D31 | — | 2 | 4 | no | — |
| 78 | C2×D31 | — | 4 | 2 | no | — |
| 79 | C2×D31 | — | 4 | 2 | no | — |
| 80 | C2×D31 | — | 4 | 4 | no | — |
| 81 | C2×D31 | — | 4 | 1 | no | — |
| 82 | C2×D31 | — | 8 | 2 | no | — |
| 83 | C2×D31 | — | 8 | 4 | no | — |
| 84 | C2×D31 | — | 8 | 2 | no | — |
| 85 | C4×C31 | — | 2 | 2 | no | — |
| 86 | C4×C31 | — | 2 | 1 | no | — |
| 87 | C4×C31 | — | 8 | 2 | no | — |
| 88 | Dic31 | — | 4 | 2 | no | — |
| 89 | Dic31 | — | 4 | 1 | no | — |
| 90 | D42 | — | 31 | 1 | no | — |
| 91 | B(4,𝔽2) | — | 31 | 2 | no | — |
| 92 | #453 | — | 31 | 1 | no | — |
| 93 | #353 | D31 | 1 | 1 | yes | characteristic, complemented |
| 94 | AΣL(1,𝔽16) | — | 31 | 2 | no | — |
| 95 | C2×C31 | — | 2 | 1 | no | — |
| 96 | C2×C31 | — | 4 | 2 | no | — |
| 97 | C2×C31 | — | 8 | 1 | no | — |
| 98 | C2×C31 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 99 | D31 | — | 4 | 4 | no | — |
| 100 | D31 | — | 8 | 1 | no | — |
| 101 | H(5,𝔽2) | — | 31 | 1 | no | — |
| 102 | C22×D4 | — | 62 | 2 | no | — |
| 103 | AGL(1,ℤ/8) | — | 62 | 1 | no | — |
| 104 | #127 | C2×D31 | 1 | 1 | yes | characteristic |
| 105 | #121 | — | 62 | 2 | no | — |
| 106 | #120 | — | 31 | 2 | no | — |
| 107 | #120 | — | 62 | 2 | no | — |
| 108 | #120 | — | 62 | 1 | no | — |
| 109 | C4×D4 | — | 62 | 1 | no | — |
| 110 | #104 | — | 62 | 1 | no | — |
| 111 | #100 | — | 31 | 1 | no | — |
| 112 | #99 | — | 62 | 2 | no | — |
| 113 | #99 | C2×D31 | 1 | 2 | yes | complemented |
| 114 | C31 | #2196 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 115 | C24 | — | 31 | 2 | no | — |
| 116 | C24 | — | 62 | 1 | no | — |
| 117 | #41 | — | 62 | 1 | no | — |
| 118 | #41 | — | 124 | 1 | no | — |
| 119 | C2×D4 | — | 2 | 2 | no | — |
| 120 | C2×D4 | — | 62 | 1 | no | — |
| 121 | C2×D4 | — | 62 | 2 | no | — |
| 122 | C2×D4 | — | 62 | 2 | no | — |
| 123 | C2×D4 | — | 62 | 2 | no | — |
| 124 | C2×D4 | — | 124 | 2 | no | — |
| 125 | C2×D4 | — | 124 | 1 | no | — |
| 126 | C2×D4 | — | 124 | 8 | no | — |
| 127 | C2×D4 | D124 | 1 | 1 | yes | characteristic, complemented |
| 128 | C2×D4 | C22×D31 | 1 | 1 | yes | characteristic |
| 129 | C22×C4 | — | 62 | 2 | no | — |
| 130 | SD16 | — | 124 | 1 | no | — |
| 131 | D8 | — | 124 | 1 | no | — |
| 132 | M16 | — | 62 | 1 | no | — |
| 133 | C4⋊C4 | — | 62 | 1 | no | — |
| 134 | C22⋊C4 | — | 2 | 2 | no | — |
| 135 | C22⋊C4 | — | 31 | 2 | no | — |
| 136 | C22⋊C4 | — | 62 | 1 | no | — |
| 137 | C22⋊C4 | — | 62 | 2 | no | — |
| 138 | C22⋊C4 | — | 124 | 2 | no | — |
| 139 | C42 | D124 | 1 | 1 | yes | characteristic, complemented |
| 140 | C23 | — | 2 | 1 | no | — |
| 141 | C23 | — | 62 | 2 | no | — |
| 142 | C23 | — | 62 | 2 | no | — |
| 143 | C23 | — | 124 | 2 | no | — |
| 144 | C23 | — | 124 | 4 | no | — |
| 145 | C23 | — | 124 | 4 | no | — |
| 146 | C23 | — | 248 | 1 | no | — |
| 147 | C23 | D4×D31 | 1 | 2 | yes | — |
| 148 | Q8 | — | 62 | 1 | no | — |
| 149 | D4 | — | 4 | 1 | no | — |
| 150 | D4 | — | 4 | 1 | no | — |
| 151 | D4 | — | 4 | 4 | no | — |
| 152 | D4 | — | 62 | 4 | no | — |
| 153 | D4 | — | 62 | 1 | no | — |
| 154 | D4 | — | 124 | 2 | no | — |
| 155 | D4 | — | 124 | 4 | no | — |
| 156 | D4 | — | 124 | 4 | no | — |
| 157 | D4 | — | 124 | 1 | no | — |
| 158 | D4 | — | 248 | 4 | no | — |
| 159 | C2×C4 | — | 2 | 1 | no | — |
| 160 | C2×C4 | — | 4 | 2 | no | — |
| 161 | C2×C4 | — | 62 | 2 | no | — |
| 162 | C2×C4 | — | 62 | 2 | no | — |
| 163 | C2×C4 | — | 124 | 4 | no | — |
| 164 | C2×C4 | — | 124 | 1 | no | — |
| 165 | C2×C4 | C2×D124 | 1 | 1 | yes | characteristic |
| 166 | C8 | — | 124 | 1 | no | — |
| 167 | C22 | — | 2 | 2 | no | — |
| 168 | C22 | — | 4 | 2 | no | — |
| 169 | C22 | — | 4 | 2 | no | — |
| 170 | C22 | — | 4 | 1 | no | — |
| 171 | C22 | — | 62 | 4 | no | — |
| 172 | C22 | — | 124 | 2 | no | — |
| 173 | C22 | — | 124 | 4 | no | — |
| 174 | C22 | — | 124 | 2 | no | — |
| 175 | C22 | — | 124 | 1 | no | — |
| 176 | C22 | — | 248 | 2 | no | — |
| 177 | C22 | — | 248 | 2 | no | — |
| 178 | C22 | — | 248 | 4 | no | — |
| 179 | C22 | #11758459 | 1 | 1 | yes | characteristic |
| 180 | C4 | — | 2 | 2 | no | — |
| 181 | C4 | — | 2 | 1 | no | — |
| 182 | C4 | — | 8 | 2 | no | — |
| 183 | C4 | — | 124 | 2 | no | — |
| 184 | C4 | — | 124 | 1 | no | — |
| 185 | C2 | — | 2 | 1 | no | — |
| 186 | C2 | — | 4 | 2 | no | — |
| 187 | C2 | — | 8 | 1 | no | — |
| 188 | C2 | — | 124 | 4 | no | — |
| 189 | C2 | — | 248 | 1 | no | — |
| 190 | C2 | #14521237 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #14521684 | C2 | semidirect |
| 2. #14521237 | C2 | semidirect |
| 3. #14521216 | C2 | semidirect |
| 4. #14520835 | C2 | semidirect |
| 5. C31×#353 | C2 | semidirect |
| 6. #11758561 | C22 | semidirect |
| 15. C31×#99 | C22 | semidirect |
| 16. #11758421 | C22 | semidirect |
| 20. C23×D31 | D4 | semidirect |
| 22. C2×D4×C31 | D4 | semidirect |
| 35. C42×C31 | D4 | semidirect |
| 37. C22⋊Dic31 | D4 | semidirect |
| 93. #353 | D31 | semidirect |
| 113. #99 | C2×D31 | semidirect |
| 114. C31 | #2196 | semidirect |
| 127. C2×D4 | D124 | semidirect |
| 139. C42 | D124 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #14521684 | 2 |
| #14521237 | 2 |
| #14521216 | 2 |
| #14520835 | 2 |
| C31×#353 | 2 |
| #2196 | 31 |