← #87166
Subgroups — #87166
Order 384.
1424 conjugacy classes of subgroups.
The subgroup lattice has 186 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #5964 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#568 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#948 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #5750 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | D3×#131 | C2 | 1 | 8 | yes | maximal, complemented |
| 6 | #5745 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | C2×#907 | C2 | 1 | 6 | yes | maximal, complemented |
| 8 | #3573 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C2×C3×#41 | C22 | 1 | 1 | yes | characteristic |
| 10 | #1010 | C4 | 1 | 4 | yes | — |
| 11 | #1009 | C4 | 1 | 4 | yes | — |
| 12 | D3×#41 | C22 | 1 | 8 | yes | — |
| 13 | #1007 | C4 | 1 | 2 | yes | — |
| 14 | #1004 | C4 | 1 | 6 | yes | — |
| 15 | C2×#235 | C22 | 1 | 6 | yes | — |
| 16 | C3×#131 | C22 | 1 | 8 | yes | complemented |
| 17 | C2×C3×M16 | C22 | 1 | 6 | yes | complemented |
| 18 | #948 | C22 | 1 | 4 | yes | complemented |
| 19 | #948 | C22 | 1 | 4 | yes | complemented |
| 20 | C2×#208 | C22 | 1 | 3 | yes | complemented |
| 21 | C22×C3⋊C8 | C22 | 1 | 3 | yes | complemented |
| 22 | #907 | C22 | 1 | 4 | yes | complemented |
| 23 | #907 | C22 | 1 | 12 | yes | complemented |
| 24 | #907 | C22 | 1 | 12 | yes | complemented |
| 25 | #907 | C22 | 1 | 12 | yes | complemented |
| 26 | D3×M16 | C22 | 1 | 24 | yes | complemented |
| 27 | #901 | C22 | 1 | 24 | yes | complemented |
| 28 | C2×ΓL(1,𝔽25) | C22 | 1 | 12 | yes | complemented |
| 29 | C2×D3×C8 | C22 | 1 | 12 | yes | complemented |
| 30 | #585 | — | 3 | 1 | no | — |
| 31 | #568 | — | 3 | 8 | no | — |
| 32 | #568 | — | 3 | 6 | no | — |
| 33 | #568 | D3 | 1 | 1 | yes | characteristic, complemented |
| 34 | C2×#131 | — | 3 | 1 | no | — |
| 35 | C2×#131 | — | 3 | 8 | no | — |
| 36 | C2×#131 | — | 3 | 6 | no | — |
| 37 | C3×#41 | C2×C4 | 1 | 4 | yes | — |
| 38 | C3×#41 | C23 | 1 | 4 | yes | — |
| 39 | C2×C3×Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | C2×C3×D4 | C2×C4 | 1 | 3 | yes | — |
| 41 | C22×C3×C4 | C23 | 1 | 3 | yes | — |
| 42 | C2×#67 | C2×C4 | 1 | 12 | yes | — |
| 43 | #239 | C2×C4 | 1 | 8 | yes | — |
| 44 | D3×Q8 | C2×C4 | 1 | 8 | yes | — |
| 45 | #237 | C2×C4 | 1 | 24 | yes | — |
| 46 | D3×D4 | C2×C4 | 1 | 24 | yes | — |
| 47 | #235 | C2×C4 | 1 | 12 | yes | — |
| 48 | #235 | C2×C4 | 1 | 12 | yes | — |
| 49 | #235 | C23 | 1 | 12 | yes | — |
| 50 | #235 | C23 | 1 | 4 | yes | — |
| 51 | C2×D12 | C2×C4 | 1 | 6 | yes | — |
| 52 | C2×C4×D3 | C23 | 1 | 12 | yes | — |
| 53 | C2×Dic6 | C2×C4 | 1 | 6 | yes | — |
| 54 | C3×M16 | C23 | 1 | 12 | yes | — |
| 55 | C3×M16 | C23 | 1 | 4 | yes | — |
| 56 | C2×C3×C8 | C23 | 1 | 12 | yes | — |
| 57 | #208 | C23 | 1 | 12 | yes | — |
| 58 | C2×C3⋊C8 | C23 | 1 | 6 | yes | — |
| 59 | C2×C3⋊C8 | C23 | 1 | 6 | yes | — |
| 60 | C2×C3⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 61 | C2×C3⋊C8 | C23 | 1 | 3 | yes | — |
| 62 | ΓL(1,𝔽25) | C23 | 1 | 8 | yes | — |
| 63 | ΓL(1,𝔽25) | C23 | 1 | 24 | yes | — |
| 64 | D3×C8 | C23 | 1 | 8 | yes | — |
| 65 | D3×C8 | C23 | 1 | 24 | yes | — |
| 66 | 2−1+4 | — | 3 | 4 | no | — |
| 67 | 2−1+4 | — | 3 | 2 | no | — |
| 68 | H(5,𝔽2) | — | 3 | 4 | no | — |
| 69 | H(5,𝔽2) | — | 3 | 6 | no | — |
| 70 | C2×#41 | — | 3 | 8 | no | — |
| 71 | C2×#41 | — | 3 | 6 | no | — |
| 72 | C2×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 73 | #131 | — | 3 | 4 | no | — |
| 74 | #131 | — | 3 | 4 | no | — |
| 75 | #131 | — | 3 | 12 | no | — |
| 76 | #131 | — | 3 | 4 | no | — |
| 77 | #131 | — | 3 | 12 | no | — |
| 78 | #131 | — | 3 | 12 | no | — |
| 79 | #131 | — | 3 | 24 | no | — |
| 80 | #131 | C2×D3 | 1 | 8 | yes | complemented |
| 81 | C2×M16 | — | 3 | 3 | no | — |
| 82 | C2×M16 | — | 3 | 24 | no | — |
| 83 | C2×M16 | — | 3 | 12 | no | — |
| 84 | C2×M16 | C2×D3 | 1 | 6 | yes | complemented |
| 85 | C22×C8 | — | 3 | 3 | no | — |
| 86 | C22×C8 | — | 3 | 12 | no | — |
| 87 | C23×C3 | C22×C4 | 1 | 3 | yes | — |
| 88 | C22×D3 | C22×C4 | 1 | 12 | yes | — |
| 89 | C3×Q8 | C22×C4 | 1 | 4 | yes | — |
| 90 | C3×D4 | C22×C4 | 1 | 12 | yes | — |
| 91 | C2×C3×C4 | C22×C4 | 1 | 6 | yes | — |
| 92 | C2×C3×C4 | C22×C4 | 1 | 3 | yes | — |
| 93 | C2×C3×C4 | C24 | 1 | 1 | yes | characteristic |
| 94 | C2×C3×C4 | C24 | 1 | 6 | yes | — |
| 95 | #67 | C22×C4 | 1 | 8 | yes | — |
| 96 | #67 | C22×C4 | 1 | 24 | yes | — |
| 97 | C2×Dic3 | C22×C4 | 1 | 12 | yes | — |
| 98 | D12 | C22×C4 | 1 | 4 | yes | — |
| 99 | D12 | C22×C4 | 1 | 12 | yes | — |
| 100 | C4×D3 | C22×C4 | 1 | 24 | yes | — |
| 101 | C4×D3 | C24 | 1 | 8 | yes | — |
| 102 | Dic6 | C22×C4 | 1 | 4 | yes | — |
| 103 | Dic6 | C22×C4 | 1 | 12 | yes | — |
| 104 | C3×C8 | C24 | 1 | 8 | yes | — |
| 105 | C3⋊C8 | C24 | 1 | 1 | yes | characteristic |
| 106 | C3⋊C8 | C24 | 1 | 6 | yes | — |
| 107 | C3⋊C8 | C24 | 1 | 1 | yes | characteristic |
| 108 | #41 | — | 3 | 8 | no | — |
| 109 | #41 | — | 3 | 24 | no | — |
| 110 | #41 | — | 3 | 12 | no | — |
| 111 | #41 | — | 3 | 12 | no | — |
| 112 | #41 | — | 3 | 4 | no | — |
| 113 | #41 | — | 3 | 12 | no | — |
| 114 | #41 | C4×D3 | 1 | 4 | yes | — |
| 115 | #41 | C22×D3 | 1 | 4 | yes | — |
| 116 | C2×Q8 | — | 3 | 8 | no | — |
| 117 | C2×Q8 | — | 3 | 6 | no | — |
| 118 | C2×Q8 | C4×D3 | 1 | 1 | yes | characteristic |
| 119 | C2×D4 | — | 3 | 12 | no | — |
| 120 | C2×D4 | — | 3 | 24 | no | — |
| 121 | C2×D4 | — | 3 | 6 | no | — |
| 122 | C2×D4 | C4×D3 | 1 | 3 | yes | — |
| 123 | C22×C4 | — | 3 | 12 | no | — |
| 124 | C22×C4 | C22×D3 | 1 | 3 | yes | — |
| 125 | M16 | — | 3 | 12 | no | — |
| 126 | M16 | — | 3 | 24 | no | — |
| 127 | M16 | — | 3 | 8 | no | — |
| 128 | M16 | C22×D3 | 1 | 12 | yes | — |
| 129 | M16 | C22×D3 | 1 | 4 | yes | — |
| 130 | C2×C8 | — | 3 | 1 | no | — |
| 131 | C2×C8 | — | 3 | 6 | no | — |
| 132 | C2×C8 | — | 3 | 3 | no | — |
| 133 | C2×C8 | — | 3 | 6 | no | — |
| 134 | C2×C8 | — | 3 | 8 | no | — |
| 135 | C2×C8 | — | 3 | 24 | no | — |
| 136 | C2×C8 | C22×D3 | 1 | 12 | yes | — |
| 137 | C22×C3 | — | 4 | 3 | no | — |
| 138 | C22×C3 | C23×C4 | 1 | 6 | yes | — |
| 139 | C22×C3 | C23×C4 | 1 | 1 | yes | characteristic |
| 140 | C2×D3 | — | 4 | 12 | no | — |
| 141 | C2×D3 | C23×C4 | 1 | 8 | yes | — |
| 142 | C3×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 143 | C3×C4 | C23×C4 | 1 | 6 | yes | — |
| 144 | C3×C4 | C25 | 1 | 1 | yes | characteristic |
| 145 | Dic3 | C23×C4 | 1 | 8 | yes | — |
| 146 | C23 | — | 3 | 12 | no | — |
| 147 | C23 | C2×C4×D3 | 1 | 3 | yes | — |
| 148 | Q8 | — | 3 | 12 | no | — |
| 149 | Q8 | — | 3 | 4 | no | — |
| 150 | Q8 | C2×C4×D3 | 1 | 4 | yes | — |
| 151 | D4 | — | 3 | 24 | no | — |
| 152 | D4 | — | 3 | 8 | no | — |
| 153 | D4 | — | 3 | 12 | no | — |
| 154 | D4 | — | 3 | 4 | no | — |
| 155 | D4 | C2×C4×D3 | 1 | 12 | yes | — |
| 156 | C2×C4 | — | 3 | 12 | no | — |
| 157 | C2×C4 | — | 3 | 24 | no | — |
| 158 | C2×C4 | — | 3 | 8 | no | — |
| 159 | C2×C4 | C2×C4×D3 | 1 | 6 | yes | — |
| 160 | C2×C4 | C2×C4×D3 | 1 | 3 | yes | — |
| 161 | C2×C4 | C23×D3 | 1 | 1 | yes | characteristic |
| 162 | C2×C4 | C23×D3 | 1 | 6 | yes | — |
| 163 | C8 | — | 3 | 1 | no | — |
| 164 | C8 | — | 3 | 1 | no | — |
| 165 | C8 | — | 3 | 6 | no | — |
| 166 | C8 | C23×D3 | 1 | 8 | yes | — |
| 167 | C2×C3 | — | 2 | 6 | no | — |
| 168 | C2×C3 | — | 2 | 1 | no | — |
| 169 | C2×C3 | C24×C4 | 1 | 1 | yes | characteristic |
| 170 | D3 | — | 2 | 8 | no | — |
| 171 | C22 | — | 3 | 8 | no | — |
| 172 | C22 | — | 4 | 3 | no | — |
| 173 | C22 | — | 12 | 12 | no | — |
| 174 | C22 | C22×C4×D3 | 1 | 6 | yes | — |
| 175 | C22 | C22×C4×D3 | 1 | 1 | yes | characteristic |
| 176 | C4 | — | 3 | 8 | no | — |
| 177 | C4 | C22×C4×D3 | 1 | 6 | yes | — |
| 178 | C4 | C22×C4×D3 | 1 | 1 | yes | characteristic |
| 179 | C4 | C24×D3 | 1 | 1 | yes | characteristic |
| 180 | C3 | #3573 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 181 | C2 | — | 2 | 1 | no | — |
| 182 | C2 | — | 2 | 6 | no | — |
| 183 | C2 | — | 6 | 8 | no | — |
| 184 | C2 | C23×C4×D3 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta