Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 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 | #5964 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#413 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #5234 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #5132 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #5129 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #1791 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C2×C3×#41 | C22 | 1 | 1 | yes | characteristic |
| 8 | #1010 | — | 2 | 2 | no | — |
| 9 | #1009 | — | 2 | 2 | no | — |
| 10 | D3×#41 | — | 2 | 4 | no | — |
| 11 | #1007 | — | 2 | 1 | no | — |
| 12 | #1004 | — | 2 | 1 | no | — |
| 13 | #1004 | C4 | 1 | 4 | yes | — |
| 14 | C2×#235 | — | 2 | 2 | no | — |
| 15 | C2×#235 | C22 | 1 | 2 | yes | — |
| 16 | C2×C3×M16 | C22 | 1 | 2 | yes | complemented |
| 17 | C2×#208 | C22 | 1 | 1 | yes | characteristic, complemented |
| 18 | C22×C3⋊C8 | C22 | 1 | 1 | yes | characteristic, complemented |
| 19 | C3×#101 | — | 2 | 1 | no | — |
| 20 | C3×#100 | — | 2 | 1 | no | — |
| 21 | #830 | — | 2 | 2 | no | — |
| 22 | #824 | — | 2 | 2 | no | — |
| 23 | #823 | — | 2 | 2 | no | — |
| 24 | (C2×D3)⋊C8 | — | 2 | 4 | no | — |
| 25 | #585 | — | 3 | 1 | no | — |
| 26 | #413 | — | 3 | 2 | no | — |
| 27 | #413 | D3 | 1 | 1 | yes | characteristic, complemented |
| 28 | #408 | — | 3 | 1 | no | — |
| 29 | #408 | — | 3 | 2 | no | — |
| 30 | C3×#41 | — | 2 | 2 | no | — |
| 31 | C3×#41 | — | 2 | 2 | no | — |
| 32 | C2×C3×Q8 | D4 | 1 | 1 | yes | characteristic |
| 33 | C2×C3×D4 | C2×C4 | 1 | 2 | yes | — |
| 34 | C2×C3×D4 | D4 | 1 | 1 | yes | characteristic |
| 35 | C22×C3×C4 | D4 | 1 | 2 | yes | — |
| 36 | C22×C3×C4 | C23 | 1 | 1 | yes | characteristic |
| 37 | C2×#67 | — | 2 | 4 | no | — |
| 38 | C2×#67 | C2×C4 | 1 | 4 | yes | — |
| 39 | #239 | — | 2 | 4 | no | — |
| 40 | D3×Q8 | — | 2 | 4 | no | — |
| 41 | #237 | — | 2 | 4 | no | — |
| 42 | #237 | — | 2 | 8 | no | — |
| 43 | D3×D4 | — | 2 | 4 | no | — |
| 44 | D3×D4 | — | 2 | 8 | no | — |
| 45 | #235 | — | 2 | 4 | no | — |
| 46 | #235 | — | 2 | 4 | no | — |
| 47 | #235 | — | 2 | 4 | no | — |
| 48 | #235 | — | 2 | 2 | no | — |
| 49 | #235 | — | 2 | 2 | no | — |
| 50 | #235 | — | 2 | 2 | no | — |
| 51 | #235 | — | 2 | 2 | no | — |
| 52 | C2×D12 | — | 2 | 2 | no | — |
| 53 | C2×D12 | D4 | 1 | 2 | yes | — |
| 54 | C2×C4×D3 | — | 2 | 4 | no | — |
| 55 | C2×C4×D3 | D4 | 1 | 4 | yes | — |
| 56 | C2×Dic6 | — | 2 | 2 | no | — |
| 57 | C2×Dic6 | D4 | 1 | 2 | yes | — |
| 58 | C3×M16 | — | 2 | 2 | no | — |
| 59 | C3×M16 | — | 2 | 2 | no | — |
| 60 | C2×C3×C8 | — | 2 | 2 | no | — |
| 61 | #208 | — | 2 | 2 | no | — |
| 62 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 63 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 64 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 65 | 2−1+4 | — | 6 | 2 | no | — |
| 66 | 2−1+4 | — | 6 | 1 | no | — |
| 67 | H(5,𝔽2) | — | 3 | 4 | no | — |
| 68 | H(5,𝔽2) | — | 6 | 2 | no | — |
| 69 | H(5,𝔽2) | — | 6 | 1 | no | — |
| 70 | C2×#41 | — | 3 | 2 | no | — |
| 71 | C2×#41 | — | 6 | 4 | no | — |
| 72 | C2×#41 | — | 6 | 2 | no | — |
| 73 | C2×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 74 | C2×M16 | — | 3 | 1 | no | — |
| 75 | C2×M16 | C2×D3 | 1 | 2 | yes | complemented |
| 76 | C22×C8 | — | 3 | 1 | no | — |
| 77 | #101 | — | 2 | 1 | no | — |
| 78 | #101 | — | 6 | 2 | no | — |
| 79 | #100 | — | 2 | 1 | no | — |
| 80 | #100 | — | 6 | 2 | no | — |
| 81 | C22⋊C8 | — | 6 | 4 | no | — |
| 82 | C22⋊C8 | — | 6 | 2 | no | — |
| 83 | C23×C3 | C22⋊C4 | 1 | 2 | yes | — |
| 84 | C23×C3 | C22×C4 | 1 | 1 | yes | characteristic |
| 85 | C22×D3 | — | 2 | 4 | no | — |
| 86 | C22×D3 | C22⋊C4 | 1 | 4 | yes | — |
| 87 | C3×Q8 | — | 2 | 2 | no | — |
| 88 | C3×D4 | — | 2 | 4 | no | — |
| 89 | C3×D4 | — | 2 | 2 | no | — |
| 90 | C2×C3×C4 | — | 2 | 2 | no | — |
| 91 | C2×C3×C4 | — | 2 | 2 | no | — |
| 92 | C2×C3×C4 | C22⋊C4 | 1 | 2 | yes | — |
| 93 | C2×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 94 | C2×C3×C4 | C2×D4 | 1 | 2 | yes | — |
| 95 | C2×C3×C4 | C2×D4 | 1 | 2 | yes | — |
| 96 | C2×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 97 | #67 | — | 2 | 8 | no | — |
| 98 | #67 | — | 2 | 4 | no | — |
| 99 | #67 | — | 2 | 4 | no | — |
| 100 | C2×Dic3 | — | 2 | 4 | no | — |
| 101 | C2×Dic3 | C22⋊C4 | 1 | 4 | yes | — |
| 102 | D12 | — | 2 | 4 | no | — |
| 103 | D12 | — | 2 | 2 | no | — |
| 104 | D12 | — | 2 | 2 | no | — |
| 105 | C4×D3 | — | 2 | 8 | no | — |
| 106 | C4×D3 | — | 2 | 4 | no | — |
| 107 | C4×D3 | — | 2 | 4 | no | — |
| 108 | Dic6 | — | 2 | 4 | no | — |
| 109 | Dic6 | — | 2 | 2 | no | — |
| 110 | Dic6 | — | 2 | 2 | no | — |
| 111 | C3×C8 | — | 4 | 2 | no | — |
| 112 | C3⋊C8 | — | 4 | 1 | no | — |
| 113 | C3⋊C8 | — | 4 | 1 | no | — |
| 114 | #41 | — | 2 | 2 | no | — |
| 115 | #41 | — | 2 | 2 | no | — |
| 116 | #41 | — | 6 | 4 | no | — |
| 117 | #41 | — | 6 | 4 | no | — |
| 118 | #41 | — | 6 | 8 | no | — |
| 119 | #41 | — | 6 | 4 | no | — |
| 120 | #41 | — | 6 | 4 | no | — |
| 121 | #41 | — | 6 | 4 | no | — |
| 122 | #41 | — | 6 | 2 | no | — |
| 123 | #41 | — | 6 | 2 | no | — |
| 124 | #41 | — | 6 | 2 | no | — |
| 125 | #41 | — | 6 | 2 | no | — |
| 126 | C2×Q8 | — | 3 | 2 | no | — |
| 127 | C2×Q8 | — | 6 | 2 | no | — |
| 128 | C2×Q8 | — | 6 | 4 | no | — |
| 129 | C2×Q8 | #67 | 1 | 1 | yes | characteristic |
| 130 | C2×D4 | — | 3 | 4 | no | — |
| 131 | C2×D4 | — | 3 | 2 | no | — |
| 132 | C2×D4 | — | 6 | 2 | no | — |
| 133 | C2×D4 | — | 6 | 8 | no | — |
| 134 | C2×D4 | — | 6 | 4 | no | — |
| 135 | C2×D4 | — | 6 | 4 | no | — |
| 136 | C2×D4 | C4×D3 | 1 | 2 | yes | — |
| 137 | C2×D4 | #67 | 1 | 1 | yes | characteristic |
| 138 | C22×C4 | — | 3 | 4 | no | — |
| 139 | C22×C4 | — | 6 | 4 | no | — |
| 140 | C22×C4 | D12 | 1 | 2 | yes | — |
| 141 | C22×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 142 | M16 | — | 2 | 2 | no | — |
| 143 | M16 | — | 2 | 2 | no | — |
| 144 | M16 | — | 6 | 2 | no | — |
| 145 | C2×C8 | — | 2 | 2 | no | — |
| 146 | C2×C8 | — | 6 | 1 | no | — |
| 147 | C2×C8 | — | 6 | 1 | no | — |
| 148 | C2×C8 | — | 6 | 2 | no | — |
| 149 | C22×C3 | — | 2 | 2 | no | — |
| 150 | C22×C3 | — | 4 | 1 | no | — |
| 151 | C22×C3 | — | 4 | 2 | no | — |
| 152 | C22×C3 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 153 | C22×C3 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 154 | C2×D3 | — | 2 | 4 | no | — |
| 155 | C2×D3 | — | 4 | 4 | no | — |
| 156 | C2×D3 | — | 8 | 4 | no | — |
| 157 | C3×C4 | — | 2 | 2 | no | — |
| 158 | C3×C4 | #120 | 1 | 1 | yes | characteristic |
| 159 | C3×C4 | #120 | 1 | 1 | yes | characteristic |
| 160 | C3×C4 | #120 | 1 | 2 | yes | — |
| 161 | Dic3 | — | 2 | 4 | no | — |
| 162 | C23 | — | 3 | 4 | no | — |
| 163 | C23 | — | 6 | 4 | no | — |
| 164 | C23 | (C2×D3)⋊C4 | 1 | 2 | yes | — |
| 165 | C23 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 166 | Q8 | — | 2 | 2 | no | — |
| 167 | Q8 | — | 6 | 4 | no | — |
| 168 | Q8 | — | 6 | 2 | no | — |
| 169 | Q8 | — | 6 | 2 | no | — |
| 170 | D4 | — | 2 | 4 | no | — |
| 171 | D4 | — | 2 | 2 | no | — |
| 172 | D4 | — | 6 | 4 | no | — |
| 173 | D4 | — | 6 | 8 | no | — |
| 174 | D4 | — | 6 | 4 | no | — |
| 175 | D4 | — | 6 | 4 | no | — |
| 176 | D4 | — | 6 | 2 | no | — |
| 177 | D4 | — | 6 | 2 | no | — |
| 178 | C2×C4 | — | 2 | 2 | no | — |
| 179 | C2×C4 | — | 2 | 2 | no | — |
| 180 | C2×C4 | — | 3 | 4 | no | — |
| 181 | C2×C4 | — | 6 | 8 | no | — |
| 182 | C2×C4 | — | 6 | 4 | no | — |
| 183 | C2×C4 | — | 6 | 4 | no | — |
| 184 | C2×C4 | — | 6 | 4 | no | — |
| 185 | C2×C4 | (C2×D3)⋊C4 | 1 | 2 | yes | — |
| 186 | C2×C4 | C2×D12 | 1 | 1 | yes | characteristic |
| 187 | C2×C4 | D3×D4 | 1 | 2 | yes | — |
| 188 | C2×C4 | D3×D4 | 1 | 2 | yes | — |
| 189 | C2×C4 | C2×#67 | 1 | 1 | yes | characteristic |
| 190 | C8 | — | 4 | 2 | no | — |
| 191 | C8 | — | 12 | 1 | no | — |
| 192 | C8 | — | 12 | 1 | no | — |
| 193 | C2×C3 | — | 2 | 2 | no | — |
| 194 | C2×C3 | — | 2 | 1 | no | — |
| 195 | C2×C3 | — | 4 | 2 | no | — |
| 196 | C2×C3 | #379 | 1 | 1 | yes | characteristic |
| 197 | D3 | — | 4 | 4 | no | — |
| 198 | C22 | — | 2 | 2 | no | — |
| 199 | C22 | — | 4 | 1 | no | — |
| 200 | C22 | — | 4 | 2 | no | — |
| 201 | C22 | — | 6 | 4 | no | — |
| 202 | C22 | — | 12 | 4 | no | — |
| 203 | C22 | — | 24 | 4 | no | — |
| 204 | C22 | D3×C22⋊C4 | 1 | 2 | yes | — |
| 205 | C22 | C2×(C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 206 | C4 | — | 2 | 2 | no | — |
| 207 | C4 | — | 6 | 4 | no | — |
| 208 | C4 | #882 | 1 | 1 | yes | characteristic |
| 209 | C4 | #882 | 1 | 1 | yes | characteristic |
| 210 | C4 | #937 | 1 | 2 | yes | — |
| 211 | C3 | #1791 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 212 | C2 | — | 2 | 1 | no | — |
| 213 | C2 | — | 2 | 2 | no | — |
| 214 | C2 | — | 4 | 2 | no | — |
| 215 | C2 | — | 12 | 4 | no | — |
| 216 | C2 | #4955 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#413 | C2 | semidirect |
| 3. #5234 | C2 | semidirect |
| 4. #5132 | C2 | semidirect |
| 5. #5129 | C2 | semidirect |
| 16. C2×C3×M16 | C22 | semidirect |
| 17. C2×#208 | C22 | semidirect |
| 18. C22×C3⋊C8 | C22 | semidirect |
| 27. #413 | D3 | semidirect |
| 75. C2×M16 | C2×D3 | semidirect |
| 211. C3 | #1791 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5964 | 2 |
| C3×#413 | 2 |
| #5234 | 2 |
| #5132 | 2 |
| #5129 | 2 |
| #1791 | 3 |