Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
368 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 57 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 | C3×#534 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #5036 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #4960 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #4537 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #4525 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #1595 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C22×C3×D4 | — | 2 | 1 | no | — |
| 8 | C22×#67 | — | 2 | 1 | no | — |
| 9 | C22×D12 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | C3×#123 | — | 2 | 2 | no | — |
| 11 | C3×#121 | — | 2 | 1 | no | — |
| 12 | C3×#121 | C22 | 1 | 2 | yes | — |
| 13 | C3×#120 | — | 2 | 2 | no | — |
| 14 | C2×C3×C22⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#213 | — | 2 | 2 | no | — |
| 16 | C2×(C2×D3)⋊C4 | — | 2 | 1 | no | — |
| 17 | C3⋊#95 | C22 | 1 | 1 | yes | characteristic, complemented |
| 18 | #830 | C22 | 1 | 2 | yes | complemented |
| 19 | #809 | — | 2 | 2 | no | — |
| 20 | #534 | D3 | 1 | 1 | yes | characteristic, complemented |
| 21 | #447 | — | 3 | 2 | no | — |
| 22 | #392 | — | 3 | 1 | no | — |
| 23 | #327 | — | 3 | 2 | no | — |
| 24 | #323 | — | 3 | 1 | no | — |
| 25 | C24×C3 | — | 2 | 1 | no | — |
| 26 | C24×C3 | D4 | 1 | 1 | yes | characteristic |
| 27 | C23×D3 | — | 2 | 1 | no | — |
| 28 | C2×C3×D4 | — | 2 | 2 | no | — |
| 29 | C2×C3×D4 | — | 2 | 2 | no | — |
| 30 | C2×C3×D4 | — | 2 | 1 | no | — |
| 31 | C2×C3×D4 | — | 4 | 2 | no | — |
| 32 | C2×C3×D4 | — | 4 | 2 | no | — |
| 33 | C2×C3×D4 | D4 | 1 | 2 | yes | complemented |
| 34 | C22×C3×C4 | — | 2 | 1 | no | — |
| 35 | C22×C3×C4 | D4 | 1 | 1 | yes | characteristic |
| 36 | C22×C3×C4 | C23 | 1 | 1 | yes | characteristic |
| 37 | C2×#67 | — | 4 | 2 | no | — |
| 38 | C2×#67 | — | 4 | 2 | no | — |
| 39 | C2×#67 | — | 4 | 2 | no | — |
| 40 | C22×Dic3 | — | 2 | 1 | no | — |
| 41 | C2×D12 | — | 2 | 2 | no | — |
| 42 | C2×D12 | — | 4 | 2 | no | — |
| 43 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 44 | C3×C4⋊C4 | D4 | 1 | 2 | yes | complemented |
| 45 | C3×C22⋊C4 | — | 2 | 2 | no | — |
| 46 | C3×C22⋊C4 | — | 2 | 2 | no | — |
| 47 | C3×C22⋊C4 | — | 2 | 2 | no | — |
| 48 | #213 | — | 4 | 4 | no | — |
| 49 | (C2×D3)⋊C4 | — | 4 | 2 | no | — |
| 50 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 51 | C22×D4 | — | 2 | 1 | no | — |
| 52 | C22×D4 | — | 3 | 1 | no | — |
| 53 | C22×D4 | — | 6 | 1 | no | — |
| 54 | C2×D8 | — | 6 | 2 | no | — |
| 55 | #123 | — | 2 | 2 | no | — |
| 56 | #121 | — | 2 | 1 | no | — |
| 57 | #121 | C2×D3 | 1 | 2 | yes | — |
| 58 | #120 | — | 2 | 2 | no | — |
| 59 | C2×C22⋊C4 | — | 6 | 1 | no | — |
| 60 | C2×C22⋊C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 61 | #102 | — | 6 | 2 | no | — |
| 62 | C22⋊C8 | — | 3 | 2 | no | — |
| 63 | #95 | — | 3 | 1 | no | — |
| 64 | C23×C3 | — | 2 | 1 | no | — |
| 65 | C23×C3 | — | 2 | 1 | no | — |
| 66 | C23×C3 | — | 2 | 2 | no | — |
| 67 | C23×C3 | — | 4 | 1 | no | — |
| 68 | C23×C3 | — | 4 | 1 | no | — |
| 69 | C23×C3 | — | 4 | 2 | no | — |
| 70 | C23×C3 | — | 8 | 1 | no | — |
| 71 | C23×C3 | — | 8 | 1 | no | — |
| 72 | C23×C3 | D8 | 1 | 2 | yes | complemented |
| 73 | C23×C3 | C2×D4 | 1 | 1 | yes | characteristic |
| 74 | C22×D3 | — | 2 | 2 | no | — |
| 75 | C22×D3 | — | 4 | 1 | no | — |
| 76 | C22×D3 | — | 4 | 1 | no | — |
| 77 | C22×D3 | — | 4 | 1 | no | — |
| 78 | C22×D3 | — | 4 | 1 | no | — |
| 79 | C22×D3 | — | 4 | 1 | no | — |
| 80 | C22×D3 | — | 4 | 1 | no | — |
| 81 | C3×D4 | — | 4 | 2 | no | — |
| 82 | C3×D4 | — | 4 | 4 | no | — |
| 83 | C3×D4 | — | 4 | 2 | no | — |
| 84 | C3×D4 | — | 4 | 2 | no | — |
| 85 | C3×D4 | — | 8 | 1 | no | — |
| 86 | C2×C3×C4 | — | 2 | 1 | no | — |
| 87 | C2×C3×C4 | — | 2 | 2 | no | — |
| 88 | C2×C3×C4 | — | 4 | 1 | no | — |
| 89 | C2×C3×C4 | — | 4 | 2 | no | — |
| 90 | C2×C3×C4 | — | 4 | 1 | no | — |
| 91 | C2×C3×C4 | C2×D4 | 1 | 2 | yes | — |
| 92 | #67 | — | 4 | 8 | no | — |
| 93 | C2×Dic3 | — | 2 | 2 | no | — |
| 94 | C2×Dic3 | — | 4 | 1 | no | — |
| 95 | C2×Dic3 | — | 4 | 1 | no | — |
| 96 | D12 | — | 2 | 4 | no | — |
| 97 | D12 | — | 4 | 2 | no | — |
| 98 | C3⋊C8 | — | 4 | 2 | no | — |
| 99 | C24 | — | 2 | 1 | no | — |
| 100 | C24 | — | 6 | 1 | no | — |
| 101 | C24 | #67 | 1 | 1 | yes | characteristic |
| 102 | C2×D4 | — | 2 | 2 | no | — |
| 103 | C2×D4 | — | 2 | 2 | no | — |
| 104 | C2×D4 | — | 2 | 1 | no | — |
| 105 | C2×D4 | — | 4 | 2 | no | — |
| 106 | C2×D4 | — | 4 | 2 | no | — |
| 107 | C2×D4 | — | 6 | 2 | no | — |
| 108 | C2×D4 | — | 12 | 2 | no | — |
| 109 | C2×D4 | — | 12 | 2 | no | — |
| 110 | C2×D4 | — | 12 | 2 | no | — |
| 111 | C2×D4 | — | 12 | 2 | no | — |
| 112 | C2×D4 | #67 | 1 | 2 | yes | complemented |
| 113 | C22×C4 | — | 2 | 1 | no | — |
| 114 | C22×C4 | — | 6 | 1 | no | — |
| 115 | C22×C4 | #67 | 1 | 1 | yes | characteristic |
| 116 | C22×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 117 | D8 | — | 12 | 4 | no | — |
| 118 | C2×C8 | — | 6 | 2 | no | — |
| 119 | C4⋊C4 | — | 2 | 1 | no | — |
| 120 | C4⋊C4 | #67 | 1 | 2 | yes | complemented |
| 121 | C22⋊C4 | — | 2 | 2 | no | — |
| 122 | C22⋊C4 | — | 2 | 2 | no | — |
| 123 | C22⋊C4 | — | 2 | 2 | no | — |
| 124 | C22⋊C4 | — | 12 | 2 | no | — |
| 125 | C22×C3 | — | 2 | 1 | no | — |
| 126 | C22×C3 | — | 2 | 1 | no | — |
| 127 | C22×C3 | — | 2 | 2 | no | — |
| 128 | C22×C3 | — | 4 | 2 | no | — |
| 129 | C22×C3 | — | 4 | 1 | no | — |
| 130 | C22×C3 | — | 4 | 1 | no | — |
| 131 | C22×C3 | — | 4 | 2 | no | — |
| 132 | C22×C3 | — | 4 | 2 | no | — |
| 133 | C22×C3 | — | 4 | 2 | no | — |
| 134 | C22×C3 | — | 8 | 1 | no | — |
| 135 | C22×C3 | — | 8 | 4 | no | — |
| 136 | C22×C3 | #120 | 1 | 1 | yes | characteristic |
| 137 | C22×C3 | C2×D8 | 1 | 1 | yes | characteristic |
| 138 | C22×C3 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 139 | C2×D3 | — | 4 | 2 | no | — |
| 140 | C2×D3 | — | 4 | 2 | no | — |
| 141 | C2×D3 | — | 4 | 2 | no | — |
| 142 | C2×D3 | — | 8 | 1 | no | — |
| 143 | C2×D3 | — | 8 | 1 | no | — |
| 144 | C2×D3 | — | 8 | 1 | no | — |
| 145 | C2×D3 | — | 8 | 1 | no | — |
| 146 | C3×C4 | — | 2 | 2 | no | — |
| 147 | C3×C4 | — | 4 | 2 | no | — |
| 148 | C3×C4 | — | 4 | 1 | no | — |
| 149 | Dic3 | — | 4 | 2 | no | — |
| 150 | C23 | — | 2 | 1 | no | — |
| 151 | C23 | — | 2 | 1 | no | — |
| 152 | C23 | — | 2 | 2 | no | — |
| 153 | C23 | — | 4 | 1 | no | — |
| 154 | C23 | — | 4 | 1 | no | — |
| 155 | C23 | — | 4 | 2 | no | — |
| 156 | C23 | — | 6 | 2 | no | — |
| 157 | C23 | — | 8 | 1 | no | — |
| 158 | C23 | — | 8 | 1 | no | — |
| 159 | C23 | — | 12 | 1 | no | — |
| 160 | C23 | — | 12 | 1 | no | — |
| 161 | C23 | — | 12 | 1 | no | — |
| 162 | C23 | — | 12 | 1 | no | — |
| 163 | C23 | — | 12 | 1 | no | — |
| 164 | C23 | — | 12 | 1 | no | — |
| 165 | C23 | #213 | 1 | 2 | yes | complemented |
| 166 | C23 | C2×#67 | 1 | 1 | yes | characteristic |
| 167 | D4 | — | 4 | 2 | no | — |
| 168 | D4 | — | 4 | 4 | no | — |
| 169 | D4 | — | 4 | 2 | no | — |
| 170 | D4 | — | 4 | 2 | no | — |
| 171 | D4 | — | 6 | 4 | no | — |
| 172 | D4 | — | 8 | 1 | no | — |
| 173 | D4 | — | 12 | 2 | no | — |
| 174 | D4 | — | 12 | 8 | no | — |
| 175 | C2×C4 | — | 2 | 1 | no | — |
| 176 | C2×C4 | — | 2 | 2 | no | — |
| 177 | C2×C4 | — | 4 | 1 | no | — |
| 178 | C2×C4 | — | 4 | 2 | no | — |
| 179 | C2×C4 | — | 4 | 1 | no | — |
| 180 | C2×C4 | — | 6 | 2 | no | — |
| 181 | C2×C4 | — | 12 | 1 | no | — |
| 182 | C2×C4 | — | 12 | 1 | no | — |
| 183 | C2×C4 | C2×#67 | 1 | 2 | yes | — |
| 184 | C8 | — | 12 | 2 | no | — |
| 185 | C2×C3 | — | 2 | 1 | no | — |
| 186 | C2×C3 | — | 2 | 1 | no | — |
| 187 | C2×C3 | — | 4 | 2 | no | — |
| 188 | C2×C3 | — | 8 | 2 | no | — |
| 189 | C2×C3 | #447 | 1 | 1 | yes | characteristic |
| 190 | C2×C3 | #453 | 1 | 1 | yes | characteristic |
| 191 | C2×C3 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 192 | D3 | — | 8 | 2 | no | — |
| 193 | C22 | — | 2 | 1 | no | — |
| 194 | C22 | — | 2 | 1 | no | — |
| 195 | C22 | — | 2 | 2 | no | — |
| 196 | C22 | — | 4 | 2 | no | — |
| 197 | C22 | — | 4 | 1 | no | — |
| 198 | C22 | — | 4 | 1 | no | — |
| 199 | C22 | — | 4 | 2 | no | — |
| 200 | C22 | — | 4 | 2 | no | — |
| 201 | C22 | — | 4 | 2 | no | — |
| 202 | C22 | — | 8 | 1 | no | — |
| 203 | C22 | — | 8 | 4 | no | — |
| 204 | C22 | — | 12 | 2 | no | — |
| 205 | C22 | — | 12 | 2 | no | — |
| 206 | C22 | — | 12 | 2 | no | — |
| 207 | C22 | — | 24 | 1 | no | — |
| 208 | C22 | — | 24 | 1 | no | — |
| 209 | C22 | — | 24 | 1 | no | — |
| 210 | C22 | — | 24 | 1 | no | — |
| 211 | C22 | C2×#213 | 1 | 1 | yes | characteristic |
| 212 | C22 | #932 | 1 | 1 | yes | characteristic |
| 213 | C22 | #953 | 1 | 1 | yes | characteristic |
| 214 | C4 | — | 2 | 2 | no | — |
| 215 | C4 | — | 4 | 2 | no | — |
| 216 | C4 | — | 4 | 1 | no | — |
| 217 | C4 | — | 12 | 2 | no | — |
| 218 | C3 | #1595 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 219 | C2 | — | 2 | 1 | no | — |
| 220 | C2 | — | 2 | 1 | no | — |
| 221 | C2 | — | 4 | 2 | no | — |
| 222 | C2 | — | 8 | 2 | no | — |
| 223 | C2 | — | 24 | 2 | no | — |
| 224 | C2 | #5217 | 1 | 1 | yes | characteristic |
| 225 | C2 | #5241 | 1 | 1 | yes | characteristic |
| 226 | C2 | #5244 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #953 |
| commutator, cosocle | C22×C3×C4 | C23 |
| Frattini | C22×C4 | C22×D3 |
| Fitting | C3×#534 | C2 |
| socle | C22×C3 | #120 |
| hypercenter | #534 | D3 |
| hypocenter | C3 | #1595 |
| LCS | G ▷ C22×C3×C4 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #534 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#534 | C2 | semidirect |
| 2. #5036 | C2 | semidirect |
| 3. #4960 | C2 | semidirect |
| 4. #4537 | C2 | semidirect |
| 5. #4525 | C2 | semidirect |
| 9. C22×D12 | C22 | semidirect |
| 17. C3⋊#95 | C22 | semidirect |
| 18. #830 | C22 | semidirect |
| 20. #534 | D3 | semidirect |
| 33. C2×C3×D4 | D4 | semidirect |
| 44. C3×C4⋊C4 | D4 | semidirect |
| 72. C23×C3 | D8 | semidirect |
| 112. C2×D4 | #67 | semidirect |
| 120. C4⋊C4 | #67 | semidirect |
| 165. C23 | #213 | semidirect |
| 218. C3 | #1595 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#534 | 2 |
| #5036 | 2 |
| #4960 | 2 |
| #4537 | 2 |
| #4525 | 2 |
| #1595 | 3 |