Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
907 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 907 conjugacy classes of subgroups — too many to draw. Only its 102 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 | C2×D4×A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×GL(2,ℤ/4) | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #5914 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | D4×S4 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #5912 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #3484 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C23×A4 | C22 | 1 | 2 | yes | complemented |
| 8 | C22×S4 | C22 | 1 | 4 | yes | complemented |
| 9 | #1004 | — | 4 | 1 | no | maximal |
| 10 | D4×A4 | C22 | 1 | 4 | yes | complemented |
| 11 | C2×C4×A4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 12 | GL(2,ℤ/4) | C22 | 1 | 4 | yes | complemented |
| 13 | GL(2,ℤ/4) | C22 | 1 | 8 | yes | complemented |
| 14 | C2×SL(2,ℤ/4) | C22 | 1 | 4 | yes | complemented |
| 15 | C4⋊S4 | C22 | 1 | 2 | yes | complemented |
| 16 | C4×S4 | C22 | 1 | 4 | yes | complemented |
| 17 | A4⋊Q8 | C22 | 1 | 2 | yes | complemented |
| 18 | C2×H(5,𝔽2) | — | 3 | 1 | no | — |
| 19 | C23×D4 | D3 | 1 | 1 | yes | characteristic, complemented |
| 20 | #546 | — | 3 | 4 | no | — |
| 21 | #546 | — | 3 | 4 | no | — |
| 22 | D42 | — | 3 | 4 | no | — |
| 23 | D42 | — | 3 | 4 | no | — |
| 24 | #535 | — | 3 | 2 | no | — |
| 25 | #534 | — | 3 | 4 | no | — |
| 26 | #525 | — | 3 | 2 | no | — |
| 27 | C2×#120 | — | 3 | 4 | no | — |
| 28 | #518 | — | 3 | 1 | no | — |
| 29 | C22×A4 | — | 4 | 2 | no | — |
| 30 | C22×A4 | C23 | 1 | 4 | yes | — |
| 31 | C22×A4 | C23 | 1 | 1 | yes | characteristic |
| 32 | C2×S4 | — | 4 | 4 | no | — |
| 33 | C2×S4 | C23 | 1 | 4 | yes | — |
| 34 | C2×C3×D4 | — | 4 | 1 | no | — |
| 35 | C2×#67 | — | 4 | 4 | no | — |
| 36 | #237 | — | 4 | 4 | no | — |
| 37 | D3×D4 | — | 4 | 4 | no | — |
| 38 | #235 | — | 4 | 2 | no | — |
| 39 | C4×A4 | C23 | 1 | 2 | yes | — |
| 40 | SL(2,ℤ/4) | C23 | 1 | 4 | yes | — |
| 41 | C25 | C2×D3 | 1 | 2 | yes | complemented |
| 42 | H(5,𝔽2) | — | 6 | 2 | no | — |
| 43 | H(5,𝔽2) | — | 6 | 4 | no | — |
| 44 | H(5,𝔽2) | — | 6 | 1 | no | — |
| 45 | H(5,𝔽2) | — | 6 | 1 | no | — |
| 46 | C2×#41 | — | 3 | 2 | no | — |
| 47 | C2×#41 | — | 3 | 4 | no | — |
| 48 | C22×D4 | — | 3 | 1 | no | — |
| 49 | C22×D4 | — | 3 | 2 | no | — |
| 50 | C22×D4 | — | 3 | 1 | no | — |
| 51 | C22×D4 | — | 3 | 4 | no | — |
| 52 | C22×D4 | — | 3 | 4 | no | — |
| 53 | C22×D4 | — | 3 | 4 | no | — |
| 54 | C22×D4 | — | 3 | 4 | no | — |
| 55 | C22×D4 | — | 3 | 4 | no | — |
| 56 | C22×D4 | C2×D3 | 1 | 4 | yes | complemented |
| 57 | C23×C4 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 58 | #127 | — | 3 | 4 | no | — |
| 59 | #124 | — | 3 | 4 | no | — |
| 60 | #123 | — | 3 | 4 | no | — |
| 61 | #123 | — | 3 | 8 | no | — |
| 62 | C22⋊Q8 | — | 3 | 2 | no | — |
| 63 | C22⋊Q8 | — | 3 | 2 | no | — |
| 64 | #121 | — | 3 | 2 | no | — |
| 65 | #121 | — | 3 | 8 | no | — |
| 66 | #121 | — | 3 | 8 | no | — |
| 67 | #121 | — | 3 | 4 | no | — |
| 68 | #121 | — | 3 | 4 | no | — |
| 69 | #121 | — | 3 | 2 | no | — |
| 70 | #120 | — | 3 | 8 | no | — |
| 71 | #120 | — | 3 | 8 | no | — |
| 72 | #120 | — | 3 | 8 | no | — |
| 73 | #120 | — | 3 | 4 | no | — |
| 74 | #120 | — | 3 | 8 | no | — |
| 75 | C4×D4 | — | 3 | 4 | no | — |
| 76 | C4×D4 | — | 3 | 4 | no | — |
| 77 | C4×D4 | — | 3 | 4 | no | — |
| 78 | C4×D4 | — | 3 | 4 | no | — |
| 79 | #117 | — | 3 | 2 | no | — |
| 80 | C2×C22⋊C4 | — | 3 | 4 | no | — |
| 81 | C2×C22⋊C4 | — | 3 | 4 | no | — |
| 82 | C2×C22⋊C4 | — | 3 | 4 | no | — |
| 83 | C23×C3 | — | 4 | 2 | no | — |
| 84 | C22×D3 | — | 4 | 4 | no | — |
| 85 | C2×A4 | — | 2 | 4 | no | — |
| 86 | C2×A4 | — | 2 | 1 | no | — |
| 87 | C2×A4 | C24 | 1 | 1 | yes | characteristic |
| 88 | S4 | — | 2 | 4 | no | — |
| 89 | C3×D4 | — | 4 | 4 | no | — |
| 90 | C2×C3×C4 | — | 4 | 1 | no | — |
| 91 | #67 | — | 4 | 4 | no | — |
| 92 | #67 | — | 4 | 8 | no | — |
| 93 | C2×Dic3 | — | 4 | 4 | no | — |
| 94 | D12 | — | 4 | 2 | no | — |
| 95 | C4×D3 | — | 4 | 4 | no | — |
| 96 | Dic6 | — | 4 | 2 | no | — |
| 97 | C24 | — | 3 | 2 | no | — |
| 98 | C24 | — | 3 | 4 | no | — |
| 99 | C24 | — | 3 | 2 | no | — |
| 100 | C24 | — | 3 | 4 | no | — |
| 101 | C24 | — | 4 | 2 | no | — |
| 102 | C24 | — | 12 | 2 | no | — |
| 103 | C24 | C22×D3 | 1 | 4 | yes | — |
| 104 | C24 | C22×D3 | 1 | 1 | yes | characteristic |
| 105 | #41 | — | 6 | 2 | no | — |
| 106 | #41 | — | 6 | 2 | no | — |
| 107 | #41 | — | 6 | 2 | no | — |
| 108 | #41 | — | 6 | 8 | no | — |
| 109 | #41 | — | 6 | 4 | no | — |
| 110 | #41 | — | 6 | 4 | no | — |
| 111 | #41 | — | 6 | 2 | no | — |
| 112 | C2×Q8 | — | 3 | 2 | no | — |
| 113 | C2×D4 | — | 3 | 2 | no | — |
| 114 | C2×D4 | — | 3 | 4 | no | — |
| 115 | C2×D4 | — | 3 | 4 | no | — |
| 116 | C2×D4 | — | 3 | 8 | no | — |
| 117 | C2×D4 | — | 3 | 4 | no | — |
| 118 | C2×D4 | — | 3 | 4 | no | — |
| 119 | C2×D4 | — | 3 | 4 | no | — |
| 120 | C2×D4 | — | 3 | 8 | no | — |
| 121 | C2×D4 | — | 3 | 2 | no | — |
| 122 | C2×D4 | — | 3 | 4 | no | — |
| 123 | C2×D4 | — | 3 | 8 | no | — |
| 124 | C2×D4 | — | 3 | 8 | no | — |
| 125 | C2×D4 | — | 3 | 1 | no | — |
| 126 | C2×D4 | — | 6 | 2 | no | — |
| 127 | C2×D4 | — | 6 | 1 | no | — |
| 128 | C2×D4 | — | 6 | 4 | no | — |
| 129 | C2×D4 | — | 6 | 8 | no | — |
| 130 | C2×D4 | — | 6 | 4 | no | — |
| 131 | C2×D4 | — | 6 | 4 | no | — |
| 132 | C2×D4 | — | 6 | 4 | no | — |
| 133 | C2×D4 | — | 6 | 4 | no | — |
| 134 | C2×D4 | — | 6 | 4 | no | — |
| 135 | C2×D4 | — | 6 | 4 | no | — |
| 136 | C2×D4 | — | 12 | 4 | no | — |
| 137 | C2×D4 | — | 12 | 4 | no | — |
| 138 | C2×D4 | — | 12 | 8 | no | — |
| 139 | C2×D4 | S4 | 1 | 1 | yes | characteristic, complemented |
| 140 | C22×C4 | — | 3 | 1 | no | — |
| 141 | C22×C4 | — | 3 | 2 | no | — |
| 142 | C22×C4 | — | 3 | 1 | no | — |
| 143 | C22×C4 | — | 3 | 4 | no | — |
| 144 | C22×C4 | — | 3 | 4 | no | — |
| 145 | C22×C4 | — | 3 | 4 | no | — |
| 146 | C22×C4 | — | 3 | 4 | no | — |
| 147 | C22×C4 | C22×D3 | 1 | 2 | yes | — |
| 148 | C4⋊C4 | — | 3 | 2 | no | — |
| 149 | C4⋊C4 | — | 3 | 2 | no | — |
| 150 | C4⋊C4 | — | 3 | 4 | no | — |
| 151 | C4⋊C4 | — | 3 | 4 | no | — |
| 152 | C22⋊C4 | — | 3 | 4 | no | — |
| 153 | C22⋊C4 | — | 3 | 4 | no | — |
| 154 | C22⋊C4 | — | 3 | 8 | no | — |
| 155 | C22⋊C4 | — | 3 | 8 | no | — |
| 156 | C22⋊C4 | — | 3 | 8 | no | — |
| 157 | C22⋊C4 | — | 3 | 4 | no | — |
| 158 | C22⋊C4 | — | 3 | 4 | no | — |
| 159 | C22⋊C4 | — | 3 | 4 | no | — |
| 160 | C42 | — | 3 | 4 | no | — |
| 161 | C22×C3 | — | 4 | 4 | no | — |
| 162 | C22×C3 | — | 4 | 1 | no | — |
| 163 | C22×C3 | — | 16 | 2 | no | — |
| 164 | C2×D3 | — | 4 | 4 | no | — |
| 165 | C2×D3 | — | 16 | 4 | no | — |
| 166 | A4 | H(5,𝔽2) | 1 | 1 | yes | characteristic, complemented |
| 167 | C3×C4 | — | 4 | 2 | no | — |
| 168 | Dic3 | — | 4 | 4 | no | — |
| 169 | C23 | — | 2 | 4 | no | — |
| 170 | C23 | — | 2 | 1 | no | — |
| 171 | C23 | — | 3 | 1 | no | — |
| 172 | C23 | — | 3 | 4 | no | — |
| 173 | C23 | — | 3 | 1 | no | — |
| 174 | C23 | — | 3 | 4 | no | — |
| 175 | C23 | — | 3 | 4 | no | — |
| 176 | C23 | — | 3 | 2 | no | — |
| 177 | C23 | — | 3 | 4 | no | — |
| 178 | C23 | — | 6 | 1 | no | — |
| 179 | C23 | — | 6 | 2 | no | — |
| 180 | C23 | — | 6 | 4 | no | — |
| 181 | C23 | — | 6 | 8 | no | — |
| 182 | C23 | — | 6 | 4 | no | — |
| 183 | C23 | — | 6 | 4 | no | — |
| 184 | C23 | — | 12 | 2 | no | — |
| 185 | C23 | — | 12 | 2 | no | — |
| 186 | C23 | — | 12 | 4 | no | — |
| 187 | C23 | — | 12 | 2 | no | — |
| 188 | C23 | — | 12 | 2 | no | — |
| 189 | C23 | — | 12 | 4 | no | — |
| 190 | C23 | — | 12 | 4 | no | — |
| 191 | C23 | C2×S4 | 1 | 2 | yes | complemented |
| 192 | C23 | C23×D3 | 1 | 1 | yes | characteristic |
| 193 | Q8 | — | 6 | 2 | no | — |
| 194 | Q8 | — | 6 | 2 | no | — |
| 195 | D4 | — | 3 | 8 | no | — |
| 196 | D4 | — | 3 | 4 | no | — |
| 197 | D4 | — | 6 | 4 | no | — |
| 198 | D4 | — | 6 | 2 | no | — |
| 199 | D4 | — | 6 | 4 | no | — |
| 200 | D4 | — | 6 | 8 | no | — |
| 201 | D4 | — | 6 | 4 | no | — |
| 202 | D4 | — | 6 | 8 | no | — |
| 203 | D4 | — | 6 | 4 | no | — |
| 204 | D4 | — | 6 | 4 | no | — |
| 205 | D4 | — | 6 | 8 | no | — |
| 206 | D4 | — | 6 | 2 | no | — |
| 207 | D4 | — | 12 | 8 | no | — |
| 208 | D4 | — | 12 | 8 | no | — |
| 209 | D4 | C2×S4 | 1 | 4 | yes | complemented |
| 210 | C2×C4 | — | 3 | 1 | no | — |
| 211 | C2×C4 | — | 3 | 2 | no | — |
| 212 | C2×C4 | — | 3 | 2 | no | — |
| 213 | C2×C4 | — | 3 | 2 | no | — |
| 214 | C2×C4 | — | 3 | 4 | no | — |
| 215 | C2×C4 | — | 3 | 4 | no | — |
| 216 | C2×C4 | — | 3 | 4 | no | — |
| 217 | C2×C4 | — | 6 | 1 | no | — |
| 218 | C2×C4 | — | 6 | 4 | no | — |
| 219 | C2×C4 | — | 6 | 4 | no | — |
| 220 | C2×C4 | — | 6 | 4 | no | — |
| 221 | C2×C4 | — | 6 | 4 | no | — |
| 222 | C2×C4 | — | 12 | 4 | no | — |
| 223 | C2×C4 | — | 12 | 4 | no | — |
| 224 | C2×C4 | C2×S4 | 1 | 1 | yes | characteristic, complemented |
| 225 | C2×C3 | — | 4 | 1 | no | — |
| 226 | C2×C3 | — | 8 | 4 | no | — |
| 227 | C2×C3 | — | 8 | 1 | no | — |
| 228 | D3 | — | 8 | 4 | no | — |
| 229 | C22 | — | 3 | 1 | no | — |
| 230 | C22 | — | 3 | 1 | no | — |
| 231 | C22 | — | 3 | 2 | no | — |
| 232 | C22 | — | 3 | 1 | no | — |
| 233 | C22 | — | 3 | 4 | no | — |
| 234 | C22 | — | 4 | 2 | no | — |
| 235 | C22 | — | 6 | 4 | no | — |
| 236 | C22 | — | 6 | 4 | no | — |
| 237 | C22 | — | 6 | 1 | no | — |
| 238 | C22 | — | 6 | 1 | no | — |
| 239 | C22 | — | 6 | 4 | no | — |
| 240 | C22 | — | 6 | 4 | no | — |
| 241 | C22 | — | 6 | 1 | no | — |
| 242 | C22 | — | 6 | 4 | no | — |
| 243 | C22 | — | 6 | 4 | no | — |
| 244 | C22 | — | 6 | 4 | no | — |
| 245 | C22 | — | 12 | 4 | no | — |
| 246 | C22 | — | 12 | 2 | no | — |
| 247 | C22 | — | 12 | 4 | no | — |
| 248 | C22 | — | 24 | 4 | no | — |
| 249 | C22 | — | 24 | 4 | no | — |
| 250 | C22 | #1004 | 1 | 1 | yes | characteristic, complemented |
| 251 | C22 | C22×S4 | 1 | 1 | yes | characteristic |
| 252 | C22 | C22×S4 | 1 | 4 | yes | — |
| 253 | C4 | — | 3 | 2 | no | — |
| 254 | C4 | — | 6 | 4 | no | — |
| 255 | C4 | — | 6 | 4 | no | — |
| 256 | C4 | — | 6 | 4 | no | — |
| 257 | C4 | C22×S4 | 1 | 2 | yes | — |
| 258 | C3 | — | 4 | 1 | no | Sylow |
| 259 | C2 | — | 2 | 1 | no | — |
| 260 | C2 | — | 2 | 4 | no | — |
| 261 | C2 | — | 3 | 1 | no | — |
| 262 | C2 | — | 3 | 1 | no | — |
| 263 | C2 | — | 6 | 1 | no | — |
| 264 | C2 | — | 6 | 4 | no | — |
| 265 | C2 | — | 12 | 4 | no | — |
| 266 | C2 | C23×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×S4 |
| commutator, cosocle | C2×A4 | C24 |
| Fitting | C23×D4 | D3 |
| socle | C23 | C23×D3 |
| hypercenter | C2×D4 | S4 |
| hypocenter | A4 | H(5,𝔽2) |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ C2×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D4×A4 | C2 | semidirect |
| 2. C2×GL(2,ℤ/4) | C2 | semidirect |
| 3. #5914 | C2 | semidirect |
| 4. D4×S4 | C2 | semidirect |
| 5. #5912 | C2 | semidirect |
| 7. C23×A4 | C22 | semidirect |
| 8. C22×S4 | C22 | semidirect |
| 10. D4×A4 | C22 | semidirect |
| 11. C2×C4×A4 | C22 | semidirect |
| 12. GL(2,ℤ/4) | C22 | semidirect |
| 13. GL(2,ℤ/4) | C22 | semidirect |
| 14. C2×SL(2,ℤ/4) | C22 | semidirect |
| 15. C4⋊S4 | C22 | semidirect |
| 16. C4×S4 | C22 | semidirect |
| 17. A4⋊Q8 | C22 | semidirect |
| 19. C23×D4 | D3 | semidirect |
| 41. C25 | C2×D3 | semidirect |
| 56. C22×D4 | C2×D3 | semidirect |
| 57. C23×C4 | C2×D3 | semidirect |
| 139. C2×D4 | S4 | semidirect |
| 166. A4 | H(5,𝔽2) | semidirect |
| 191. C23 | C2×S4 | semidirect |
| 209. D4 | C2×S4 | semidirect |
| 224. C2×C4 | C2×S4 | semidirect |
| 250. C22 | #1004 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D4×A4 | 2 |
| C2×GL(2,ℤ/4) | 2 |
| #5914 | 2 |
| D4×S4 | 2 |
| #5912 | 2 |
| #3484 | 3 |
| #1004 | 4 |