Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
297 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 297 conjugacy classes of subgroups — too many to draw. Only its 48 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×C4×S4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | A4×M16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #5409 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #5400 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C8×S4 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | D4×M16 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C22×S4 | C4 | 1 | 1 | yes | characteristic |
| 8 | C2×C4×A4 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×SL(2,ℤ/4) | C4 | 1 | 1 | yes | characteristic |
| 10 | C4×S4 | C4 | 1 | 1 | yes | characteristic |
| 11 | C4×S4 | C4 | 1 | 1 | yes | characteristic |
| 12 | C4×S4 | C22 | 1 | 1 | yes | characteristic |
| 13 | C4×S4 | C22 | 1 | 1 | yes | characteristic |
| 14 | D3×M16 | — | 4 | 1 | no | maximal |
| 15 | C8×A4 | C22 | 1 | 2 | yes | complemented |
| 16 | A4⋊C8 | C22 | 1 | 2 | yes | complemented |
| 17 | C22×M16 | — | 3 | 1 | no | — |
| 18 | C22×M16 | D3 | 1 | 1 | yes | characteristic, complemented |
| 19 | C2×C4×D4 | — | 3 | 1 | no | — |
| 20 | #436 | — | 3 | 2 | no | — |
| 21 | #435 | — | 3 | 2 | no | — |
| 22 | #435 | — | 3 | 2 | no | — |
| 23 | C8×D4 | — | 3 | 2 | no | — |
| 24 | #423 | — | 3 | 1 | no | — |
| 25 | #407 | — | 3 | 1 | no | — |
| 26 | #407 | — | 3 | 1 | no | — |
| 27 | C4×M16 | — | 3 | 1 | no | — |
| 28 | C22×A4 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | C2×S4 | — | 2 | 2 | no | — |
| 30 | C2×S4 | C2×C4 | 1 | 1 | yes | characteristic |
| 31 | C2×S4 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | C2×C4×D3 | — | 4 | 1 | no | — |
| 33 | C4×A4 | C2×C4 | 1 | 1 | yes | characteristic |
| 34 | C4×A4 | C23 | 1 | 1 | yes | characteristic |
| 35 | SL(2,ℤ/4) | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | SL(2,ℤ/4) | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C3×M16 | — | 4 | 1 | no | — |
| 38 | #208 | — | 4 | 1 | no | — |
| 39 | ΓL(1,𝔽25) | — | 4 | 2 | no | — |
| 40 | D3×C8 | — | 4 | 2 | no | — |
| 41 | C22×D4 | — | 3 | 1 | no | — |
| 42 | C23×C4 | — | 3 | 1 | no | — |
| 43 | C23×C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 44 | C2×M16 | — | 3 | 1 | no | — |
| 45 | C2×M16 | — | 3 | 1 | no | — |
| 46 | C2×M16 | — | 3 | 2 | no | — |
| 47 | C2×M16 | — | 3 | 1 | no | — |
| 48 | C2×M16 | — | 3 | 2 | no | — |
| 49 | C2×M16 | — | 3 | 1 | no | — |
| 50 | C2×M16 | — | 6 | 1 | no | — |
| 51 | C2×M16 | — | 6 | 2 | no | — |
| 52 | C2×M16 | — | 6 | 1 | no | — |
| 53 | C22×C8 | — | 3 | 2 | no | — |
| 54 | C22×C8 | C2×D3 | 1 | 2 | yes | complemented |
| 55 | C4×D4 | — | 3 | 1 | no | — |
| 56 | C4×D4 | — | 3 | 1 | no | — |
| 57 | C4×D4 | — | 3 | 1 | no | — |
| 58 | C4×D4 | — | 3 | 1 | no | — |
| 59 | C4×D4 | — | 3 | 1 | no | — |
| 60 | C4×D4 | — | 3 | 1 | no | — |
| 61 | C4×D4 | — | 3 | 1 | no | — |
| 62 | C4×D4 | — | 3 | 1 | no | — |
| 63 | C2×C4⋊C4 | — | 3 | 1 | no | — |
| 64 | C2×C22⋊C4 | — | 3 | 1 | no | — |
| 65 | C2×C22⋊C4 | — | 3 | 1 | no | — |
| 66 | C2×C42 | — | 3 | 1 | no | — |
| 67 | C4⋊C8 | — | 3 | 2 | no | — |
| 68 | C4⋊C8 | — | 3 | 2 | no | — |
| 69 | C22⋊C8 | — | 3 | 2 | no | — |
| 70 | C22⋊C8 | — | 3 | 2 | no | — |
| 71 | C22⋊C8 | — | 3 | 2 | no | — |
| 72 | C22⋊C8 | — | 3 | 2 | no | — |
| 73 | #97 | — | 3 | 2 | no | — |
| 74 | C4×C8 | — | 3 | 2 | no | — |
| 75 | C22×D3 | — | 4 | 1 | no | — |
| 76 | C2×A4 | — | 2 | 1 | no | — |
| 77 | C2×A4 | C22×C4 | 1 | 1 | yes | characteristic |
| 78 | S4 | — | 2 | 1 | no | — |
| 79 | S4 | M16 | 1 | 2 | yes | direct product factor |
| 80 | C2×C3×C4 | — | 4 | 1 | no | — |
| 81 | C2×Dic3 | — | 4 | 1 | no | — |
| 82 | C4×D3 | — | 4 | 1 | no | — |
| 83 | C4×D3 | — | 4 | 1 | no | — |
| 84 | C4×D3 | — | 4 | 1 | no | — |
| 85 | C4×D3 | — | 4 | 1 | no | — |
| 86 | C3×C8 | — | 4 | 2 | no | — |
| 87 | C3⋊C8 | — | 4 | 2 | no | — |
| 88 | C24 | — | 3 | 1 | no | — |
| 89 | C24 | C4×D3 | 1 | 1 | yes | characteristic |
| 90 | C2×D4 | — | 3 | 1 | no | — |
| 91 | C2×D4 | — | 3 | 1 | no | — |
| 92 | C2×D4 | — | 3 | 1 | no | — |
| 93 | C2×D4 | — | 3 | 1 | no | — |
| 94 | C2×D4 | — | 6 | 2 | no | — |
| 95 | C2×D4 | — | 6 | 2 | no | — |
| 96 | C22×C4 | — | 3 | 1 | no | — |
| 97 | C22×C4 | — | 3 | 1 | no | — |
| 98 | C22×C4 | — | 3 | 1 | no | — |
| 99 | C22×C4 | — | 3 | 1 | no | — |
| 100 | C22×C4 | — | 3 | 1 | no | — |
| 101 | C22×C4 | — | 3 | 1 | no | — |
| 102 | C22×C4 | — | 3 | 1 | no | — |
| 103 | C22×C4 | — | 3 | 1 | no | — |
| 104 | C22×C4 | — | 3 | 1 | no | — |
| 105 | C22×C4 | — | 3 | 1 | no | — |
| 106 | C22×C4 | — | 3 | 1 | no | — |
| 107 | C22×C4 | — | 6 | 1 | no | — |
| 108 | C22×C4 | — | 6 | 1 | no | — |
| 109 | C22×C4 | — | 6 | 1 | no | — |
| 110 | C22×C4 | — | 6 | 1 | no | — |
| 111 | C22×C4 | C4×D3 | 1 | 1 | yes | characteristic |
| 112 | C22×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 113 | M16 | — | 3 | 1 | no | — |
| 114 | M16 | — | 3 | 2 | no | — |
| 115 | M16 | — | 6 | 1 | no | — |
| 116 | M16 | — | 6 | 2 | no | — |
| 117 | M16 | — | 6 | 1 | no | — |
| 118 | M16 | — | 6 | 1 | no | — |
| 119 | M16 | — | 6 | 1 | no | — |
| 120 | M16 | — | 6 | 1 | no | — |
| 121 | M16 | — | 6 | 2 | no | — |
| 122 | M16 | S4 | 1 | 1 | yes | characteristic, direct product factor |
| 123 | C2×C8 | — | 3 | 2 | no | — |
| 124 | C2×C8 | — | 3 | 2 | no | — |
| 125 | C2×C8 | — | 3 | 2 | no | — |
| 126 | C2×C8 | — | 3 | 2 | no | — |
| 127 | C2×C8 | — | 6 | 2 | no | — |
| 128 | C2×C8 | — | 6 | 2 | no | — |
| 129 | C4⋊C4 | — | 3 | 1 | no | — |
| 130 | C4⋊C4 | — | 3 | 1 | no | — |
| 131 | C4⋊C4 | — | 3 | 1 | no | — |
| 132 | C4⋊C4 | — | 3 | 1 | no | — |
| 133 | C22⋊C4 | — | 3 | 1 | no | — |
| 134 | C22⋊C4 | — | 3 | 1 | no | — |
| 135 | C22⋊C4 | — | 3 | 1 | no | — |
| 136 | C22⋊C4 | — | 3 | 1 | no | — |
| 137 | C22⋊C4 | — | 3 | 1 | no | — |
| 138 | C22⋊C4 | — | 3 | 1 | no | — |
| 139 | C22⋊C4 | — | 3 | 1 | no | — |
| 140 | C22⋊C4 | — | 3 | 1 | no | — |
| 141 | C42 | — | 3 | 1 | no | — |
| 142 | C42 | — | 3 | 1 | no | — |
| 143 | C42 | — | 3 | 1 | no | — |
| 144 | C42 | — | 3 | 1 | no | — |
| 145 | C22×C3 | — | 4 | 1 | no | — |
| 146 | C2×D3 | — | 4 | 1 | no | — |
| 147 | C2×D3 | — | 4 | 1 | no | — |
| 148 | C2×D3 | — | 8 | 2 | no | — |
| 149 | A4 | C2×M16 | 1 | 1 | yes | characteristic, complemented |
| 150 | C3×C4 | — | 4 | 1 | no | — |
| 151 | C3×C4 | — | 4 | 1 | no | — |
| 152 | Dic3 | — | 4 | 1 | no | — |
| 153 | Dic3 | — | 4 | 1 | no | — |
| 154 | C23 | — | 2 | 1 | no | — |
| 155 | C23 | — | 3 | 1 | no | — |
| 156 | C23 | — | 3 | 1 | no | — |
| 157 | C23 | — | 3 | 1 | no | — |
| 158 | C23 | — | 3 | 1 | no | — |
| 159 | C23 | — | 6 | 1 | no | — |
| 160 | C23 | — | 6 | 1 | no | — |
| 161 | C23 | — | 6 | 2 | no | — |
| 162 | C23 | — | 6 | 1 | no | — |
| 163 | C23 | — | 12 | 1 | no | — |
| 164 | C23 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 165 | D4 | — | 3 | 2 | no | — |
| 166 | D4 | — | 3 | 2 | no | — |
| 167 | D4 | — | 6 | 2 | no | — |
| 168 | D4 | — | 6 | 1 | no | — |
| 169 | D4 | — | 6 | 2 | no | — |
| 170 | D4 | — | 6 | 1 | no | — |
| 171 | C2×C4 | — | 3 | 1 | no | — |
| 172 | C2×C4 | — | 3 | 1 | no | — |
| 173 | C2×C4 | — | 3 | 1 | no | — |
| 174 | C2×C4 | — | 3 | 1 | no | — |
| 175 | C2×C4 | — | 3 | 1 | no | — |
| 176 | C2×C4 | — | 3 | 1 | no | — |
| 177 | C2×C4 | — | 3 | 1 | no | — |
| 178 | C2×C4 | — | 3 | 1 | no | — |
| 179 | C2×C4 | — | 3 | 1 | no | — |
| 180 | C2×C4 | — | 3 | 1 | no | — |
| 181 | C2×C4 | — | 3 | 1 | no | — |
| 182 | C2×C4 | — | 3 | 1 | no | — |
| 183 | C2×C4 | — | 3 | 1 | no | — |
| 184 | C2×C4 | — | 6 | 1 | no | — |
| 185 | C2×C4 | — | 6 | 1 | no | — |
| 186 | C2×C4 | — | 6 | 1 | no | — |
| 187 | C2×C4 | — | 6 | 1 | no | — |
| 188 | C2×C4 | — | 6 | 1 | no | — |
| 189 | C2×C4 | — | 6 | 2 | no | — |
| 190 | C2×C4 | — | 6 | 1 | no | — |
| 191 | C2×C4 | — | 6 | 1 | no | — |
| 192 | C2×C4 | — | 6 | 1 | no | — |
| 193 | C2×C4 | — | 6 | 1 | no | — |
| 194 | C2×C4 | — | 6 | 1 | no | — |
| 195 | C2×C4 | — | 6 | 1 | no | — |
| 196 | C2×C4 | — | 12 | 1 | no | — |
| 197 | C2×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 198 | C8 | — | 3 | 2 | no | — |
| 199 | C8 | — | 6 | 2 | no | — |
| 200 | C8 | — | 6 | 2 | no | — |
| 201 | C8 | C2×S4 | 1 | 2 | yes | complemented |
| 202 | C2×C3 | — | 4 | 1 | no | — |
| 203 | C2×C3 | — | 8 | 1 | no | — |
| 204 | D3 | — | 4 | 2 | no | — |
| 205 | D3 | — | 8 | 1 | no | — |
| 206 | C22 | — | 3 | 1 | no | — |
| 207 | C22 | — | 3 | 1 | no | — |
| 208 | C22 | — | 3 | 2 | no | — |
| 209 | C22 | — | 3 | 1 | no | — |
| 210 | C22 | — | 6 | 1 | no | — |
| 211 | C22 | — | 6 | 1 | no | — |
| 212 | C22 | — | 6 | 1 | no | — |
| 213 | C22 | — | 6 | 1 | no | — |
| 214 | C22 | — | 6 | 1 | no | — |
| 215 | C22 | — | 6 | 1 | no | — |
| 216 | C22 | — | 6 | 1 | no | — |
| 217 | C22 | — | 6 | 1 | no | — |
| 218 | C22 | — | 6 | 1 | no | — |
| 219 | C22 | — | 12 | 2 | no | — |
| 220 | C22 | — | 12 | 2 | no | — |
| 221 | C22 | D3×M16 | 1 | 1 | yes | characteristic, complemented |
| 222 | C22 | C4×S4 | 1 | 1 | yes | characteristic |
| 223 | C4 | — | 3 | 2 | no | — |
| 224 | C4 | — | 3 | 1 | no | — |
| 225 | C4 | — | 3 | 1 | no | — |
| 226 | C4 | — | 6 | 1 | no | — |
| 227 | C4 | — | 6 | 1 | no | — |
| 228 | C4 | — | 6 | 1 | no | — |
| 229 | C4 | — | 6 | 1 | no | — |
| 230 | C4 | — | 6 | 1 | no | — |
| 231 | C4 | C4×S4 | 1 | 1 | yes | characteristic |
| 232 | C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 233 | C3 | — | 4 | 1 | no | Sylow |
| 234 | C2 | — | 2 | 1 | no | — |
| 235 | C2 | — | 3 | 1 | no | — |
| 236 | C2 | — | 3 | 1 | no | — |
| 237 | C2 | — | 6 | 1 | no | — |
| 238 | C2 | — | 6 | 2 | no | — |
| 239 | C2 | — | 12 | 1 | no | — |
| 240 | C2 | C2×C4×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×S4 |
| commutator | C2×A4 | C22×C4 |
| Fitting | C22×M16 | D3 |
| socle | C23 | C2×C4×D3 |
| prime radical | C22×S4 | C4 |
| cosocle | C4×A4 | C23 |
| hypercenter | M16 | S4 |
| hypocenter | A4 | C2×M16 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C4 ◁ M16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. A4×M16 | C2 | semidirect |
| 3. #5409 | C2 | semidirect |
| 4. #5400 | C2 | semidirect |
| 5. C8×S4 | C2 | semidirect |
| 15. C8×A4 | C22 | semidirect |
| 16. A4⋊C8 | C22 | semidirect |
| 18. C22×M16 | D3 | semidirect |
| 54. C22×C8 | C2×D3 | semidirect |
| 79. S4 | M16 | direct (N × H) |
| 122. M16 | S4 | direct (N × H) |
| 149. A4 | C2×M16 | semidirect |
| 201. C8 | C2×S4 | semidirect |
| 221. C22 | D3×M16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×S4 | 2 |
| A4×M16 | 2 |
| #5409 | 2 |
| #5400 | 2 |
| C8×S4 | 2 |
| D4×M16 | 3 |
| D3×M16 | 4 |