Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
797 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 797 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 | C2×2−1+4⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #5926 | C2 | 1 | 8 | yes | maximal, direct product factor |
| 3 | C2×#986 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#985 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C2×#983 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C22×GL(2,𝔽3) | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C2×#577 | — | 3 | 1 | no | maximal, Sylow |
| 8 | C2×D3×D4 | — | 4 | 1 | no | maximal |
| 9 | 2−1+4⋊C3 | C22 | 1 | 4 | yes | complemented |
| 10 | C2×#41⋊C3 | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | C22×SL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 12 | #986 | C22 | 1 | 4 | yes | complemented |
| 13 | #985 | C22 | 1 | 4 | yes | complemented |
| 14 | #983 | C22 | 1 | 8 | yes | complemented |
| 15 | C2×GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 16 | C2×GL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 17 | C2×GL(2,𝔽3) | C22 | 1 | 8 | yes | complemented |
| 18 | C2×#226 | C22 | 1 | 1 | yes | characteristic, complemented |
| 19 | C2×2−1+4 | D3 | 1 | 1 | yes | characteristic, complemented |
| 20 | C2×H(5,𝔽2) | — | 3 | 1 | no | — |
| 21 | #577 | — | 3 | 8 | no | — |
| 22 | #577 | — | 3 | 8 | no | — |
| 23 | C2×#137 | — | 3 | 1 | no | — |
| 24 | C2×#137 | — | 3 | 2 | no | — |
| 25 | C2×AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 26 | C2×AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 27 | C2×#135 | — | 3 | 2 | no | — |
| 28 | C2×#135 | — | 3 | 1 | no | — |
| 29 | C22×SD16 | — | 3 | 1 | no | — |
| 30 | C22×SD16 | — | 3 | 2 | no | — |
| 31 | C2×#131 | — | 3 | 1 | no | — |
| 32 | C23×D3 | — | 4 | 2 | no | — |
| 33 | C2×C3×D4 | — | 4 | 1 | no | — |
| 34 | C2×#67 | — | 4 | 2 | no | — |
| 35 | D3×D4 | — | 4 | 8 | no | — |
| 36 | C2×D12 | — | 4 | 1 | no | — |
| 37 | C2×C4×D3 | — | 4 | 1 | no | — |
| 38 | #41⋊C3 | C23 | 1 | 2 | yes | complemented |
| 39 | C2×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 40 | C2×SL(2,𝔽3) | C23 | 1 | 4 | yes | complemented |
| 41 | GL(2,𝔽3) | C23 | 1 | 2 | yes | complemented |
| 42 | GL(2,𝔽3) | C23 | 1 | 4 | yes | complemented |
| 43 | #226 | C23 | 1 | 2 | yes | complemented |
| 44 | 2−1+4 | — | 3 | 4 | no | — |
| 45 | 2−1+4 | C2×D3 | 1 | 4 | yes | complemented |
| 46 | H(5,𝔽2) | — | 3 | 8 | no | — |
| 47 | H(5,𝔽2) | — | 6 | 4 | no | — |
| 48 | C2×#41 | — | 3 | 1 | no | — |
| 49 | C2×#41 | — | 3 | 2 | no | — |
| 50 | C2×#41 | — | 3 | 2 | no | — |
| 51 | C2×#41 | — | 3 | 1 | no | — |
| 52 | C2×#41 | — | 6 | 1 | no | — |
| 53 | C2×#41 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 54 | C22×Q8 | — | 3 | 1 | no | — |
| 55 | C22×Q8 | C2×D3 | 1 | 2 | yes | complemented |
| 56 | C22×D4 | — | 3 | 2 | no | — |
| 57 | C22×D4 | — | 3 | 1 | no | — |
| 58 | C22×D4 | — | 6 | 2 | no | — |
| 59 | C22×D4 | — | 6 | 1 | no | — |
| 60 | #137 | — | 3 | 8 | no | — |
| 61 | #137 | — | 3 | 8 | no | — |
| 62 | #137 | — | 3 | 8 | no | — |
| 63 | AGL(1,ℤ/8) | — | 3 | 4 | no | — |
| 64 | AGL(1,ℤ/8) | — | 3 | 16 | no | — |
| 65 | AGL(1,ℤ/8) | — | 3 | 4 | no | — |
| 66 | #135 | — | 3 | 4 | no | — |
| 67 | #135 | — | 3 | 16 | no | — |
| 68 | #135 | — | 3 | 4 | no | — |
| 69 | C2×Q16 | — | 3 | 2 | no | — |
| 70 | C2×Q16 | — | 3 | 1 | no | — |
| 71 | C2×SD16 | — | 3 | 2 | no | — |
| 72 | C2×SD16 | — | 3 | 2 | no | — |
| 73 | C2×SD16 | — | 3 | 2 | no | — |
| 74 | C2×SD16 | — | 3 | 8 | no | — |
| 75 | C2×SD16 | — | 3 | 8 | no | — |
| 76 | C2×SD16 | — | 3 | 1 | no | — |
| 77 | C2×SD16 | — | 3 | 1 | no | — |
| 78 | C2×SD16 | — | 3 | 2 | no | — |
| 79 | C2×SD16 | — | 3 | 8 | no | — |
| 80 | C2×D8 | — | 3 | 2 | no | — |
| 81 | C2×D8 | — | 3 | 1 | no | — |
| 82 | #131 | — | 3 | 8 | no | — |
| 83 | C2×M16 | — | 3 | 2 | no | — |
| 84 | C2×M16 | — | 3 | 1 | no | — |
| 85 | C22×C8 | — | 3 | 2 | no | — |
| 86 | C22×C8 | — | 3 | 1 | no | — |
| 87 | C23×C3 | — | 4 | 2 | no | — |
| 88 | C22×D3 | — | 4 | 1 | no | — |
| 89 | C22×D3 | — | 4 | 2 | no | — |
| 90 | C22×D3 | — | 4 | 8 | no | — |
| 91 | C22×D3 | — | 8 | 8 | no | — |
| 92 | C3×D4 | — | 4 | 4 | no | — |
| 93 | C2×C3×C4 | — | 4 | 1 | no | — |
| 94 | #67 | — | 4 | 8 | no | — |
| 95 | C2×Dic3 | — | 4 | 1 | no | — |
| 96 | D12 | — | 4 | 4 | no | — |
| 97 | C4×D3 | — | 4 | 4 | no | — |
| 98 | SL(2,𝔽3) | C24 | 1 | 1 | yes | characteristic |
| 99 | C24 | — | 6 | 1 | no | — |
| 100 | C24 | — | 6 | 2 | no | — |
| 101 | #41 | — | 3 | 4 | no | — |
| 102 | #41 | — | 3 | 4 | no | — |
| 103 | #41 | — | 3 | 2 | no | — |
| 104 | #41 | — | 3 | 8 | no | — |
| 105 | #41 | — | 3 | 8 | no | — |
| 106 | #41 | — | 3 | 4 | no | — |
| 107 | #41 | — | 6 | 4 | no | — |
| 108 | #41 | — | 6 | 4 | no | — |
| 109 | #41 | — | 6 | 2 | no | — |
| 110 | #41 | — | 6 | 8 | no | — |
| 111 | #41 | C22×D3 | 1 | 2 | yes | complemented |
| 112 | C2×Q8 | — | 3 | 1 | no | — |
| 113 | C2×Q8 | — | 3 | 2 | no | — |
| 114 | C2×Q8 | — | 3 | 4 | no | — |
| 115 | C2×Q8 | — | 3 | 4 | no | — |
| 116 | C2×Q8 | — | 3 | 1 | no | — |
| 117 | C2×Q8 | — | 6 | 2 | no | — |
| 118 | C2×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 119 | C2×Q8 | C22×D3 | 1 | 4 | yes | complemented |
| 120 | C2×D4 | — | 3 | 2 | no | — |
| 121 | C2×D4 | — | 3 | 1 | no | — |
| 122 | C2×D4 | — | 3 | 2 | no | — |
| 123 | C2×D4 | — | 3 | 1 | no | — |
| 124 | C2×D4 | — | 3 | 8 | no | — |
| 125 | C2×D4 | — | 3 | 4 | no | — |
| 126 | C2×D4 | — | 6 | 2 | no | — |
| 127 | C2×D4 | — | 6 | 4 | no | — |
| 128 | C2×D4 | — | 6 | 2 | no | — |
| 129 | C2×D4 | — | 6 | 16 | no | — |
| 130 | C2×D4 | — | 6 | 1 | no | — |
| 131 | C2×D4 | — | 6 | 2 | no | — |
| 132 | C2×D4 | — | 6 | 8 | no | — |
| 133 | C2×D4 | — | 6 | 1 | no | — |
| 134 | C2×D4 | S4 | 1 | 1 | yes | characteristic |
| 135 | C22×C4 | — | 3 | 2 | no | — |
| 136 | C22×C4 | — | 3 | 1 | no | — |
| 137 | C22×C4 | — | 6 | 1 | no | — |
| 138 | C22×C4 | — | 6 | 2 | no | — |
| 139 | C22×C4 | — | 6 | 1 | no | — |
| 140 | Q16 | — | 3 | 2 | no | — |
| 141 | Q16 | — | 3 | 8 | no | — |
| 142 | Q16 | — | 3 | 2 | no | — |
| 143 | SD16 | — | 3 | 2 | no | — |
| 144 | SD16 | — | 3 | 8 | no | — |
| 145 | SD16 | — | 3 | 4 | no | — |
| 146 | SD16 | — | 3 | 8 | no | — |
| 147 | SD16 | — | 3 | 8 | no | — |
| 148 | SD16 | — | 3 | 4 | no | — |
| 149 | SD16 | — | 3 | 2 | no | — |
| 150 | SD16 | — | 3 | 4 | no | — |
| 151 | D8 | — | 3 | 8 | no | — |
| 152 | D8 | — | 3 | 4 | no | — |
| 153 | M16 | — | 3 | 8 | no | — |
| 154 | M16 | — | 3 | 4 | no | — |
| 155 | C2×C8 | — | 3 | 1 | no | — |
| 156 | C2×C8 | — | 3 | 1 | no | — |
| 157 | C2×C8 | — | 3 | 2 | no | — |
| 158 | C2×C8 | — | 3 | 8 | no | — |
| 159 | C2×C8 | — | 3 | 4 | no | — |
| 160 | C22×C3 | — | 4 | 4 | no | — |
| 161 | C22×C3 | — | 4 | 1 | no | — |
| 162 | C22×C3 | — | 8 | 4 | no | — |
| 163 | C2×D3 | — | 4 | 2 | no | — |
| 164 | C2×D3 | — | 4 | 4 | no | — |
| 165 | C2×D3 | — | 4 | 4 | no | — |
| 166 | C2×D3 | — | 8 | 4 | no | — |
| 167 | C2×D3 | — | 8 | 16 | no | — |
| 168 | C3×C4 | — | 4 | 2 | no | — |
| 169 | Dic3 | — | 4 | 2 | no | — |
| 170 | C23 | — | 3 | 1 | no | — |
| 171 | C23 | — | 6 | 4 | no | — |
| 172 | C23 | — | 6 | 1 | no | — |
| 173 | C23 | — | 6 | 2 | no | — |
| 174 | C23 | — | 6 | 8 | no | — |
| 175 | C23 | — | 24 | 4 | no | — |
| 176 | C23 | — | 24 | 2 | no | — |
| 177 | C23 | C2×S4 | 1 | 2 | yes | — |
| 178 | Q8 | — | 3 | 1 | no | — |
| 179 | Q8 | — | 3 | 4 | no | — |
| 180 | Q8 | — | 3 | 4 | no | — |
| 181 | Q8 | — | 3 | 2 | no | — |
| 182 | Q8 | — | 6 | 2 | no | — |
| 183 | Q8 | — | 6 | 1 | no | — |
| 184 | Q8 | C23×D3 | 1 | 1 | yes | characteristic |
| 185 | D4 | — | 3 | 2 | no | — |
| 186 | D4 | — | 3 | 8 | no | — |
| 187 | D4 | — | 3 | 4 | no | — |
| 188 | D4 | — | 3 | 2 | no | — |
| 189 | D4 | — | 6 | 1 | no | — |
| 190 | D4 | — | 6 | 1 | no | — |
| 191 | D4 | — | 6 | 2 | no | — |
| 192 | D4 | — | 6 | 8 | no | — |
| 193 | D4 | — | 6 | 4 | no | — |
| 194 | D4 | — | 6 | 8 | no | — |
| 195 | D4 | — | 6 | 4 | no | — |
| 196 | D4 | C2×S4 | 1 | 4 | yes | — |
| 197 | C2×C4 | — | 3 | 4 | no | — |
| 198 | C2×C4 | — | 3 | 4 | no | — |
| 199 | C2×C4 | — | 3 | 1 | no | — |
| 200 | C2×C4 | — | 3 | 2 | no | — |
| 201 | C2×C4 | — | 3 | 2 | no | — |
| 202 | C2×C4 | — | 3 | 2 | no | — |
| 203 | C2×C4 | — | 6 | 4 | no | — |
| 204 | C2×C4 | — | 6 | 8 | no | — |
| 205 | C2×C4 | — | 6 | 1 | no | — |
| 206 | C2×C4 | — | 6 | 4 | no | — |
| 207 | C2×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 208 | C8 | — | 3 | 2 | no | — |
| 209 | C8 | — | 3 | 2 | no | — |
| 210 | C8 | — | 3 | 4 | no | — |
| 211 | C2×C3 | — | 4 | 2 | no | — |
| 212 | C2×C3 | — | 4 | 1 | no | — |
| 213 | C2×C3 | — | 8 | 4 | no | — |
| 214 | D3 | — | 4 | 4 | no | — |
| 215 | D3 | — | 8 | 4 | no | — |
| 216 | C22 | — | 2 | 4 | no | — |
| 217 | C22 | — | 3 | 2 | no | — |
| 218 | C22 | — | 6 | 2 | no | — |
| 219 | C22 | — | 6 | 2 | no | — |
| 220 | C22 | — | 6 | 4 | no | — |
| 221 | C22 | — | 12 | 2 | no | — |
| 222 | C22 | — | 12 | 4 | no | — |
| 223 | C22 | — | 24 | 8 | no | — |
| 224 | C22 | — | 24 | 4 | no | — |
| 225 | C22 | C22×S4 | 1 | 4 | yes | — |
| 226 | C22 | C22×S4 | 1 | 1 | yes | characteristic |
| 227 | C4 | — | 3 | 1 | no | — |
| 228 | C4 | — | 3 | 1 | no | — |
| 229 | C4 | — | 3 | 4 | no | — |
| 230 | C4 | — | 6 | 2 | no | — |
| 231 | C4 | C22×S4 | 1 | 2 | yes | — |
| 232 | C3 | — | 4 | 1 | no | Sylow |
| 233 | C2 | — | 2 | 4 | no | — |
| 234 | C2 | — | 6 | 2 | no | — |
| 235 | C2 | — | 12 | 2 | no | — |
| 236 | C2 | — | 12 | 4 | no | — |
| 237 | C2 | #5926 | 1 | 2 | yes | direct product factor |
| 238 | C2 | C23×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C24 |
| Frattini | C2 | C23×S4 |
| Fitting | C2×2−1+4 | D3 |
| hypercenter | C2×D4 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×2−1+4⋊C3 | C2 | semidirect |
| 2. #5926 | C2 | direct (N × H) |
| 3. C2×#986 | C2 | semidirect |
| 4. C2×#985 | C2 | semidirect |
| 5. C2×#983 | C2 | semidirect |
| 6. C22×GL(2,𝔽3) | C2 | semidirect |
| 9. 2−1+4⋊C3 | C22 | semidirect |
| 10. C2×#41⋊C3 | C22 | semidirect |
| 11. C22×SL(2,𝔽3) | C22 | semidirect |
| 12. #986 | C22 | semidirect |
| 13. #985 | C22 | semidirect |
| 14. #983 | C22 | semidirect |
| 15. C2×GL(2,𝔽3) | C22 | semidirect |
| 16. C2×GL(2,𝔽3) | C22 | semidirect |
| 17. C2×GL(2,𝔽3) | C22 | semidirect |
| 18. C2×#226 | C22 | semidirect |
| 19. C2×2−1+4 | D3 | semidirect |
| 38. #41⋊C3 | C23 | semidirect |
| 40. C2×SL(2,𝔽3) | C23 | semidirect |
| 41. GL(2,𝔽3) | C23 | semidirect |
| 42. GL(2,𝔽3) | C23 | semidirect |
| 43. #226 | C23 | semidirect |
| 45. 2−1+4 | C2×D3 | semidirect |
| 53. C2×#41 | C2×D3 | semidirect |
| 55. C22×Q8 | C2×D3 | semidirect |
| 111. #41 | C22×D3 | semidirect |
| 119. C2×Q8 | C22×D3 | semidirect |
| 237. C2 | #5926 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×2−1+4⋊C3 | 2 |
| #5926 | 2 |
| C2×#986 | 2 |
| C2×#985 | 2 |
| C2×#983 | 2 |
| C22×GL(2,𝔽3) | 2 |
| C2×#577 | 3 |
| C2×D3×D4 | 4 |