Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
347 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 42 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 | C23×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#983 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #5421 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #5420 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #5419 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #3004 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C22×SL(2,𝔽3) | — | 2 | 2 | no | — |
| 8 | C22×SL(2,𝔽3) | C22 | 1 | 2 | yes | — |
| 9 | C22×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 10 | #983 | — | 2 | 2 | no | — |
| 11 | C2×GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 12 | C2×#226 | C22 | 1 | 1 | yes | characteristic, complemented |
| 13 | #953 | — | 4 | 1 | no | maximal |
| 14 | #859 | C22 | 1 | 2 | yes | complemented |
| 15 | C23×Q8 | D3 | 1 | 1 | yes | characteristic, complemented |
| 16 | C2×#137 | — | 3 | 1 | no | — |
| 17 | C2×#137 | — | 3 | 1 | no | — |
| 18 | #525 | — | 3 | 1 | no | — |
| 19 | #451 | — | 3 | 2 | no | — |
| 20 | #451 | — | 3 | 2 | no | — |
| 21 | #448 | — | 3 | 2 | no | — |
| 22 | #448 | — | 3 | 2 | no | — |
| 23 | #419 | — | 3 | 1 | no | — |
| 24 | #419 | — | 3 | 1 | no | — |
| 25 | #407 | — | 3 | 1 | no | — |
| 26 | C24×C3 | — | 4 | 1 | no | — |
| 27 | C2×#67 | — | 4 | 1 | no | — |
| 28 | C2×#67 | — | 4 | 2 | no | — |
| 29 | C2×SL(2,𝔽3) | — | 2 | 2 | no | — |
| 30 | C2×SL(2,𝔽3) | D4 | 1 | 2 | yes | complemented |
| 31 | C2×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 32 | GL(2,𝔽3) | — | 2 | 1 | no | — |
| 33 | #226 | — | 2 | 1 | no | — |
| 34 | #217 | — | 4 | 2 | no | — |
| 35 | #217 | — | 4 | 1 | no | — |
| 36 | C2×#41 | — | 3 | 1 | no | — |
| 37 | C22×Q8 | — | 2 | 2 | no | — |
| 38 | C22×Q8 | — | 3 | 1 | no | — |
| 39 | C22×Q8 | — | 3 | 2 | no | — |
| 40 | C22×Q8 | — | 6 | 2 | no | — |
| 41 | C22×Q8 | C2×D3 | 1 | 2 | yes | — |
| 42 | C22×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 43 | C23×C4 | — | 3 | 1 | no | — |
| 44 | #137 | — | 6 | 2 | no | — |
| 45 | #137 | — | 6 | 4 | no | — |
| 46 | #137 | — | 6 | 2 | no | — |
| 47 | C2×Q16 | — | 3 | 2 | no | — |
| 48 | C2×Q16 | — | 3 | 1 | no | — |
| 49 | C2×Q16 | — | 3 | 1 | no | — |
| 50 | C2×SD16 | — | 3 | 2 | no | — |
| 51 | C2×SD16 | — | 3 | 1 | no | — |
| 52 | C2×SD16 | — | 3 | 1 | no | — |
| 53 | C2×M16 | — | 3 | 1 | no | — |
| 54 | C2×M16 | — | 3 | 1 | no | — |
| 55 | #123 | — | 6 | 1 | no | — |
| 56 | C22⋊Q8 | — | 3 | 2 | no | — |
| 57 | #121 | — | 3 | 2 | no | — |
| 58 | #120 | — | 6 | 1 | no | — |
| 59 | C4×D4 | — | 6 | 2 | no | — |
| 60 | #117 | — | 3 | 1 | no | — |
| 61 | #103 | — | 3 | 2 | no | — |
| 62 | #103 | — | 3 | 2 | no | — |
| 63 | #103 | — | 3 | 2 | no | — |
| 64 | #103 | — | 3 | 2 | no | — |
| 65 | C22⋊C8 | — | 3 | 2 | no | — |
| 66 | C22⋊C8 | — | 3 | 2 | no | — |
| 67 | C23×C3 | — | 4 | 2 | no | — |
| 68 | C23×C3 | — | 4 | 1 | no | — |
| 69 | C23×C3 | — | 8 | 2 | no | — |
| 70 | C23×C3 | — | 8 | 4 | no | — |
| 71 | C22×D3 | — | 4 | 1 | no | — |
| 72 | #67 | — | 8 | 2 | no | — |
| 73 | #67 | — | 8 | 4 | no | — |
| 74 | C2×Dic3 | — | 4 | 2 | no | — |
| 75 | C2×Dic3 | — | 4 | 1 | no | — |
| 76 | SL(2,𝔽3) | C2×D4 | 1 | 1 | yes | characteristic |
| 77 | C24 | S4 | 1 | 1 | yes | characteristic |
| 78 | #41 | — | 6 | 2 | no | — |
| 79 | #41 | — | 6 | 2 | no | — |
| 80 | C2×Q8 | — | 2 | 2 | no | — |
| 81 | C2×Q8 | — | 3 | 1 | no | — |
| 82 | C2×Q8 | — | 3 | 2 | no | — |
| 83 | C2×Q8 | — | 3 | 2 | no | — |
| 84 | C2×Q8 | — | 3 | 4 | no | — |
| 85 | C2×Q8 | — | 3 | 1 | no | — |
| 86 | C2×Q8 | — | 6 | 1 | no | — |
| 87 | C2×Q8 | — | 6 | 2 | no | — |
| 88 | C2×Q8 | — | 6 | 4 | no | — |
| 89 | C2×Q8 | — | 6 | 2 | no | — |
| 90 | C2×Q8 | — | 6 | 4 | no | — |
| 91 | C2×Q8 | #67 | 1 | 2 | yes | complemented |
| 92 | C2×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 93 | C2×D4 | — | 3 | 1 | no | — |
| 94 | C2×D4 | — | 6 | 2 | no | — |
| 95 | C2×D4 | — | 6 | 1 | no | — |
| 96 | C22×C4 | — | 3 | 1 | no | — |
| 97 | C22×C4 | — | 3 | 2 | no | — |
| 98 | C22×C4 | — | 3 | 2 | no | — |
| 99 | C22×C4 | — | 3 | 1 | no | — |
| 100 | C22×C4 | — | 6 | 2 | no | — |
| 101 | C22×C4 | — | 6 | 2 | no | — |
| 102 | C22×C4 | — | 6 | 1 | no | — |
| 103 | Q16 | — | 6 | 2 | no | — |
| 104 | Q16 | — | 6 | 1 | no | — |
| 105 | Q16 | — | 6 | 4 | no | — |
| 106 | Q16 | — | 6 | 1 | no | — |
| 107 | SD16 | — | 6 | 2 | no | — |
| 108 | SD16 | — | 6 | 1 | no | — |
| 109 | SD16 | — | 6 | 4 | no | — |
| 110 | SD16 | — | 6 | 1 | no | — |
| 111 | M16 | — | 6 | 2 | no | — |
| 112 | M16 | — | 6 | 2 | no | — |
| 113 | C2×C8 | — | 3 | 1 | no | — |
| 114 | C2×C8 | — | 3 | 1 | no | — |
| 115 | C2×C8 | — | 3 | 2 | no | — |
| 116 | C4⋊C4 | — | 3 | 2 | no | — |
| 117 | C4⋊C4 | — | 6 | 2 | no | — |
| 118 | C22⋊C4 | — | 6 | 2 | no | — |
| 119 | C22⋊C4 | — | 6 | 1 | no | — |
| 120 | C22⋊C4 | — | 6 | 2 | no | — |
| 121 | C42 | — | 6 | 1 | no | — |
| 122 | C22×C3 | — | 4 | 4 | no | — |
| 123 | C22×C3 | — | 4 | 2 | no | — |
| 124 | C22×C3 | — | 4 | 1 | no | — |
| 125 | C22×C3 | — | 8 | 2 | no | — |
| 126 | C22×C3 | — | 8 | 2 | no | — |
| 127 | C22×C3 | — | 8 | 2 | no | — |
| 128 | C22×C3 | — | 8 | 8 | no | — |
| 129 | C2×D3 | — | 8 | 2 | no | — |
| 130 | C2×D3 | — | 8 | 1 | no | — |
| 131 | Dic3 | — | 8 | 2 | no | — |
| 132 | Dic3 | — | 8 | 1 | no | — |
| 133 | C23 | — | 2 | 4 | no | — |
| 134 | C23 | — | 2 | 2 | no | — |
| 135 | C23 | — | 6 | 1 | no | — |
| 136 | C23 | C2×S4 | 1 | 2 | yes | — |
| 137 | C23 | C2×S4 | 1 | 1 | yes | characteristic |
| 138 | Q8 | — | 3 | 2 | no | — |
| 139 | Q8 | — | 3 | 1 | no | — |
| 140 | Q8 | — | 3 | 4 | no | — |
| 141 | Q8 | — | 6 | 1 | no | — |
| 142 | Q8 | — | 6 | 2 | no | — |
| 143 | Q8 | — | 6 | 4 | no | — |
| 144 | Q8 | — | 6 | 1 | no | — |
| 145 | Q8 | — | 6 | 1 | no | — |
| 146 | Q8 | C2×#67 | 1 | 1 | yes | characteristic |
| 147 | D4 | — | 6 | 1 | no | — |
| 148 | D4 | — | 6 | 1 | no | — |
| 149 | D4 | — | 12 | 4 | no | — |
| 150 | D4 | — | 12 | 2 | no | — |
| 151 | C2×C4 | — | 3 | 2 | no | — |
| 152 | C2×C4 | — | 3 | 2 | no | — |
| 153 | C2×C4 | — | 3 | 1 | no | — |
| 154 | C2×C4 | — | 3 | 1 | no | — |
| 155 | C2×C4 | — | 3 | 2 | no | — |
| 156 | C2×C4 | — | 6 | 2 | no | — |
| 157 | C2×C4 | — | 6 | 2 | no | — |
| 158 | C2×C4 | — | 6 | 4 | no | — |
| 159 | C2×C4 | — | 6 | 2 | no | — |
| 160 | C2×C4 | — | 6 | 2 | no | — |
| 161 | C2×C4 | — | 6 | 1 | no | — |
| 162 | C2×C4 | — | 12 | 2 | no | — |
| 163 | C8 | — | 6 | 2 | no | — |
| 164 | C8 | — | 6 | 1 | no | — |
| 165 | C8 | — | 6 | 1 | no | — |
| 166 | C2×C3 | — | 4 | 2 | no | — |
| 167 | C2×C3 | — | 4 | 1 | no | — |
| 168 | C2×C3 | — | 8 | 4 | no | — |
| 169 | C2×C3 | — | 8 | 2 | no | — |
| 170 | D3 | — | 16 | 1 | no | — |
| 171 | C22 | — | 2 | 8 | no | — |
| 172 | C22 | — | 2 | 2 | no | — |
| 173 | C22 | — | 2 | 2 | no | — |
| 174 | C22 | — | 2 | 2 | no | — |
| 175 | C22 | — | 12 | 2 | no | — |
| 176 | C22 | — | 12 | 1 | no | — |
| 177 | C22 | #983 | 1 | 4 | yes | complemented |
| 178 | C22 | GL(2,ℤ/4) | 1 | 2 | yes | — |
| 179 | C22 | C22×S4 | 1 | 1 | yes | characteristic |
| 180 | C4 | — | 3 | 1 | no | — |
| 181 | C4 | — | 3 | 1 | no | — |
| 182 | C4 | — | 3 | 2 | no | — |
| 183 | C4 | — | 6 | 2 | no | — |
| 184 | C4 | — | 12 | 2 | no | — |
| 185 | C4 | — | 12 | 1 | no | — |
| 186 | C3 | — | 4 | 1 | no | Sylow |
| 187 | C2 | — | 2 | 4 | no | — |
| 188 | C2 | — | 2 | 2 | no | — |
| 189 | C2 | — | 24 | 1 | no | — |
| 190 | C2 | C2×#983 | 1 | 2 | yes | — |
| 191 | C2 | C2×GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×S4 |
| commutator, cosocle | C2×SL(2,𝔽3) | C23 |
| Fitting | C23×Q8 | D3 |
| hypercenter | C24 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×SL(2,𝔽3) | C2 | semidirect |
| 2. C2×#983 | C2 | semidirect |
| 3. #5421 | C2 | semidirect |
| 4. #5420 | C2 | semidirect |
| 5. #5419 | C2 | semidirect |
| 11. C2×GL(2,𝔽3) | C22 | semidirect |
| 12. C2×#226 | C22 | semidirect |
| 14. #859 | C22 | semidirect |
| 15. C23×Q8 | D3 | semidirect |
| 30. C2×SL(2,𝔽3) | D4 | semidirect |
| 91. C2×Q8 | #67 | semidirect |
| 177. C22 | #983 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×SL(2,𝔽3) | 2 |
| C2×#983 | 2 |
| #5421 | 2 |
| #5420 | 2 |
| #5419 | 2 |
| #3004 | 3 |
| #953 | 4 |