Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
250 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 44 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 | #5923 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #568⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×GU(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #5405 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #5404 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #2956 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C2×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 8 | #986 | C4 | 1 | 1 | yes | characteristic, complemented |
| 9 | #985 | C22 | 1 | 2 | yes | — |
| 10 | #984 | C4 | 1 | 1 | yes | characteristic, complemented |
| 11 | #983 | C4 | 1 | 2 | yes | complemented |
| 12 | #907 | — | 4 | 1 | no | maximal |
| 13 | #131⋊C3 | C22 | 1 | 2 | yes | complemented |
| 14 | GU(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | GU(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 16 | #575 | — | 3 | 1 | no | — |
| 17 | #568 | — | 3 | 1 | no | — |
| 18 | #568 | D3 | 1 | 1 | yes | characteristic, complemented |
| 19 | #444 | — | 3 | 2 | no | — |
| 20 | #444 | — | 3 | 2 | no | — |
| 21 | #443 | — | 3 | 2 | no | — |
| 22 | #443 | — | 3 | 2 | no | — |
| 23 | #430 | — | 3 | 1 | no | — |
| 24 | C2×#104 | — | 3 | 1 | no | — |
| 25 | C2×#104 | — | 3 | 1 | no | — |
| 26 | C4×M16 | — | 3 | 1 | no | — |
| 27 | #235 | — | 4 | 1 | no | — |
| 28 | #41⋊C3 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | #41⋊C3 | C23 | 1 | 1 | yes | characteristic |
| 30 | C2×SL(2,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 31 | GL(2,𝔽3) | C2×C4 | 1 | 2 | yes | complemented |
| 32 | #226 | C2×C4 | 1 | 2 | yes | complemented |
| 33 | C3×M16 | — | 4 | 1 | no | — |
| 34 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 35 | ΓL(1,𝔽25) | — | 4 | 2 | no | — |
| 36 | D3×C8 | — | 4 | 2 | no | — |
| 37 | C2×#41 | — | 3 | 1 | no | — |
| 38 | C2×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 39 | #137 | — | 3 | 1 | no | — |
| 40 | #137 | — | 3 | 1 | no | — |
| 41 | #137 | — | 3 | 2 | no | — |
| 42 | AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 43 | AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 44 | AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 45 | #135 | — | 3 | 2 | no | — |
| 46 | #135 | — | 3 | 2 | no | — |
| 47 | #131 | — | 3 | 2 | no | — |
| 48 | #131 | — | 3 | 2 | no | — |
| 49 | #131 | — | 3 | 2 | no | — |
| 50 | #131 | — | 6 | 1 | no | — |
| 51 | #131 | — | 6 | 1 | no | — |
| 52 | #131 | C2×D3 | 1 | 2 | yes | — |
| 53 | C2×M16 | — | 3 | 1 | no | — |
| 54 | C2×M16 | — | 3 | 1 | no | — |
| 55 | C2×M16 | — | 3 | 1 | no | — |
| 56 | C2×M16 | — | 3 | 1 | no | — |
| 57 | C2×M16 | — | 6 | 2 | no | — |
| 58 | C2×C42 | — | 3 | 1 | no | — |
| 59 | #108 | — | 3 | 2 | no | — |
| 60 | #108 | — | 3 | 2 | no | — |
| 61 | #104 | — | 3 | 2 | no | — |
| 62 | #104 | — | 3 | 1 | no | — |
| 63 | #104 | — | 3 | 2 | no | — |
| 64 | #104 | — | 3 | 1 | no | — |
| 65 | #104 | — | 3 | 1 | no | — |
| 66 | #104 | — | 3 | 1 | no | — |
| 67 | #97 | — | 3 | 2 | no | — |
| 68 | C4×C8 | — | 3 | 2 | no | — |
| 69 | C2×C3×C4 | — | 4 | 1 | no | — |
| 70 | #67 | — | 4 | 2 | no | — |
| 71 | D12 | — | 4 | 1 | no | — |
| 72 | C4×D3 | — | 4 | 2 | no | — |
| 73 | Dic6 | — | 4 | 1 | no | — |
| 74 | SL(2,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 75 | C3×C8 | — | 4 | 2 | no | — |
| 76 | C3⋊C8 | — | 4 | 1 | no | — |
| 77 | C3⋊C8 | — | 4 | 1 | no | — |
| 78 | #41 | — | 3 | 1 | no | — |
| 79 | #41 | — | 3 | 1 | no | — |
| 80 | #41 | — | 3 | 2 | no | — |
| 81 | #41 | — | 3 | 2 | no | — |
| 82 | #41 | — | 6 | 1 | no | — |
| 83 | #41 | — | 6 | 1 | no | — |
| 84 | #41 | C4×D3 | 1 | 1 | yes | characteristic |
| 85 | #41 | C22×D3 | 1 | 1 | yes | characteristic |
| 86 | C2×Q8 | — | 3 | 1 | no | — |
| 87 | C2×Q8 | C4×D3 | 1 | 1 | yes | characteristic |
| 88 | C2×D4 | — | 3 | 1 | no | — |
| 89 | C2×D4 | — | 3 | 1 | no | — |
| 90 | C2×D4 | — | 6 | 1 | no | — |
| 91 | C22×C4 | — | 3 | 1 | no | — |
| 92 | C22×C4 | — | 6 | 1 | no | — |
| 93 | C22×C4 | — | 6 | 1 | no | — |
| 94 | Q16 | — | 3 | 2 | no | — |
| 95 | Q16 | — | 3 | 2 | no | — |
| 96 | SD16 | — | 3 | 2 | no | — |
| 97 | SD16 | — | 3 | 2 | no | — |
| 98 | SD16 | — | 3 | 2 | no | — |
| 99 | SD16 | — | 3 | 2 | no | — |
| 100 | D8 | — | 3 | 2 | no | — |
| 101 | D8 | — | 3 | 2 | no | — |
| 102 | M16 | — | 3 | 2 | no | — |
| 103 | M16 | — | 3 | 1 | no | — |
| 104 | M16 | — | 3 | 1 | no | — |
| 105 | M16 | — | 3 | 1 | no | — |
| 106 | M16 | — | 3 | 2 | no | — |
| 107 | M16 | — | 3 | 2 | no | — |
| 108 | M16 | — | 3 | 1 | no | — |
| 109 | M16 | — | 3 | 1 | no | — |
| 110 | M16 | — | 6 | 2 | no | — |
| 111 | M16 | — | 6 | 1 | no | — |
| 112 | M16 | — | 6 | 2 | no | — |
| 113 | M16 | S4 | 1 | 1 | yes | characteristic |
| 114 | C2×C8 | — | 3 | 2 | no | — |
| 115 | C2×C8 | — | 3 | 2 | no | — |
| 116 | C2×C8 | — | 6 | 1 | no | — |
| 117 | C2×C8 | — | 6 | 2 | no | — |
| 118 | C2×C8 | — | 6 | 1 | no | — |
| 119 | C2×C8 | — | 6 | 2 | no | — |
| 120 | C42 | — | 3 | 1 | no | — |
| 121 | C42 | — | 3 | 1 | no | — |
| 122 | C42 | — | 6 | 1 | no | — |
| 123 | C22×C3 | — | 4 | 1 | no | — |
| 124 | C2×D3 | — | 4 | 2 | no | — |
| 125 | C3×C4 | — | 4 | 1 | no | — |
| 126 | C3×C4 | — | 4 | 1 | no | — |
| 127 | Dic3 | — | 4 | 2 | no | — |
| 128 | C23 | — | 3 | 1 | no | — |
| 129 | C23 | — | 6 | 1 | no | — |
| 130 | Q8 | — | 3 | 1 | no | — |
| 131 | Q8 | — | 3 | 2 | no | — |
| 132 | Q8 | — | 6 | 1 | no | — |
| 133 | Q8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 134 | D4 | — | 3 | 1 | no | — |
| 135 | D4 | — | 3 | 1 | no | — |
| 136 | D4 | — | 3 | 2 | no | — |
| 137 | D4 | — | 6 | 1 | no | — |
| 138 | D4 | — | 6 | 2 | no | — |
| 139 | D4 | — | 6 | 2 | no | — |
| 140 | D4 | — | 6 | 1 | no | — |
| 141 | C2×C4 | — | 3 | 1 | no | — |
| 142 | C2×C4 | — | 3 | 1 | no | — |
| 143 | C2×C4 | — | 3 | 1 | no | — |
| 144 | C2×C4 | — | 3 | 1 | no | — |
| 145 | C2×C4 | — | 3 | 1 | no | — |
| 146 | C2×C4 | — | 6 | 2 | no | — |
| 147 | C2×C4 | — | 6 | 1 | no | — |
| 148 | C2×C4 | — | 6 | 1 | no | — |
| 149 | C2×C4 | — | 6 | 2 | no | — |
| 150 | C2×C4 | — | 6 | 1 | no | — |
| 151 | C2×C4 | — | 12 | 2 | no | — |
| 152 | C2×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 153 | C8 | — | 3 | 2 | no | — |
| 154 | C8 | — | 3 | 2 | no | — |
| 155 | C8 | — | 3 | 2 | no | — |
| 156 | C8 | — | 6 | 1 | no | — |
| 157 | C8 | — | 6 | 1 | no | — |
| 158 | C8 | C2×S4 | 1 | 2 | yes | — |
| 159 | C2×C3 | — | 4 | 1 | no | — |
| 160 | C2×C3 | — | 8 | 1 | no | — |
| 161 | D3 | — | 8 | 2 | no | — |
| 162 | C22 | — | 3 | 1 | no | — |
| 163 | C22 | — | 3 | 1 | no | — |
| 164 | C22 | — | 6 | 2 | no | — |
| 165 | C22 | — | 12 | 1 | no | — |
| 166 | C22 | — | 24 | 1 | no | — |
| 167 | C22 | C4×S4 | 1 | 1 | yes | characteristic |
| 168 | C4 | — | 3 | 1 | no | — |
| 169 | C4 | — | 3 | 1 | no | — |
| 170 | C4 | — | 6 | 2 | no | — |
| 171 | C4 | — | 6 | 2 | no | — |
| 172 | C4 | — | 12 | 1 | no | — |
| 173 | C4 | C4×S4 | 1 | 1 | yes | characteristic |
| 174 | C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 175 | C3 | — | 4 | 1 | no | Sylow |
| 176 | C2 | — | 2 | 1 | no | — |
| 177 | C2 | — | 6 | 1 | no | — |
| 178 | C2 | — | 6 | 1 | no | — |
| 179 | C2 | — | 12 | 2 | no | — |
| 180 | C2 | C2×C4×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C4 |
| Fitting | #568 | D3 |
| socle | C2 | C2×C4×S4 |
| prime radical | #5923 | C2 |
| cosocle | #41⋊C3 | C23 |
| hypercenter | M16 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C4 ◁ M16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #568⋊C3 | C2 | semidirect |
| 3. C2×GU(2,𝔽3) | C2 | semidirect |
| 4. #5405 | C2 | semidirect |
| 5. #5404 | C2 | semidirect |
| 8. #986 | C4 | semidirect |
| 10. #984 | C4 | semidirect |
| 11. #983 | C4 | semidirect |
| 13. #131⋊C3 | C22 | semidirect |
| 14. GU(2,𝔽3) | C22 | semidirect |
| 15. GU(2,𝔽3) | C22 | semidirect |
| 18. #568 | D3 | semidirect |
| 31. GL(2,𝔽3) | C2×C4 | semidirect |
| 32. #226 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5923 | 2 |
| #568⋊C3 | 2 |
| C2×GU(2,𝔽3) | 2 |
| #5405 | 2 |
| #5404 | 2 |
| #2956 | 3 |
| #907 | 4 |