Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
123 conjugacy classes of subgroups; only the 13 normal subgroups are shown.
This group has 123 conjugacy classes of subgroups — too many to draw. Only its 13 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 | #5450 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #1020⋊C2 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #4625 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #2199 | — | 3 | 1 | no | maximal, Sylow |
| 5 | #1020 | — | 2 | 1 | no | — |
| 6 | GL(2,ℤ/4) | — | 4 | 1 | no | maximal |
| 7 | #120⋊C3 | — | 2 | 1 | no | — |
| 8 | #120⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 9 | H(3,𝔽4) | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | B(4,𝔽2) | — | 3 | 1 | no | — |
| 11 | B(4,𝔽2) | — | 3 | 1 | no | — |
| 12 | #455 | — | 3 | 1 | no | — |
| 13 | #354 | — | 3 | 1 | no | — |
| 14 | AΣL(1,𝔽16) | — | 3 | 1 | no | — |
| 15 | AΣL(1,𝔽16) | — | 3 | 1 | no | — |
| 16 | C22⋊A4 | D4 | 1 | 1 | yes | characteristic, complemented |
| 17 | C22×A4 | — | 4 | 1 | no | — |
| 18 | C2×S4 | — | 4 | 1 | no | — |
| 19 | SL(2,ℤ/4) | — | 4 | 1 | no | — |
| 20 | H(5,𝔽2) | — | 3 | 1 | no | — |
| 21 | #137 | — | 6 | 1 | no | — |
| 22 | #124 | — | 3 | 1 | no | — |
| 23 | #124 | — | 6 | 1 | no | — |
| 24 | #120 | — | 2 | 1 | no | — |
| 25 | #120 | — | 3 | 1 | no | — |
| 26 | #120 | — | 6 | 1 | no | — |
| 27 | #120 | — | 6 | 1 | no | — |
| 28 | #120 | C2×D3 | 1 | 1 | yes | characteristic |
| 29 | #104 | — | 6 | 1 | no | — |
| 30 | #100 | — | 3 | 1 | no | — |
| 31 | #99 | — | 3 | 1 | no | — |
| 32 | #99 | — | 3 | 1 | no | — |
| 33 | #99 | — | 6 | 1 | no | — |
| 34 | #99 | — | 6 | 1 | no | — |
| 35 | C2×A4 | — | 4 | 1 | no | — |
| 36 | C2×A4 | — | 8 | 1 | no | — |
| 37 | S4 | — | 8 | 1 | no | — |
| 38 | S4 | — | 16 | 1 | no | — |
| 39 | #67 | — | 16 | 1 | no | — |
| 40 | C24 | #67 | 1 | 1 | yes | characteristic, complemented |
| 41 | C24 | S4 | 1 | 1 | yes | characteristic, complemented |
| 42 | #41 | — | 6 | 1 | no | — |
| 43 | #41 | — | 12 | 1 | no | — |
| 44 | C2×Q8 | — | 3 | 1 | no | — |
| 45 | C2×D4 | — | 3 | 1 | no | — |
| 46 | C2×D4 | — | 6 | 1 | no | — |
| 47 | C2×D4 | — | 6 | 1 | no | — |
| 48 | C2×D4 | — | 6 | 1 | no | — |
| 49 | C2×D4 | — | 12 | 1 | no | — |
| 50 | C2×D4 | — | 12 | 1 | no | — |
| 51 | C2×D4 | — | 12 | 1 | no | — |
| 52 | Q16 | — | 12 | 1 | no | — |
| 53 | SD16 | — | 12 | 1 | no | — |
| 54 | M16 | — | 6 | 1 | no | — |
| 55 | C22⋊C4 | — | 3 | 1 | no | — |
| 56 | C22⋊C4 | — | 3 | 1 | no | — |
| 57 | C22⋊C4 | — | 6 | 1 | no | — |
| 58 | C22⋊C4 | — | 6 | 1 | no | — |
| 59 | C22⋊C4 | — | 6 | 1 | no | — |
| 60 | C22⋊C4 | — | 6 | 1 | no | — |
| 61 | C22⋊C4 | — | 12 | 1 | no | — |
| 62 | C22⋊C4 | — | 12 | 1 | no | — |
| 63 | C42 | — | 3 | 1 | no | — |
| 64 | C22×C3 | — | 16 | 1 | no | — |
| 65 | C2×D3 | — | 16 | 1 | no | — |
| 66 | A4 | — | 4 | 1 | no | — |
| 67 | A4 | — | 16 | 1 | no | — |
| 68 | Dic3 | — | 16 | 1 | no | — |
| 69 | C23 | — | 2 | 1 | no | — |
| 70 | C23 | — | 3 | 1 | no | — |
| 71 | C23 | — | 6 | 1 | no | — |
| 72 | C23 | — | 6 | 1 | no | — |
| 73 | C23 | — | 12 | 1 | no | — |
| 74 | C23 | — | 12 | 1 | no | — |
| 75 | C23 | C2×S4 | 1 | 1 | yes | characteristic |
| 76 | Q8 | — | 6 | 1 | no | — |
| 77 | Q8 | — | 12 | 1 | no | — |
| 78 | D4 | — | 6 | 1 | no | — |
| 79 | D4 | — | 12 | 1 | no | — |
| 80 | D4 | — | 12 | 1 | no | — |
| 81 | D4 | — | 12 | 1 | no | — |
| 82 | D4 | — | 12 | 1 | no | — |
| 83 | D4 | — | 24 | 1 | no | — |
| 84 | D4 | — | 24 | 1 | no | — |
| 85 | D4 | — | 24 | 1 | no | — |
| 86 | D4 | — | 24 | 1 | no | — |
| 87 | D4 | — | 24 | 1 | no | — |
| 88 | C2×C4 | — | 3 | 1 | no | — |
| 89 | C2×C4 | — | 6 | 1 | no | — |
| 90 | C2×C4 | — | 6 | 1 | no | — |
| 91 | C2×C4 | — | 6 | 1 | no | — |
| 92 | C2×C4 | — | 12 | 1 | no | — |
| 93 | C2×C4 | — | 12 | 1 | no | — |
| 94 | C2×C4 | — | 12 | 1 | no | — |
| 95 | C8 | — | 12 | 1 | no | — |
| 96 | C2×C3 | — | 16 | 1 | no | — |
| 97 | C2×C3 | — | 32 | 1 | no | — |
| 98 | D3 | — | 32 | 1 | no | — |
| 99 | C22 | — | 4 | 1 | no | — |
| 100 | C22 | — | 4 | 1 | no | — |
| 101 | C22 | — | 6 | 1 | no | — |
| 102 | C22 | — | 6 | 1 | no | — |
| 103 | C22 | — | 12 | 1 | no | — |
| 104 | C22 | — | 12 | 1 | no | — |
| 105 | C22 | — | 12 | 1 | no | — |
| 106 | C22 | — | 12 | 1 | no | — |
| 107 | C22 | — | 12 | 1 | no | — |
| 108 | C22 | — | 24 | 1 | no | — |
| 109 | C22 | — | 24 | 1 | no | — |
| 110 | C22 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 111 | C4 | — | 6 | 1 | no | — |
| 112 | C4 | — | 12 | 1 | no | — |
| 113 | C4 | — | 12 | 1 | no | — |
| 114 | C4 | — | 24 | 1 | no | — |
| 115 | C4 | — | 24 | 1 | no | — |
| 116 | C3 | — | 16 | 1 | no | Sylow |
| 117 | C2 | — | 3 | 1 | no | — |
| 118 | C2 | — | 4 | 1 | no | — |
| 119 | C2 | — | 8 | 1 | no | — |
| 120 | C2 | — | 12 | 1 | no | — |
| 121 | C2 | — | 24 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle | #120⋊C3 | C22 |
| Frattini | C23 | C2×S4 |
| Fitting | H(3,𝔽4) | D3 |
| socle | C22 | GL(2,ℤ/4) |
| hypocenter | C22⋊A4 | D4 |
| LCS | G ▷ #120⋊C3 ▷ C22⋊A4 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5450 | C2 | semidirect |
| 2. #1020⋊C2 | C2 | semidirect |
| 3. #4625 | C2 | semidirect |
| 9. H(3,𝔽4) | D3 | semidirect |
| 16. C22⋊A4 | D4 | semidirect |
| 40. C24 | #67 | semidirect |
| 41. C24 | S4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5450 | 2 |
| #1020⋊C2 | 2 |
| #4625 | 2 |
| #2199 | 3 |
| GL(2,ℤ/4) | 4 |