Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
248 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 45 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 | C7×H(3,𝔽4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #89834 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #89337 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #6551 | C22 | 1 | 1 | yes | characteristic, complemented |
| 5 | C7×#124 | — | 2 | 3 | no | — |
| 6 | C7×#124 | C22 | 1 | 3 | yes | — |
| 7 | C7×#120 | — | 2 | 3 | no | — |
| 8 | #6495 | — | 2 | 3 | no | — |
| 9 | #6410 | C22 | 1 | 3 | yes | complemented |
| 10 | #6379 | — | 2 | 3 | no | — |
| 11 | #2202 | — | 7 | 1 | no | maximal, Sylow |
| 12 | C24×C7 | — | 2 | 1 | no | — |
| 13 | C2×C7×Q8 | — | 2 | 1 | no | — |
| 14 | C2×C7×Q8 | C23 | 1 | 1 | yes | characteristic |
| 15 | C2×C7×D4 | — | 2 | 3 | no | — |
| 16 | C2×C7×D4 | D4 | 1 | 3 | yes | — |
| 17 | #1161 | — | 4 | 1 | no | — |
| 18 | Q8×D7 | — | 4 | 1 | no | — |
| 19 | #1157 | — | 2 | 3 | no | — |
| 20 | C7×C22⋊C4 | — | 2 | 3 | no | — |
| 21 | C7×C22⋊C4 | — | 2 | 6 | no | — |
| 22 | C42×C7 | D4 | 1 | 3 | yes | — |
| 23 | #1142 | — | 4 | 3 | no | — |
| 24 | #1141 | — | 4 | 3 | no | — |
| 25 | #1136 | — | 2 | 3 | no | — |
| 26 | H(3,𝔽4) | D7 | 1 | 1 | yes | characteristic, complemented |
| 27 | #454 | — | 7 | 3 | no | — |
| 28 | #355 | — | 7 | 3 | no | — |
| 29 | C23×C7 | — | 2 | 3 | no | — |
| 30 | C23×C7 | — | 8 | 2 | no | — |
| 31 | C23×C7 | — | 8 | 1 | no | — |
| 32 | C7×Q8 | — | 4 | 1 | no | — |
| 33 | C7×Q8 | — | 8 | 1 | no | — |
| 34 | C7×D4 | — | 4 | 3 | no | — |
| 35 | C7×D4 | — | 8 | 3 | no | — |
| 36 | C2×C4×C7 | — | 2 | 3 | no | — |
| 37 | C2×C4×C7 | C2×D4 | 1 | 3 | yes | — |
| 38 | C22⋊D7 | — | 4 | 1 | no | — |
| 39 | D28 | — | 2 | 3 | no | — |
| 40 | C4×D7 | — | 4 | 3 | no | — |
| 41 | Dic14 | — | 2 | 3 | no | — |
| 42 | C7⋊C8 | — | 4 | 3 | no | — |
| 43 | 2−1+4 | — | 7 | 1 | no | — |
| 44 | AGL(1,ℤ/8) | — | 14 | 3 | no | — |
| 45 | #124 | — | 2 | 3 | no | — |
| 46 | #124 | C2×D7 | 1 | 3 | yes | — |
| 47 | #120 | — | 2 | 3 | no | — |
| 48 | #104 | — | 14 | 3 | no | — |
| 49 | #101 | — | 7 | 3 | no | — |
| 50 | C22×C7 | — | 4 | 6 | no | — |
| 51 | C22×C7 | — | 4 | 3 | no | — |
| 52 | C22×C7 | — | 8 | 4 | no | — |
| 53 | C22×C7 | #120 | 1 | 1 | yes | characteristic |
| 54 | C2×D7 | — | 4 | 1 | no | — |
| 55 | C4×C7 | — | 2 | 3 | no | — |
| 56 | C4×C7 | — | 4 | 3 | no | — |
| 57 | Dic7 | — | 4 | 1 | no | — |
| 58 | C24 | — | 2 | 1 | no | — |
| 59 | #41 | — | 14 | 3 | no | — |
| 60 | #41 | — | 28 | 1 | no | — |
| 61 | C2×Q8 | — | 2 | 1 | no | — |
| 62 | C2×Q8 | — | 28 | 1 | no | — |
| 63 | C2×Q8 | C22×D7 | 1 | 1 | yes | characteristic |
| 64 | C2×D4 | — | 2 | 3 | no | — |
| 65 | C2×D4 | C22⋊D7 | 1 | 3 | yes | — |
| 66 | SD16 | — | 28 | 3 | no | — |
| 67 | D8 | — | 28 | 3 | no | — |
| 68 | M16 | — | 14 | 3 | no | — |
| 69 | C22⋊C4 | — | 2 | 3 | no | — |
| 70 | C22⋊C4 | — | 2 | 6 | no | — |
| 71 | C42 | C22⋊D7 | 1 | 3 | yes | — |
| 72 | C2×C7 | — | 2 | 1 | no | — |
| 73 | C2×C7 | — | 8 | 3 | no | — |
| 74 | C2×C7 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 75 | D7 | — | 8 | 1 | no | — |
| 76 | C23 | — | 2 | 3 | no | — |
| 77 | C23 | — | 8 | 2 | no | — |
| 78 | C23 | — | 8 | 1 | no | — |
| 79 | Q8 | — | 4 | 1 | no | — |
| 80 | Q8 | — | 8 | 1 | no | — |
| 81 | Q8 | — | 14 | 3 | no | — |
| 82 | D4 | — | 4 | 3 | no | — |
| 83 | D4 | — | 8 | 3 | no | — |
| 84 | D4 | — | 14 | 3 | no | — |
| 85 | D4 | — | 28 | 1 | no | — |
| 86 | C2×C4 | — | 2 | 3 | no | — |
| 87 | C2×C4 | — | 28 | 3 | no | — |
| 88 | C2×C4 | C2×C22⋊D7 | 1 | 3 | yes | — |
| 89 | C8 | — | 28 | 3 | no | — |
| 90 | C7 | #2202 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 91 | C22 | — | 4 | 6 | no | — |
| 92 | C22 | — | 4 | 3 | no | — |
| 93 | C22 | — | 8 | 4 | no | — |
| 94 | C22 | — | 28 | 1 | no | — |
| 95 | C22 | #6516 | 1 | 1 | yes | characteristic |
| 96 | C4 | — | 2 | 3 | no | — |
| 97 | C4 | — | 4 | 3 | no | — |
| 98 | C4 | — | 28 | 1 | no | — |
| 99 | C2 | — | 2 | 1 | no | — |
| 100 | C2 | — | 8 | 3 | no | — |
| 101 | C2 | — | 56 | 1 | no | — |
| 102 | C2 | #90016 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×H(3,𝔽4) | C2 | semidirect |
| 2. #89834 | C2 | semidirect |
| 3. #89337 | C2 | semidirect |
| 4. #6551 | C22 | semidirect |
| 9. #6410 | C22 | semidirect |
| 26. H(3,𝔽4) | D7 | semidirect |
| 90. C7 | #2202 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×H(3,𝔽4) | 2 |
| #89834 | 2 |
| #89337 | 2 |
| #2202 | 7 |