Finite Group Databasebeta

← #22562371

Subgroups — #22562371

Order 8576.

124 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16863797C211yesmaximal, characteristic
2Q16×Dic67C211yesmaximal, characteristic
3#16863224C211yesmaximal, characteristic
4#16863221C211yesmaximal, characteristic
5Dic67⋊C16C211yesmaximal, characteristic
6#16863200C211yesmaximal, characteristic
7#358×C67C211yesmaximal, characteristic
8#14524034—21no—
9Q8×Dic67—21no—
10C2×C67⋊C16C2211yescharacteristic
11#14523977C2211yescharacteristic
12#14523977C2211yescharacteristic
13#14523974—21no—
14#14523967—21no—
15C8×Dic67C2211yescharacteristic
16C2×Q16×C67C2211yescharacteristic
17C2×C16×C67C2211yescharacteristic
18#107×C67C2211yescharacteristic
19C2×C67⋊C8D411yescharacteristic
20C4⋊Dic67—21no—
21C4⋊Dic67—21no—
22C4⋊Dic67—41no—
23C4⋊Dic67—41no—
24Dic67⋊C4—41no—
25C4×Dic67—41no—
26C4×Dic67D411yescharacteristic
27C67⋊C16—21no—
28C2×Q8×C67—21no—
29Q16×C67—21no—
30Q16×C67Q812yes—
31C2×C8×C67C2311yescharacteristic
32C4⋊C4×C67—21no—
33C16×C67—21no—
34Q8×C67—22no—
35Q8×C67—41no—
36C2×C4×C67—41no—
37C2×C4×C67—41no—
38C2×C4×C67C2×D411yescharacteristic
39C2×Dic67—41no—
40C2×Dic67—41no—
41C2×Dic67D811yescharacteristic
42C2×Dic67D811yescharacteristic
43C8×C67C2×Q811yescharacteristic
44C8×C67#4111yescharacteristic
45C67⋊C8—21no—
46C22×C67C2×D811yescharacteristic
47C4×C67—42no—
48C4×C67—81no—
49C4×C67C22⋊Q811yescharacteristic
50C4×C67#13711yescharacteristic
51Dic67—21no—
52Dic67—21no—
53Dic67—81no—
54Dic67—81no—
55C2×C67#47411yescharacteristic
56C2×C67#50811yescharacteristic
57C2×C67#51011yescharacteristic
58#2229—671nomaximal, Sylow
59C67#222911yescharacteristic, complemented, Sylow
60#499—671no—
61C4×Q16—671no—
62#367—671no—
63#366—671no—
64C4⋊C16—671no—
65#358—671no—
66#358D6711yescharacteristic
67C2×Q16C2×D6711yescharacteristic
68#125—1341no—
69C4×Q8—1341no—
70C2×C16—671no—
71C2×C16C2×D6711yescharacteristic
72#107—671no—
73#107—671no—
74#107C2×D6711yescharacteristic
75#106—1341no—
76#103—1341no—
77C4×C8—671no—
78C2×Q8—21no—
79Q16—21no—
80Q16Dic13412yes—
81C2×C8—671no—
82C2×C8C22×D6711yescharacteristic
83C4⋊C4—21no—
84C4⋊C4—2681no—
85C4⋊C4—2681no—
86C4⋊C4—2681no—
87C4⋊C4—1341no—
88C4⋊C4—1341no—
89C42—2681no—
90C42—671no—
91C16—21no—
92C16—1341no—
93Q8—22no—
94Q8—41no—
95C2×C4—41no—
96C2×C4—41no—
97C2×C4—2681no—
98C2×C4—2681no—
99C2×C4—671no—
100C2×C4—671no—
101C2×C4D4×D6711yescharacteristic
102C8—1341no—
103C8C2×Dic13411yescharacteristic
104C8#1176162711yescharacteristic
105C22D8×D6711yescharacteristic
106C4—42no—
107C4—81no—
108C4—5361no—
109C4—5361no—
110C4—1341no—
111C4—1341no—
112C4#1452402011yescharacteristic
113C4#1452407211yescharacteristic
114C2#1686367411yescharacteristic
115C2#1686382711yescharacteristic
116C2#1686383811yescharacteristic
subgroupG/H
centerC22D8×D67
commutatorC2×C8×C67C23
FrattiniC2×C8C22×D67
Fitting#358×C67C2
socleC22×C67C2×D8
hypercenter#358D67
hypocenterC67#2229
LCSG ▷ C2×C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ #358

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
59. C67#2229semidirect