Finite Group Databasebeta

← #22559464

Subgroups — #22559464

Order 8576.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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#16864119C211yesmaximal, characteristic
2#16863799C211yesmaximal, characteristic, complemented
3#16863598C211yesmaximal, characteristic
4#16863141C211yesmaximal, characteristic, complemented
5Q8⋊(C67⋊C8)C211yesmaximal, characteristic, complemented
6#16863123C211yesmaximal, characteristic
7#475×C67C211yesmaximal, characteristic, complemented
8C2×#11761615—21no—
9#14524043C2211yescharacteristic
10#14524038—22no—
11#14524031—21no—
12#14524019—21no—
13C4×Dic134—21no—
14C4×D268C2211yescharacteristic
15#14524001—21no—
16#14523976—22no—
17C4⋊(C67⋊C8)C2211yescharacteristic
18#14523967—21no—
19C4×C67⋊C8C2211yescharacteristic
20C4⋊Q8×C67C2211yescharacteristic
21C4×Q8×C67C2211yescharacteristic
22#106×C67—21no—
23C4⋊C8×C67C2211yescharacteristic
24#103×C67—21no—
25C2×Dic134—21no—
26C2×D268D411yescharacteristic
27C2×C4×D67—21no—
28#11761615—42no—
29C2×C67⋊C8—21no—
30C2×C67⋊C8—21no—
31C4⋊Dic67—21no—
32C4⋊Dic67D411yescharacteristic
33C4⋊Dic67D412yes—
34Dic67⋊C4—22no—
35(C2×D67)⋊C4—22no—
36(C2×D67)⋊C4—22no—
37(C2×D67)⋊C4—21no—
38C4×Dic67—22no—
39C2×Q8×C67—21no—
40C2×Q8×C67D411yescharacteristic
41C2×C8×C67—21no—
42C4⋊C4×C67—21no—
43C4⋊C4×C67—21no—
44C4⋊C4×C67—21no—
45C4⋊C4×C67D411yescharacteristic
46C42×C67—21no—
47C42×C67C2311yescharacteristic
48C22×D67—21no—
49Q8×C67—21no—
50Q8×C67—21no—
51Q8×C67—41no—
52Q8×C67—41no—
53C2×C4×C67—21no—
54C2×C4×C67—21no—
55C2×C4×C67—41no—
56C2×C4×C67C2×D411yescharacteristic
57C2×C4×C67C2×D411yescharacteristic
58C2×C4×C67C2×D411yescharacteristic
59C2×Dic67—21no—
60C2×Dic67—22no—
61D268—21no—
62D268—21no—
63C4×D67—41no—
64C4×D67—41no—
65Dic134—41no—
66Dic134—41no—
67C8×C67—41no—
68C67⋊C8—22no—
69C67⋊C8—41no—
70C22×C67#12011yescharacteristic
71C2×D67—41no—
72C2×D67—41no—
73C2×D67—41no—
74C4×C67—21no—
75C4×C67—41no—
76C4×C67—41no—
77C4×C67—81no—
78C4×C67#13511yescharacteristic
79C4×C67#13511yescharacteristic
80C4×C67AGL(1,ℤ/8)11yescharacteristic
81C4×C67#13711yescharacteristic
82Dic67—42no—
83Dic67—41no—
84C2×C67#44911yescharacteristic
85C2×C67#45211yescharacteristic
86C2×C67#45611yescharacteristic
87D67—81no—
88#1650—671nomaximal, Sylow
89C67#165011yescharacteristic, complemented, Sylow
90#556—671no—
91#500—671no—
92#475D6711yescharacteristic, complemented
93#461—671no—
94#332—671no—
95Q8⋊C8—671no—
96D4⋊C8—671no—
97C2×SD16—1341no—
98C4⋊Q8—671no—
99C4⋊Q8C2×D6711yescharacteristic
100#126—1342no—
101#124—1341no—
102C22⋊Q8—1341no—
103C4×Q8—1341no—
104C4×Q8C2×D6711yescharacteristic
105C4×D4—671no—
106#117—1341no—
107#107—1342no—
108#106—21no—
109C4⋊C8—671no—
110C4⋊C8C2×D6711yescharacteristic
111#103—21no—
112#103—1341no—
113C4×C8—671no—
114C2×Q8—21no—
115C2×Q8—1341no—
116C2×Q8C22⋊D6711yescharacteristic
117C2×D4—671no—
118C22×C4—1341no—
119SD16—2682no—
120C2×C8—21no—
121C2×C8—1341no—
122C2×C8—1341no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—671no—
127C4⋊C4—672no—
128C4⋊C4—1342no—
129C4⋊C4—1341no—
130C4⋊C4C22⋊D6711yescharacteristic
131C22⋊C4—1342no—
132C22⋊C4—1342no—
133C22⋊C4—1341no—
134C42—21no—
135C42—1342no—
136C42C22×D6711yescharacteristic
137C23—1341no—
138Q8—21no—
139Q8—21no—
140Q8—41no—
141Q8—41no—
142Q8—2681no—
143Q8—2681no—
144D4—1341no—
145D4—1341no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—2681no—
150C2×C4—2681no—
151C2×C4—1341no—
152C2×C4—1342no—
153C2×C4D4×D6711yescharacteristic
154C2×C4D4×D6711yescharacteristic
155C2×C4C2×C22⋊D6711yescharacteristic
156C8—41no—
157C8—2681no—
158C8—1342no—
159C22—2681no—
160C22—2681no—
161C22—2681no—
162C22#1452401111yescharacteristic
163C4—21no—
164C4—41no—
165C4—41no—
166C4—81no—
167C4—2682no—
168C4—2681no—
169C4#1452406311yescharacteristic
170C4#1452406411yescharacteristic
171C4#1452407011yescharacteristic
172C4#1452407411yescharacteristic
173C2—5361no—
174C2#1686354011yescharacteristic
175C2#1686355611yescharacteristic
176C2#1686357711yescharacteristic
subgroupG/H
centerC22#14524011
commutatorC42×C67C23
FrattiniC42C22×D67
Fitting#475×C67C2
socleC22×C67#120
hypercenter#475D67
hypocenterC67#1650
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #16863799C2semidirect
4. #16863141C2semidirect
5. Q8⋊(C67⋊C8)C2semidirect
7. #475×C67C2semidirect
89. C67#1650semidirect
92. #475D67semidirect