Finite Group Databasebeta

← #22570582

Subgroups — #22570582

Order 8576.

484 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 121 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#16863952C211yesmaximal, characteristic, complemented
2#127×D67C211yesmaximal, characteristic
3#16863762C211yesmaximal, characteristic, complemented
4#16863761C211yesmaximal, characteristic, complemented
5#16863760C211yesmaximal, characteristic, complemented
6#16863741C214yesmaximal, complemented
7#16863399C214yesmaximal, complemented
8#16863348C211yesmaximal, characteristic, complemented
9#490×C67C211yesmaximal, characteristic, complemented
10C2×D4×D67—22no—
11C2×D4×D67C2212yes—
12C2×#11761611C2212yes—
13#14524042C2212yescomplemented
14#14524040C2211yescharacteristic
15#14524039—22no—
16#14524039C2212yes—
17#14524035C2211yescharacteristic, complemented
18#14524021—22no—
19#14524008—22no—
20#14524001C2212yes—
21C42×D67C2211yescharacteristic
22#14523965C2214yescomplemented
23#14523964C2214yescomplemented
24#14523963C2214yescomplemented
25#14523954C2211yescharacteristic, complemented
26C8×Dic67C2212yescomplemented
27#127×C67C2211yescharacteristic
28#125×C67C2211yescharacteristic, complemented
29#102×C67C2214yescomplemented
30#97×C67C2211yescharacteristic, complemented
31C23×D67—22no—
32C2×Dic134—22no—
33C2×D268—22no—
34C2×D268C2312yes—
35C2×C22⋊D67—22no—
36C2×C22⋊D67—24no—
37D4×D67—28no—
38D4×D67—48no—
39C2×C4×D67D412yes—
40C2×C4×D67C2311yescharacteristic
41#11761611—24no—
42C2×C67⋊C8C2312yes—
43C4⋊Dic67C2312yes—
44Dic67⋊C4—22no—
45Dic67⋊C4—22no—
46(C2×D67)⋊C4—22no—
47C4×Dic67—22no—
48C4×Dic67D412yes—
49C4×Dic67C2311yescharacteristic
50C2×D4×C67—22no—
51C2×D4×C67C2312yes—
52C2×C8×C67C2312yes—
53C4⋊C4×C67—22no—
54C4⋊C4×C67C2312yes—
55C42×C67C2311yescharacteristic
56C23×C67—22no—
57C22×D67—22no—
58C22×D67—48no—
59C22×D67—44no—
60C22×D67C2×D411yescharacteristic
61D4×C67—24no—
62D4×C67—44no—
63C2×C4×C67—22no—
64C2×C4×C67C2×D412yes—
65C2×C4×C67C2411yescharacteristic
66C22⋊D67—44no—
67C22⋊D67—48no—
68C2×Dic67—22no—
69C2×Dic67C2×D411yescharacteristic
70C2×Dic67C2×D412yes—
71D268—24no—
72D268—44no—
73C4×D67—24no—
74C4×D67#4114yes—
75Dic134—44no—
76C8×C67—22no—
77C67⋊C8—22no—
78C22×C67—42no—
79C22×C67—44no—
80C22×C67C22×D411yescharacteristic
81C2×D67—22no—
82C2×D67—42no—
83C2×D67—44no—
84C2×D67—88no—
85C2×D67#12412yes—
86C4×C67—22no—
87C4×C67—42no—
88C4×C67C2×#4112yes—
89Dic67—42no—
90Dic67#12412yes—
91Dic67AGL(1,ℤ/8)14yes—
92C2×C67—82no—
93C2×C67C2×#12411yescharacteristic
94C2×C67C2×AGL(1,ℤ/8)12yes—
95D67—22no—
96D67—82no—
97#3138—671nomaximal, Sylow
98C67#313811yescharacteristic, complemented, Sylow
99#533—671no—
100C2×#127—671no—
101#490—671no—
102#490—671no—
103#490—671no—
104#490D6711yescharacteristic, complemented
105#486—674no—
106#418—674no—
107C4×M16—671no—
108C22×D4—672no—
109C22×D4—1342no—
110C2×M16—672no—
111C4⋊Q8—672no—
112#127—672no—
113#127—671no—
114#127—1342no—
115#127C2×D6711yescharacteristic
116#125—671no—
117#125C2×D6711yescharacteristic, complemented
118C22⋊Q8—1342no—
119C4×Q8—1342no—
120#117—672no—
121C2×C42—671no—
122#102—674no—
123#102—674no—
124#102—674no—
125#102C2×D6714yescomplemented
126#97—671no—
127#97C2×D6711yescharacteristic, complemented
128C4×C8—672no—
129C24—1342no—
130C2×Q8—1342no—
131C2×D4—22no—
132C2×D4—2688no—
133C2×D4—672no—
134C2×D4—1348no—
135C2×D4—1344no—
136C2×D4—1342no—
137C2×D4—1342no—
138C2×D4C22×D6712yes—
139C22×C4—671no—
140C22×C4—672no—
141M16—1344no—
142C2×C8—672no—
143C2×C8C22×D6712yes—
144C4⋊C4—22no—
145C4⋊C4—672no—
146C4⋊C4—1342no—
147C4⋊C4—1342no—
148C4⋊C4C22×D6712yes—
149C22⋊C4—1342no—
150C42—672no—
151C42—671no—
152C42—1342no—
153C42C22×D6711yescharacteristic
154C23—22no—
155C23—2684no—
156C23—2688no—
157C23—671no—
158C23—1342no—
159Q8—2684no—
160D4—24no—
161D4—44no—
162D4—2684no—
163D4—2688no—
164D4—2684no—
165D4—1344no—
166C2×C4—22no—
167C2×C4—671no—
168C2×C4—672no—
169C2×C4—674no—
170C2×C4—1344no—
171C2×C4—1342no—
172C2×C4D4×D6712yes—
173C2×C4C23×D6711yescharacteristic
174C8—22no—
175C8—1342no—
176C22—42no—
177C22—44no—
178C22—2682no—
179C22—2684no—
180C22—5368no—
181C22—672no—
182C22—1342no—
183C22C2×D4×D6711yescharacteristic
184C4—22no—
185C4—42no—
186C4—2682no—
187C4—672no—
188C4—674no—
189C4#41×D6712yes—
190C2—82no—
191C2—5362no—
192C2—1342no—
193C2#124×D6711yescharacteristic
194C2AGL(1,ℤ/8)×D6712yes—
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#490×C67C2
socleC22×C67C22×D4
hypercenter#490D67
hypocenterC67#3138
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #16863952C2semidirect
3. #16863762C2semidirect
4. #16863761C2semidirect
5. #16863760C2semidirect
6. #16863741C2semidirect
7. #16863399C2semidirect
8. #16863348C2semidirect
9. #490×C67C2semidirect
13. #14524042C22semidirect
17. #14524035C22semidirect
22. #14523965C22semidirect
23. #14523964C22semidirect
24. #14523963C22semidirect
25. #14523954C22semidirect
26. C8×Dic67C22semidirect
28. #125×C67C22semidirect
29. #102×C67C22semidirect
30. #97×C67C22semidirect
98. C67#3138semidirect
104. #490D67semidirect
117. #125C2×D67semidirect
125. #102C2×D67semidirect
127. #97C2×D67semidirect