Finite Group Databasebeta

← #22570452

Subgroups — #22570452

Order 8576.

418 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 125 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#16864001C211yesmaximal, characteristic, complemented
2#16863720C212yesmaximal, complemented
3#16863696C212yesmaximal, complemented
4#16863591C212yesmaximal, complemented
5#16863530C212yesmaximal, complemented
6#16863506C212yesmaximal, complemented
7C4×#11761613C212yesmaximal, complemented
8#16863457C211yesmaximal, characteristic, complemented
9#518×C67C211yesmaximal, characteristic, complemented
10C2×#11761613C2212yes—
11C2×#11761613C2212yes—
12#14524040C2211yescharacteristic, complemented
13#14524035C2211yescharacteristic, complemented
14#14524028—22no—
15#14524017C2212yes—
16#14524016—22no—
17C4×D268C2212yescomplemented
18#14523976C2212yescomplemented
19#14523974C2212yescomplemented
20C4⋊(C67⋊C8)C2212yescomplemented
21#14523964C2212yes—
22#14523964C2212yes—
23#14523963C2212yescomplemented
24#14523963C2212yescomplemented
25C22⋊(C67⋊C8)C2212yes—
26#14523954C2211yescharacteristic, complemented
27C4×C67⋊C8C2211yescharacteristic, complemented
28C22×D4×C67C2211yescharacteristic
29C4×D4×C67—22no—
30C4×D4×C67C2212yes—
31C4×D4×C67C2212yes—
32#117×C67C2211yescharacteristic
33#117×C67C2211yescharacteristic
34C2×C22⋊C4×C67—22no—
35C2×D268—22no—
36C2×D268C2312yes—
37C2×C22⋊D67—22no—
38C2×C4×D67—22no—
39#11761613—24no—
40#11761613—24no—
41C2×C67⋊C8C2312yes—
42C2×C67⋊C8C2312yes—
43C4⋊Dic67C2312yes—
44Dic67⋊C4—22no—
45(C2×D67)⋊C4—22no—
46(C2×D67)⋊C4—22no—
47C24×C67—21no—
48C2×D4×C67—24no—
49C2×D4×C67—21no—
50C2×D4×C67C2312yes—
51C22×C4×C67—22no—
52C22×C4×C67—22no—
53C22×C4×C67C2311yescharacteristic
54C4⋊C4×C67D412yes—
55C4⋊C4×C67C2312yes—
56C22⋊C4×C67—21no—
57C22⋊C4×C67—22no—
58C22⋊C4×C67—22no—
59C22⋊C4×C67D412yes—
60C42×C67—21no—
61C42×C67C2311yescharacteristic
62C42×C67C2311yescharacteristic
63C23×C67—24no—
64C23×C67—22no—
65C23×C67—82no—
66C23×C67C2×D411yescharacteristic
67C22×D67—22no—
68D4×C67—24no—
69D4×C67—22no—
70D4×C67#4114yes—
71C2×C4×C67—22no—
72C2×C4×C67—22no—
73C2×C4×C67—82no—
74C2×C4×C67—82no—
75C2×C4×C67C2×D411yescharacteristic
76C2×C4×C67C2×D412yes—
77C2×C4×C67C2×D412yes—
78C2×C4×C67C2411yescharacteristic
79C22⋊D67—82no—
80C2×Dic67—22no—
81D268—22no—
82D268—22no—
83D268—82no—
84C4×D67—82no—
85C67⋊C8—22no—
86C67⋊C8—22no—
87C22×C67—21no—
88C22×C67—24no—
89C22×C67—22no—
90C22×C67—42no—
91C22×C67—42no—
92C22×C67—84no—
93C22×C67C22×D411yescharacteristic
94C2×D67—42no—
95C2×D67—42no—
96C2×D67—42no—
97C4×C67—21no—
98C4×C67—22no—
99C4×C67—22no—
100C4×C67—42no—
101C4×C67C2×#4112yes—
102Dic67—42no—
103C2×C67—41no—
104C2×C67—44no—
105C2×C67#52511yescharacteristic
106C2×C67#57612yes—
107D67—82no—
108#3122—671nomaximal, Sylow
109C67#312211yescharacteristic, complemented, Sylow
110#540—671no—
111#518D6711yescharacteristic, complemented
112#482—672no—
113#478—672no—
114#459—672no—
115#447—672no—
116#442—672no—
117C4×D8—672no—
118#433—671no—
119C22×D4C2×D6711yescharacteristic
120C2×D8—672no—
121C2×D8—672no—
122#127—671no—
123#125—671no—
124#123—1342no—
125#121—672no—
126#121—1342no—
127C4×D4—22no—
128C4×D4—672no—
129C4×D4C2×D6712yes—
130C4×D4C2×D6712yes—
131#117C2×D6711yescharacteristic
132#117C2×D6711yescharacteristic
133C2×C22⋊C4—22no—
134#107—672no—
135#106—672no—
136C4⋊C8—672no—
137#102—672no—
138#102—672no—
139#102—672no—
140#102—672no—
141C22⋊C8—672no—
142#97—671no—
143C4×C8—671no—
144C24—21no—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—672no—
148C2×D4—1342no—
149C2×D4—1342no—
150C2×D4C22×D6712yes—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—1342no—
154C22×C4C22×D6711yescharacteristic
155D8—1344no—
156D8—1344no—
157C2×C8—672no—
158C2×C8—672no—
159C4⋊C4—672no—
160C4⋊C4—1342no—
161C4⋊C4C22⋊D6712yes—
162C4⋊C4C22×D6712yes—
163C22⋊C4—21no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—1342no—
167C22⋊C4—1342no—
168C22⋊C4C22⋊D6712yes—
169C42—21no—
170C42C22×D6711yescharacteristic
171C42C22×D6711yescharacteristic
172C23—24no—
173C23—22no—
174C23—82no—
175C23—1342no—
176C23C2×C22⋊D6711yescharacteristic
177D4—24no—
178D4—22no—
179D4—5362no—
180D4—5362no—
181D4—1342no—
182D4—1342no—
183D4#1176162814yes—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—5362no—
189C2×C4—1342no—
190C2×C4C2×C22⋊D6711yescharacteristic
191C2×C4C2×C22⋊D6712yes—
192C2×C4C2×C22⋊D6712yes—
193C2×C4C23×D6711yescharacteristic
194C8—1342no—
195C8—1342no—
196C22—21no—
197C22—24no—
198C22—22no—
199C22—42no—
200C22—42no—
201C22—84no—
202C22—2682no—
203C22—2682no—
204C22—2682no—
205C22C22×C22⋊D6711yescharacteristic
206C4—21no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—2682no—
211C4C2×#1176162812yes—
212C2—41no—
213C2—44no—
214C2—5362no—
215C2#1686390711yescharacteristic
216C2#1686419912yes—
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#518×C67C2
socleC22×C67C22×D4
hypercenter#518D67
hypocenterC67#3122
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #16864001C2semidirect
2. #16863720C2semidirect
3. #16863696C2semidirect
4. #16863591C2semidirect
5. #16863530C2semidirect
6. #16863506C2semidirect
7. C4×#11761613C2semidirect
8. #16863457C2semidirect
9. #518×C67C2semidirect
12. #14524040C22semidirect
13. #14524035C22semidirect
17. C4×D268C22semidirect
18. #14523976C22semidirect
19. #14523974C22semidirect
20. C4⋊(C67⋊C8)C22semidirect
23. #14523963C22semidirect
24. #14523963C22semidirect
26. #14523954C22semidirect
27. C4×C67⋊C8C22semidirect
109. C67#3122semidirect
111. #518D67semidirect