Finite Group Databasebeta

← #22571214

Subgroups — #22571214

Order 8576.

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

normal subgroups only

This group has 400 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#16863953C211yesmaximal, characteristic, complemented
2C2×#14524031C211yesmaximal, characteristic
3#16863790C212yesmaximal, complemented
4#16863756C212yesmaximal, complemented
5#16863619C214yesmaximal, complemented
6#16863554C212yesmaximal, complemented
7#16863549C212yesmaximal
8#432×C67C211yesmaximal, characteristic, complemented
9C22×Dic134C2211yescharacteristic
10C22×D268C2211yescharacteristic
11C2×Dic268C2214yescomplemented
12C2×#11761617C2214yes—
13#14524043C2211yescharacteristic, complemented
14#14524043C2211yescharacteristic, complemented
15#14524035—21no—
16#14524031—21no—
17#14524031—21no—
18#14524031C2212yes—
19#14524031C2212yes—
20#14524022—21no—
21#14524022C2212yes—
22C4×Dic134—21no—
23C4×Dic134C2212yescomplemented
24#14524002—21no—
25C2×(C2×D67)⋊C4—21no—
26C2×(C2×D67)⋊C4—21no—
27#14523968C2214yescomplemented
28#14523965C2214yes—
29C2×C42×C67C2211yescharacteristic
30C4⋊C8×C67C2212yescomplemented
31C22⋊C8×C67C2212yes—
32#97×C67C2212yescomplemented
33C23×D67—21no—
34C2×Dic134—21no—
35C2×Dic134—21no—
36C2×Dic134—42no—
37C2×Dic134C2312yes—
38C2×Dic134C2312yes—
39C2×D268—21no—
40C2×D268—42no—
41C2×D268C2312yes—
42C22×Dic67—21no—
43Dic268—44no—
44#11761617—44no—
45C4⋊Dic67—21no—
46C4⋊Dic67—21no—
47C4⋊Dic67—21no—
48C4⋊Dic67C2312yes—
49Dic67⋊C4—22no—
50Dic67⋊C4—22no—
51C22⋊Dic67—22no—
52(C2×D67)⋊C4—22no—
53(C2×D67)⋊C4—22no—
54(C2×D67)⋊C4—22no—
55(C2×D67)⋊C4—22no—
56C4×Dic67—22no—
57C22×C4×C67D411yescharacteristic
58C22×C4×C67D411yescharacteristic
59C22×C4×C67C2311yescharacteristic
60C2×C8×C67C2314yes—
61C42×C67D411yescharacteristic
62C42×C67D411yescharacteristic
63C42×C67C2311yescharacteristic
64C42×C67C2311yescharacteristic
65C23×C67C2×D411yescharacteristic
66C22×D67—22no—
67C22×D67—42no—
68C22×D67—42no—
69C22×D67—42no—
70C2×C4×C67—22no—
71C2×C4×C67—22no—
72C2×C4×C67—41no—
73C2×C4×C67C2×D411yescharacteristic
74C2×C4×C67C2×D411yescharacteristic
75C2×C4×C67C2×D411yescharacteristic
76C2×C4×C67C2×D411yescharacteristic
77C2×C4×C67C2×D411yescharacteristic
78C2×C4×C67C2411yescharacteristic
79C2×Dic67—22no—
80C2×Dic67—22no—
81C2×Dic67—22no—
82C2×Dic67—42no—
83D268—24no—
84D268—42no—
85Dic134—24no—
86Dic134—24no—
87Dic134—42no—
88Dic134—42no—
89C8×C67—24no—
90C22×C67—21no—
91C22×C67—21no—
92C22×C67#13712yes—
93C22×C67C22×D411yescharacteristic
94C2×D67—42no—
95C2×D67—42no—
96C2×D67—42no—
97C2×D67—84no—
98C4×C67—21no—
99C4×C67—21no—
100C4×C67—21no—
101C4×C67—21no—
102C4×C67—21no—
103C4×C67H(5,𝔽2)12yes—
104Dic67—42no—
105Dic67—42no—
106Dic67—42no—
107C2×C67—22no—
108C2×C67#53511yescharacteristic
109C2×C67C2×#13711yescharacteristic
110C2×C67#57511yescharacteristic
111D67—82no—
112#3215—671nomaximal, Sylow
113C67#321511yescharacteristic, complemented, Sylow
114#533—671no—
115C2×#124—671no—
116#497—672no—
117#489—672no—
118#464—674no—
119#451—672no—
120#450—672no—
121#432D6711yescharacteristic, complemented
122C22×Q8—671no—
123C22×D4—671no—
124C2×Q16—674no—
125C2×SD16—674no—
126C4⋊Q8—671no—
127C4⋊Q8—671no—
128#125—1341no—
129#124—672no—
130#124—672no—
131#124—1341no—
132#124—1341no—
133C22⋊Q8—672no—
134C22⋊Q8—1341no—
135C4×Q8—672no—
136C4×Q8—1341no—
137#117—1341no—
138C2×C22⋊C4—1341no—
139C2×C22⋊C4—1341no—
140C2×C42C2×D6711yescharacteristic
141C4⋊C8C2×D6712yescomplemented
142#103—674no—
143#102—674no—
144C22⋊C8C2×D6712yes—
145#97C2×D6712yescomplemented
146C24—1341no—
147C2×Q8—2682no—
148C2×Q8—672no—
149C2×Q8—672no—
150C2×Q8—1341no—
151C2×Q8—1341no—
152C2×D4—2682no—
153C2×D4—672no—
154C2×D4—1341no—
155C22×C4—1341no—
156C22×C4D26811yescharacteristic
157C22×C4D26811yescharacteristic
158C22×C4C22×D6711yescharacteristic
159Q16—2684no—
160SD16—2684no—
161C2×C8C22×D6714yes—
162C4⋊C4—672no—
163C4⋊C4—1341no—
164C4⋊C4—1341no—
165C4⋊C4—1342no—
166C4⋊C4—1342no—
167C4⋊C4—1341no—
168C22⋊C4—1342no—
169C22⋊C4—1342no—
170C22⋊C4—1342no—
171C22⋊C4—1342no—
172C22⋊C4—1342no—
173C42—1342no—
174C42D26811yescharacteristic
175C42D26811yescharacteristic
176C42C22×D6711yescharacteristic
177C42C22×D6711yescharacteristic
178C23—2682no—
179C23—2682no—
180C23—2682no—
181C23—1342no—
182C23C2×D26811yescharacteristic
183Q8—2682no—
184Q8—2682no—
185Q8—1344no—
186Q8—1344no—
187D4—2682no—
188D4—1344no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—2682no—
193C2×C4—1342no—
194C2×C4—1342no—
195C2×C4—1342no—
196C2×C4C2×D26811yescharacteristic
197C2×C4C2×D26811yescharacteristic
198C2×C4C2×D26811yescharacteristic
199C2×C4C2×D26811yescharacteristic
200C2×C4C2×D26811yescharacteristic
201C2×C4C23×D6711yescharacteristic
202C8—24no—
203C22—21no—
204C22—21no—
205C22—2682no—
206C22—2682no—
207C22—2682no—
208C22—5364no—
209C22#1452407612yes—
210C22C22×D26811yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—21no—
215C4—21no—
216C4—2682no—
217C4—2682no—
218C4—2682no—
219C4#1452408912yes—
220C2—22no—
221C2—5362no—
222C2#1686396511yescharacteristic
223C2C2×#1452407611yescharacteristic
224C2#1686419611yescharacteristic
subgroupG/H
centerC22C22×D268
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#432×C67C2
socleC22×C67C22×D4
hypercenter#432D67
hypocenterC67#3215
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. #16863953C2semidirect
3. #16863790C2semidirect
4. #16863756C2semidirect
5. #16863619C2semidirect
6. #16863554C2semidirect
8. #432×C67C2semidirect
11. C2×Dic268C22semidirect
13. #14524043C22semidirect
14. #14524043C22semidirect
23. C4×Dic134C22semidirect
27. #14523968C22semidirect
30. C4⋊C8×C67C22semidirect
32. #97×C67C22semidirect
113. C67#3215semidirect
121. #432D67semidirect
141. C4⋊C8C2×D67semidirect
145. #97C2×D67semidirect