Finite Group Databasebeta

← #22570309

Subgroups — #22570309

Order 8576.

444 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 129 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×C4×D268C211yesmaximal, characteristic
3#16863732C212yesmaximal, complemented
4#16863677C212yesmaximal, complemented
5#16863602C212yesmaximal, complemented
6#16863549C212yesmaximal
7#16863498C214yesmaximal, complemented
8#432×C67C211yesmaximal, characteristic, complemented
9C22×D268C2211yescharacteristic
10C22×C4×D67—21no—
11C2×#11761617C2214yes—
12#14524043C2211yescharacteristic, complemented
13#14524043C2211yescharacteristic, complemented
14#14524035—21no—
15#14524022—21no—
16#14524022C2212yes—
17C4×Dic134—21no—
18C4×Dic134C2212yescomplemented
19C4×D268—21no—
20C4×D268—21no—
21C4×D268C2212yes—
22C4×D268C2212yes—
23#14524002—21no—
24C2×C4⋊Dic67C2211yescharacteristic
25C2×(C2×D67)⋊C4—21no—
26#14523977C2214yescomplemented
27#14523968C2214yescomplemented
28#14523965C2214yes—
29C2×C42×C67C2211yescharacteristic
30C4⋊C8×C67C2212yescomplemented
31C22⋊C8×C67C2212yes—
32#97×C67C2212yescomplemented
33C23×D67—21no—
34C2×Dic134—21no—
35C2×Dic134C2312yes—
36C2×D268—21no—
37C2×D268—24no—
38C2×D268C2312yes—
39C22×Dic67—21no—
40C2×C4×D67—22no—
41C2×C4×D67—22no—
42C2×C4×D67—44no—
43#11761617—28no—
44C4⋊Dic67—21no—
45C4⋊Dic67—21no—
46C4⋊Dic67—21no—
47C4⋊Dic67—21no—
48C4⋊Dic67C2312yes—
49C4⋊Dic67C2312yes—
50Dic67⋊C4—22no—
51Dic67⋊C4—22no—
52C22⋊Dic67—22no—
53(C2×D67)⋊C4—22no—
54(C2×D67)⋊C4—22no—
55C4×Dic67—22no—
56C22×C4×C67D411yescharacteristic
57C22×C4×C67D411yescharacteristic
58C22×C4×C67C2311yescharacteristic
59C2×C8×C67C2314yes—
60C42×C67D411yescharacteristic
61C42×C67D411yescharacteristic
62C42×C67C2311yescharacteristic
63C42×C67C2311yescharacteristic
64C23×C67C2×D411yescharacteristic
65C22×D67—24no—
66C22×D67—22no—
67C22×D67—42no—
68C22×D67—42no—
69C2×C4×C67—22no—
70C2×C4×C67—22no—
71C2×C4×C67—22no—
72C2×C4×C67C2×D411yescharacteristic
73C2×C4×C67C2×D411yescharacteristic
74C2×C4×C67C2×D411yescharacteristic
75C2×C4×C67C2×D411yescharacteristic
76C2×C4×C67C2×D411yescharacteristic
77C2×C4×C67C2411yescharacteristic
78C2×Dic67—22no—
79C2×Dic67—22no—
80C2×Dic67—22no—
81C2×Dic67—42no—
82D268—24no—
83D268—22no—
84D268#4114yes—
85C4×D67—44no—
86C4×D67—44no—
87Dic134—24no—
88Dic134—42no—
89C8×C67—24no—
90C22×C67—21no—
91C22×C67—21no—
92C22×C67#13712yes—
93C22×C67C22×D411yescharacteristic
94C2×D67—24no—
95C2×D67—48no—
96C2×D67—42no—
97C2×D67—42no—
98C4×C67—21no—
99C4×C67—21no—
100C4×C67—21no—
101C4×C67—21no—
102C4×C67AGL(1,ℤ/8)12yes—
103C4×C67C2×#4112yes—
104Dic67—42no—
105Dic67—42no—
106Dic67—42no—
107C2×C67—22no—
108C2×C67#52511yescharacteristic
109C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
110C2×C67C2×#13711yescharacteristic
111D67—44no—
112#3110—671nomaximal, Sylow
113C67#311011yescharacteristic, complemented, Sylow
114#533—671no—
115C2×C4×D4—671no—
116#484—672no—
117#474—672no—
118#461—672no—
119#450—672no—
120#440—674no—
121#432D6711yescharacteristic, complemented
122C22×D4—671no—
123C23×C4—1341no—
124C2×SD16—674no—
125C4⋊Q8—671no—
126C4⋊Q8—671no—
127#125—1341no—
128C22⋊Q8—672no—
129C22⋊Q8—1341no—
130C4×Q8—672no—
131C4×Q8—1341no—
132C4×D4—672no—
133C4×D4—672no—
134C4×D4—1341no—
135C4×D4—1341no—
136#117—1341no—
137C2×C4⋊C4—671no—
138C2×C22⋊C4—1341no—
139C2×C42C2×D6711yescharacteristic
140#107—674no—
141C4⋊C8C2×D6712yescomplemented
142#103—674no—
143#102—674no—
144C22⋊C8C2×D6712yes—
145#97C2×D6712yescomplemented
146C24—1341no—
147C2×Q8—672no—
148C2×Q8—1341no—
149C2×D4—672no—
150C2×D4—1344no—
151C2×D4—1341no—
152C22×C4—2684no—
153C22×C4—1341no—
154C22×C4—1342no—
155C22×C4—1342no—
156C22×C4D26811yescharacteristic
157C22×C4D26811yescharacteristic
158C22×C4C22×D6711yescharacteristic
159SD16—1348no—
160C2×C8C22×D6714yes—
161C4⋊C4—672no—
162C4⋊C4—672no—
163C4⋊C4—1341no—
164C4⋊C4—1341no—
165C4⋊C4—1341no—
166C4⋊C4—1342no—
167C4⋊C4—1342no—
168C4⋊C4—1341no—
169C22⋊C4—1342no—
170C22⋊C4—1342no—
171C22⋊C4—1342no—
172C42—1342no—
173C42D26811yescharacteristic
174C42D26811yescharacteristic
175C42C22×D6711yescharacteristic
176C42C22×D6711yescharacteristic
177C23—2682no—
178C23—2682no—
179C23—1344no—
180C23—1342no—
181C23C2×D26811yescharacteristic
182Q8—2682no—
183Q8—1344no—
184D4—674no—
185D4—1344no—
186D4—1342no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—2684no—
191C2×C4—2682no—
192C2×C4—2684no—
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—2688no—
206C22—2682no—
207C22—2682no—
208C22—1344no—
209C22#1452407612yes—
210C22C22×D26811yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—21no—
215C4—2682no—
216C4—2682no—
217C4—2682no—
218C4#1452407112yes—
219C4#41×D6712yes—
220C2—22no—
221C2—2684no—
222C2#1686390811yescharacteristic
223C2C2×#1452407111yescharacteristic
224C2C2×#1452407611yescharacteristic
subgroupG/H
centerC22C22×D268
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#432×C67C2
socleC22×C67C22×D4
hypercenter#432D67
hypocenterC67#3110
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. #16863732C2semidirect
4. #16863677C2semidirect
5. #16863602C2semidirect
7. #16863498C2semidirect
8. #432×C67C2semidirect
12. #14524043C22semidirect
13. #14524043C22semidirect
18. C4×Dic134C22semidirect
26. #14523977C22semidirect
27. #14523968C22semidirect
30. C4⋊C8×C67C22semidirect
32. #97×C67C22semidirect
113. C67#3110semidirect
121. #432D67semidirect
141. C4⋊C8C2×D67semidirect
145. #97C2×D67semidirect