Finite Group Databasebeta

← #22569449

Subgroups — #22569449

Order 8576.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#11761628C211yesmaximal, characteristic, complemented
2C2×#14524073C211yesmaximal, characteristic, complemented
3C2×#14524068C211yesmaximal, characteristic, complemented
4#16863556C214yesmaximal, complemented
5#16863539C214yesmaximal, complemented
6#16863404C211yesmaximal, characteristic, complemented
7#16863400C211yesmaximal, characteristic, complemented
8#16863360C211yesmaximal, characteristic, complemented
9#525×C67C211yesmaximal, characteristic, complemented
10C2×#11761628—24no—
11C2×#11761628—24no—
12C2×#11761628—22no—
13C2×#11761628C2214yes—
14C22×Dic134C2211yescharacteristic
15C22×D268C2211yescharacteristic
16C22×C22⋊D67—21no—
17C22×C4×D67—21no—
18#14524073—24no—
19#14524068—24no—
20C2×#11761618C2212yescomplemented
21C2×#11761616C2212yescomplemented
22C2×#11761615C2212yescomplemented
23C2×#11761613C2212yescomplemented
24C2×#11761612C2211yescharacteristic
25C2×#11761612C2211yescharacteristic
26#14523966C2214yescomplemented
27#14523963C2214yescomplemented
28C22⋊(C67⋊C8)C2214yescomplemented
29C2×#41×C67C2211yescharacteristic
30C23×C4×C67C2211yescharacteristic
31#123×C67—21no—
32C22⋊Q8×C67C2212yescomplemented
33#121×C67C2212yescomplemented
34#120×C67—21no—
35C4×D4×C67—22no—
36#117×C67C2211yescharacteristic
37C23×D67—21no—
38#11761628—216no—
39#11761628—28no—
40#11761628—28no—
41C2×Dic134—22no—
42C2×Dic134—21no—
43C2×Dic134D414yescomplemented
44C2×Dic134C2312yes—
45C2×D268—22no—
46C2×D268—21no—
47C2×D268D414yescomplemented
48C2×D268C2312yes—
49C2×C22⋊D67—28no—
50C2×C22⋊D67—24no—
51C22×Dic67—21no—
52C2×C4×D67—28no—
53C2×C4×D67—24no—
54#11761618—24no—
55#11761616—24no—
56#11761615—24no—
57#11761613—24no—
58#11761612—22no—
59#11761612—22no—
60C2×C67⋊C8C2312yes—
61C2×C67⋊C8C2312yes—
62C24×C67D411yescharacteristic
63#41×C67—22no—
64#41×C67—22no—
65C2×Q8×C67C2311yescharacteristic
66C2×D4×C67—22no—
67C2×D4×C67—21no—
68C2×D4×C67C2311yescharacteristic
69C22×C4×C67—22no—
70C22×C4×C67—22no—
71C22×C4×C67—21no—
72C22×C4×C67D412yes—
73C22×C4×C67D411yescharacteristic
74C22×C4×C67C2312yes—
75C22×C4×C67C2311yescharacteristic
76C4⋊C4×C67—22no—
77C4⋊C4×C67C2312yes—
78C22⋊C4×C67—21no—
79C22⋊C4×C67—22no—
80C22⋊C4×C67—22no—
81C42×C67—21no—
82C23×C67—22no—
83C23×C67—21no—
84C23×C67—42no—
85C23×C67C2×D412yes—
86C23×C67C2×D411yescharacteristic
87C22×D67—24no—
88C22×D67—22no—
89C22×D67—44no—
90Q8×C67—21no—
91Q8×C67—21no—
92D4×C67—21no—
93D4×C67—21no—
94D4×C67—44no—
95D4×C67—42no—
96C2×C4×C67—22no—
97C2×C4×C67—24no—
98C2×C4×C67—22no—
99C2×C4×C67—22no—
100C2×C4×C67—22no—
101C2×C4×C67—21no—
102C2×C4×C67—42no—
103C2×C4×C67C2×D412yes—
104C2×C4×C67C2×D412yes—
105C2×C4×C67C2×D412yes—
106C2×C4×C67C2×D411yescharacteristic
107C2×C4×C67C2411yescharacteristic
108C22⋊D67—216no—
109C2×Dic67—24no—
110C2×Dic67—22no—
111D268—24no—
112D268—22no—
113D268C2×D414yes—
114C4×D67—216no—
115Dic134—24no—
116Dic134—22no—
117Dic134C2×D414yes—
118C67⋊C8—22no—
119C67⋊C8—22no—
120C22×C67—22no—
121C22×C67—22no—
122C22×C67—22no—
123C22×C67—22no—
124C22×C67—44no—
125C22×C67—41no—
126C22×C67—42no—
127C22×C67#12012yes—
128C22×C67C22×D411yescharacteristic
129C2×D67—24no—
130C2×D67—48no—
131C2×D67—44no—
132C4×C67—22no—
133C4×C67—41no—
134C4×C67—42no—
135C4×C67#12012yes—
136C4×C67C22×D411yescharacteristic
137C4×C67C22×D411yescharacteristic
138Dic67—24no—
139C2×C67—22no—
140C2×C67—42no—
141C2×C67—81no—
142C2×C67C2×#12011yescharacteristic
143C2×C67#57512yes—
144D67—44no—
145#3005—671nomaximal, Sylow
146C67#300511yescharacteristic, complemented, Sylow
147C22×#41—671no—
148C2×#137—671no—
149C2×AGL(1,ℤ/8)—671no—
150#525D6711yescharacteristic, complemented
151#452—674no—
152#449—674no—
153#419—671no—
154#418—671no—
155#407—671no—
156C2×#41—674no—
157C2×#41—1344no—
158C2×#41—1344no—
159C2×#41—1342no—
160C2×#41C2×D6711yescharacteristic
161C22×Q8—671no—
162C22×D4—671no—
163C22×D4—1341no—
164C23×C4—1341no—
165C23×C4C2×D6711yescharacteristic
166#137—1344no—
167AGL(1,ℤ/8)—1344no—
168C2×Q16—672no—
169C2×SD16—672no—
170C2×SD16—672no—
171C2×D8—672no—
172C2×M16—671no—
173C2×M16—671no—
174#123—21no—
175C22⋊Q8C2×D6712yescomplemented
176#121C2×D6712yescomplemented
177#120—21no—
178C4×D4—22no—
179#117C2×D6711yescharacteristic
180#103—674no—
181#102—674no—
182C22⋊C8—674no—
183C24—1341no—
184C24C22⋊D6711yescharacteristic
185#41—22no—
186#41—22no—
187#41—1348no—
188#41—1348no—
189#41—13416no—
190C2×Q8—674no—
191C2×Q8—672no—
192C2×Q8—1342no—
193C2×Q8—1341no—
194C2×Q8C22×D6711yescharacteristic
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—674no—
198C2×D4—672no—
199C2×D4—1348no—
200C2×D4—1344no—
201C2×D4—1342no—
202C2×D4—1341no—
203C2×D4C22×D6711yescharacteristic
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—1348no—
208C22×C4—1341no—
209C22×C4—1344no—
210C22×C4C22⋊D6712yes—
211C22×C4C22⋊D6711yescharacteristic
212C22×C4C22×D6712yes—
213C22×C4C22×D6711yescharacteristic
214Q16—1344no—
215SD16—1344no—
216SD16—1344no—
217D8—1344no—
218M16—1342no—
219M16—1342no—
220C2×C8—672no—
221C2×C8—672no—
222C4⋊C4—22no—
223C4⋊C4C22×D6712yes—
224C22⋊C4—21no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C42—21no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—2684no—
232C23—1344no—
233C23—1342no—
234C23C2×C22⋊D6712yes—
235C23C2×C22⋊D6711yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—674no—
239Q8—1344no—
240Q8—1342no—
241D4—21no—
242D4—21no—
243D4—44no—
244D4—42no—
245D4—674no—
246D4—13416no—
247D4—1344no—
248D4—1342no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—13416no—
257C2×C4—1344no—
258C2×C4—1342no—
259C2×C4D4×D6712yes—
260C2×C4D4×D6712yes—
261C2×C4C2×C22⋊D6712yes—
262C2×C4C2×C22⋊D6711yescharacteristic
263C2×C4C23×D6711yescharacteristic
264C8—1342no—
265C8—1342no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—41no—
272C22—42no—
273C22—2688no—
274C22—2684no—
275C22—1344no—
276C22#1452401112yes—
277C22C22×C22⋊D6711yescharacteristic
278C4—22no—
279C4—41no—
280C4—42no—
281C4—1344no—
282C4#1452401112yes—
283C4C2×D4×D6711yescharacteristic
284C4C2×D4×D6711yescharacteristic
285C2—22no—
286C2—42no—
287C2—81no—
288C2—2684no—
289C2C2×#1452401111yescharacteristic
290C2#1686419412yes—
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#525×C67C2
socleC22×C67C22×D4
hypercenter#525D67
hypocenterC67#3005
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#11761628C2semidirect
2. C2×#14524073C2semidirect
3. C2×#14524068C2semidirect
4. #16863556C2semidirect
5. #16863539C2semidirect
6. #16863404C2semidirect
7. #16863400C2semidirect
8. #16863360C2semidirect
9. #525×C67C2semidirect
20. C2×#11761618C22semidirect
21. C2×#11761616C22semidirect
22. C2×#11761615C22semidirect
23. C2×#11761613C22semidirect
26. #14523966C22semidirect
27. #14523963C22semidirect
28. C22⋊(C67⋊C8)C22semidirect
32. C22⋊Q8×C67C22semidirect
33. #121×C67C22semidirect
43. C2×Dic134D4semidirect
47. C2×D268D4semidirect
146. C67#3005semidirect
150. #525D67semidirect
175. C22⋊Q8C2×D67semidirect
176. #121C2×D67semidirect