Finite Group Databasebeta

← #22571805

Subgroups — #22571805

Order 8576.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#14524069C212yesmaximal, complemented
2#16864051C211yesmaximal, characteristic, complemented
3#16863768C211yesmaximal, characteristic, complemented
4#16863640C212yesmaximal, complemented
5#16863601C211yesmaximal, characteristic, complemented
6#16863588C211yesmaximal, characteristic, complemented
7#16863547C212yesmaximal, complemented
8#16863539C212yesmaximal, complemented
9C4×#11761613C211yesmaximal, characteristic, complemented
10#16863474C211yesmaximal, characteristic, complemented
11D42×C67C211yesmaximal, characteristic, complemented
12C2×#11761628C2212yescomplemented
13#14524069—28no—
14C2×#11761616C2212yes—
15C2×#11761616C2212yescomplemented
16C2×#11761616C2212yescomplemented
17C2×#11761613C2211yescharacteristic
18C2×#11761613C2212yescomplemented
19C2×#11761612C2212yescomplemented
20#14524043C2211yescharacteristic, complemented
21#14524025—22no—
22#14524022C2212yescomplemented
23#14524014—21no—
24C4×D268C2211yescharacteristic, complemented
25C4⋊C4×D67—21no—
26#14523976C2211yescharacteristic, complemented
27C4⋊(C67⋊C8)C2211yescharacteristic, complemented
28#14523966C2212yescomplemented
29#14523964C2211yescharacteristic
30#14523964C2212yescomplemented
31#14523963C2211yescharacteristic, complemented
32C22⋊(C67⋊C8)C2212yescomplemented
33C4×C67⋊C8C2211yescharacteristic, complemented
34C22×D4×C67—21no—
35C22×D4×C67C2212yes—
36#127×C67C2211yescharacteristic
37#121×C67—21no—
38#121×C67C2212yes—
39#120×C67—22no—
40C4×D4×C67C2211yescharacteristic
41C4×D4×C67C2211yescharacteristic
42#11761628—24no—
43#11761628—42no—
44C2×Dic134C2312yes—
45C2×D268C2311yescharacteristic
46C2×C22⋊D67—22no—
47C2×C4×D67—21no—
48C2×C4×D67—22no—
49#11761616—24no—
50#11761616—24no—
51#11761616—42no—
52#11761613—24no—
53#11761613D414yescomplemented
54#11761612—24no—
55C2×C67⋊C8C2311yescharacteristic
56C2×C67⋊C8C2312yes—
57C2×C67⋊C8C2311yescharacteristic
58C4⋊Dic67—21no—
59C4⋊Dic67—21no—
60C4⋊Dic67C2311yescharacteristic
61Dic67⋊C4—22no—
62C22⋊Dic67—22no—
63(C2×D67)⋊C4—21no—
64C24×C67—22no—
65C2×D4×C67—24no—
66C2×D4×C67—22no—
67C2×D4×C67—22no—
68C2×D4×C67—21no—
69C2×D4×C67—21no—
70C2×D4×C67—42no—
71C2×D4×C67—44no—
72C2×D4×C67D411yescharacteristic
73C2×D4×C67C2312yes—
74C2×D4×C67C2311yescharacteristic
75C22×C4×C67—21no—
76C22×C4×C67C2312yes—
77C4⋊C4×C67D411yescharacteristic
78C4⋊C4×C67C2311yescharacteristic
79C22⋊C4×C67—21no—
80C22⋊C4×C67—22no—
81C22⋊C4×C67D412yes—
82C42×C67C2311yescharacteristic
83C23×C67—21no—
84C23×C67—22no—
85C23×C67—44no—
86C23×C67—44no—
87C23×C67—82no—
88C23×C67C2×D412yes—
89C22×D67—21no—
90D4×C67—22no—
91D4×C67—22no—
92D4×C67—21no—
93D4×C67—22no—
94D4×C67—44no—
95D4×C67—44no—
96D4×C67—42no—
97D4×C67—81no—
98D4×C67C2×D412yes—
99C2×C4×C67—22no—
100C2×C4×C67—22no—
101C2×C4×C67—21no—
102C2×C4×C67—42no—
103C2×C4×C67C2×D411yescharacteristic
104C2×C4×C67C2×D412yes—
105C2×C4×C67C2×D411yescharacteristic
106C2×C4×C67C2411yescharacteristic
107C22⋊D67—44no—
108C2×Dic67—21no—
109C2×Dic67—22no—
110D268—21no—
111D268C2×D412yescomplemented
112C4×D67—44no—
113C4×D67—42no—
114Dic134—22no—
115Dic134—22no—
116C67⋊C8—22no—
117C67⋊C8—21no—
118C67⋊C8C2×D412yescomplemented
119C22×C67—22no—
120C22×C67—22no—
121C22×C67—22no—
122C22×C67—22no—
123C22×C67—41no—
124C22×C67—41no—
125C22×C67—42no—
126C22×C67—42no—
127C22×C67—42no—
128C22×C67—88no—
129C22×C67C22×D411yescharacteristic
130C2×D67—22no—
131C2×D67—41no—
132C2×D67—41no—
133C4×C67—22no—
134C4×C67—21no—
135C4×C67—41no—
136C4×C67AGL(1,ℤ/8)12yescomplemented
137C4×C67C22×D411yescharacteristic
138C4×C67H(5,𝔽2)11yescharacteristic
139Dic67—42no—
140Dic67—41no—
141C2×C67—42no—
142C2×C67—42no—
143C2×C67—82no—
144C2×C67D4211yescharacteristic
145C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C67#57711yescharacteristic
147D67—42no—
148#3272—671nomaximal, Sylow
149C67#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—672no—
151#547—671no—
152D42D6711yescharacteristic, complemented
153#492—671no—
154#468—672no—
155#461—671no—
156#459—671no—
157#450—672no—
158#449—672no—
159C4×D8—671no—
160#436—671no—
161C2×#41—672no—
162C22×D4—21no—
163C22×D4C2×D6712yes—
164AGL(1,ℤ/8)—1348no—
165C2×SD16—672no—
166C2×SD16—672no—
167C2×SD16—672no—
168C2×D8—671no—
169C2×D8—672no—
170C2×M16—672no—
171C4⋊Q8—671no—
172#127C2×D6711yescharacteristic
173#123—1342no—
174C22⋊Q8—672no—
175#121—21no—
176#121—1341no—
177#121C2×D6712yes—
178#120—22no—
179C4×D4—671no—
180C4×D4C2×D6711yescharacteristic
181C4×D4C2×D6711yescharacteristic
182C2×C4⋊C4—1341no—
183#107—671no—
184C4⋊C8—671no—
185#103—672no—
186#102—671no—
187#102—672no—
188#102—671no—
189C22⋊C8—672no—
190C4×C8—671no—
191C24—22no—
192#41—2682no—
193#41—1344no—
194C2×Q8—672no—
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—42no—
201C2×D4—44no—
202C2×D4—671no—
203C2×D4—1342no—
204C2×D4C22⋊D6711yescharacteristic
205C2×D4C22×D6712yes—
206C2×D4C22×D6711yescharacteristic
207C22×C4—21no—
208C22×C4—1341no—
209C22×C4—1342no—
210C22×C4C22×D6712yes—
211SD16—2682no—
212SD16—1344no—
213SD16—1344no—
214D8—674no—
215D8—1344no—
216M16—1344no—
217C2×C8—671no—
218C2×C8—671no—
219C2×C8—672no—
220C4⋊C4—671no—
221C4⋊C4—1341no—
222C4⋊C4—1342no—
223C4⋊C4—1341no—
224C4⋊C4C22⋊D6711yescharacteristic
225C4⋊C4C22×D6711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—22no—
228C22⋊C4—1342no—
229C22⋊C4—1341no—
230C22⋊C4C22⋊D6712yes—
231C42C22×D6711yescharacteristic
232C23—21no—
233C23—22no—
234C23—44no—
235C23—44no—
236C23—82no—
237C23—1341no—
238C23C2×C22⋊D6712yes—
239Q8—1342no—
240Q8—1342no—
241D4—22no—
242D4—22no—
243D4—21no—
244D4—22no—
245D4—44no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—2684no—
250D4—672no—
251D4—1341no—
252D4D4×D6712yes—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—2684no—
258C2×C4—2682no—
259C2×C4—1341no—
260C2×C4—1342no—
261C2×C4C2×C22⋊D6711yescharacteristic
262C2×C4C2×C22⋊D6712yes—
263C2×C4C2×C22⋊D6711yescharacteristic
264C2×C4C23×D6711yescharacteristic
265C8—672no—
266C8—1342no—
267C8—1341no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—88no—
278C22—2681no—
279C22—2681no—
280C22—1342no—
281C22C22×C22⋊D6711yescharacteristic
282C4—22no—
283C4—21no—
284C4—41no—
285C4—2682no—
286C4—2681no—
287C4#1452406912yescomplemented
288C4C2×D4×D6711yescharacteristic
289C4#1452408811yescharacteristic
290C2—42no—
291C2—42no—
292C2—82no—
293C2—2682no—
294C2D4×C22⋊D6711yescharacteristic
295C2C2×#1452406911yescharacteristic
296C2#1686420711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
FittingD42×C67C2
socleC22×C67C22×D4
hypercenterD42D67
hypocenterC67#3272
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#14524069C2semidirect
2. #16864051C2semidirect
3. #16863768C2semidirect
4. #16863640C2semidirect
5. #16863601C2semidirect
6. #16863588C2semidirect
7. #16863547C2semidirect
8. #16863539C2semidirect
9. C4×#11761613C2semidirect
10. #16863474C2semidirect
11. D42×C67C2semidirect
12. C2×#11761628C22semidirect
15. C2×#11761616C22semidirect
16. C2×#11761616C22semidirect
18. C2×#11761613C22semidirect
19. C2×#11761612C22semidirect
20. #14524043C22semidirect
22. #14524022C22semidirect
24. C4×D268C22semidirect
26. #14523976C22semidirect
27. C4⋊(C67⋊C8)C22semidirect
28. #14523966C22semidirect
30. #14523964C22semidirect
31. #14523963C22semidirect
32. C22⋊(C67⋊C8)C22semidirect
33. C4×C67⋊C8C22semidirect
53. #11761613D4semidirect
111. D268C2×D4semidirect
118. C67⋊C8C2×D4semidirect
136. C4×C67AGL(1,ℤ/8)semidirect
149. C67#3272semidirect
152. D42D67semidirect
287. C4#14524069semidirect