Finite Group Databasebeta

← #22570004

Subgroups — #22570004

Order 8576.

440 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16863952C211yesmaximal, characteristic, complemented
2C4×D4×D67C211yesmaximal, characteristic
3#16863697C211yesmaximal, characteristic, complemented
4#16863696C211yesmaximal, characteristic, complemented
5#16863693C211yesmaximal, characteristic, complemented
6#16863685C211yesmaximal, characteristic, complemented
7#16863684C211yesmaximal, characteristic, complemented
8#16863675C211yesmaximal, characteristic, complemented
9#16863674C211yesmaximal, characteristic, complemented
10#16863437C211yesmaximal, characteristic, complemented
11#16863435C211yesmaximal, characteristic, complemented
12#16863418C211yesmaximal, characteristic, complemented
13#16863399C211yesmaximal, characteristic
14#16863399C211yesmaximal, characteristic
15#478×C67C211yesmaximal, characteristic, complemented
16C2×D4×D67C2211yescharacteristic
17C22×C4×D67—21no—
18C2×#11761611C2211yescharacteristic
19C2×#11761611C2211yescharacteristic
20#14524042C2211yescharacteristic, complemented
21#14524042C2211yescharacteristic, complemented
22#14524035C2211yescharacteristic, complemented
23#14524021—21no—
24#14524021—21no—
25#14524008—21no—
26#14524008—21no—
27C4×D268C2211yescharacteristic
28D4×Dic67C2211yescharacteristic
29#14524005C2211yescharacteristic
30C4×C22⋊D67—21no—
31(C22⋊D67)⋊C4—21no—
32#14524001C2211yescharacteristic
33#14524001C2211yescharacteristic
34C4⋊C4×D67C2211yescharacteristic
35C22⋊C4×D67—21no—
36C42×D67C2211yescharacteristic
37#14523977C2211yescharacteristic, complemented
38#14523976C2211yescharacteristic, complemented
39#14523976C2211yescharacteristic, complemented
40#14523975C2211yescharacteristic, complemented
41#14523974C2211yescharacteristic, complemented
42#14523974C2211yescharacteristic, complemented
43C4⋊(C67⋊C8)C2211yescharacteristic, complemented
44Dic67⋊C8C2211yescharacteristic, complemented
45Dic67⋊C8C2211yescharacteristic, complemented
46#14523965C2211yescharacteristic
47#14523965C2211yescharacteristic
48#14523964C2211yescharacteristic
49#14523964C2211yescharacteristic
50#14523963C2211yescharacteristic
51#14523963C2211yescharacteristic
52#125×C67C2211yescharacteristic, complemented
53C4×D4×C67C2211yescharacteristic
54#107×C67C2211yescharacteristic, complemented
55#106×C67C2211yescharacteristic, complemented
56C4⋊C8×C67C2211yescharacteristic, complemented
57#102×C67C2211yescharacteristic
58#102×C67C2211yescharacteristic
59C23×D67—21no—
60C2×Dic134—21no—
61C2×Dic134—21no—
62C2×D268C2311yescharacteristic
63C2×C22⋊D67—21no—
64D4×D67—22no—
65D4×D67Q814yescomplemented
66C22×Dic67—21no—
67C2×C4×D67—21no—
68C2×C4×D67—21no—
69C2×C4×D67—21no—
70C2×C4×D67—44no—
71C2×C4×D67D411yescharacteristic
72C2×C4×D67D411yescharacteristic
73C2×C4×D67C2311yescharacteristic
74#11761611—22no—
75#11761611—22no—
76C2×C67⋊C8C2311yescharacteristic
77C2×C67⋊C8C2311yescharacteristic
78C4⋊Dic67C2311yescharacteristic
79C4⋊Dic67C2311yescharacteristic
80C4⋊Dic67C2311yescharacteristic
81Dic67⋊C4—21no—
82Dic67⋊C4—21no—
83Dic67⋊C4—21no—
84Dic67⋊C4—21no—
85Dic67⋊C4—21no—
86C22⋊Dic67—21no—
87(C2×D67)⋊C4—21no—
88(C2×D67)⋊C4—21no—
89(C2×D67)⋊C4—21no—
90(C2×D67)⋊C4—21no—
91C4×Dic67—21no—
92C4×Dic67—21no—
93C4×Dic67D411yescharacteristic
94C4×Dic67D411yescharacteristic
95C4×Dic67C2311yescharacteristic
96C2×D4×C67C2311yescharacteristic
97C22×C4×C67—21no—
98C2×C8×C67C2311yescharacteristic
99C2×C8×C67C2311yescharacteristic
100C4⋊C4×C67—21no—
101C4⋊C4×C67—21no—
102C4⋊C4×C67C2311yescharacteristic
103C4⋊C4×C67C2311yescharacteristic
104C4⋊C4×C67C2311yescharacteristic
105C22⋊C4×C67—21no—
106C42×C67C2311yescharacteristic
107C23×C67—21no—
108C22×D67—21no—
109C22×D67—24no—
110C22×D67—42no—
111C22×D67—42no—
112C22×D67C2×D411yescharacteristic
113D4×C67—21no—
114D4×C67C2×Q812yes—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—42no—
119C2×C4×C67C2×D411yescharacteristic
120C2×C4×C67C2×D411yescharacteristic
121C2×C4×C67C2411yescharacteristic
122C22⋊D67—24no—
123C2×Dic67—21no—
124C2×Dic67—21no—
125C2×Dic67—21no—
126C2×Dic67—42no—
127C2×Dic67C2×D411yescharacteristic
128C2×Dic67C2×D411yescharacteristic
129C2×Dic67C2×D411yescharacteristic
130D268—21no—
131D268C2×Q812yes—
132C4×D67—22no—
133C4×D67—41no—
134C4×D67—42no—
135C4×D67—42no—
136C4×D67—42no—
137C4×D67C2×Q812yes—
138C4×D67#4112yes—
139Dic134—42no—
140Dic134—42no—
141C8×C67—21no—
142C8×C67—21no—
143C67⋊C8—21no—
144C67⋊C8—21no—
145C22×C67—22no—
146C22×C67—41no—
147C22×C67—41no—
148C22×C67C22×D411yescharacteristic
149C2×D67—21no—
150C2×D67—21no—
151C2×D67—22no—
152C2×D67—48no—
153C2×D67—41no—
154C2×D67—41no—
155C2×D67C22⋊Q812yes—
156C4×C67—21no—
157C4×C67—21no—
158C4×C67—41no—
159C4×C67—41no—
160C4×C67—41no—
161C4×C67C22×Q811yescharacteristic
162C4×C67C2×#4111yescharacteristic
163Dic67—21no—
164Dic67—41no—
165Dic67—41no—
166Dic67—41no—
167Dic67C22⋊Q812yes—
168Dic67AGL(1,ℤ/8)12yes—
169C2×C67—42no—
170C2×C67C2×C22⋊Q811yescharacteristic
171C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
172C2×C67#57511yescharacteristic
173D67—22no—
174D67—42no—
175#3077—671nomaximal, Sylow
176C67#307711yescharacteristic, complemented, Sylow
177#533—671no—
178C2×C4×D4—671no—
179#478—671no—
180#478—671no—
181#478—671no—
182#478D6711yescharacteristic, complemented
183#476—671no—
184#476—671no—
185#474—671no—
186#474—671no—
187#428—671no—
188#428—671no—
189#423—671no—
190#418—671no—
191#418—671no—
192C22×D4—671no—
193C23×C4—1341no—
194C2×M16—671no—
195C2×M16—671no—
196C4⋊Q8—671no—
197C4⋊Q8—671no—
198#125—671no—
199#125C2×D6711yescharacteristic, complemented
200C22⋊Q8—1341no—
201C22⋊Q8—1341no—
202C4×Q8—1341no—
203C4×Q8—1341no—
204C4×D4—671no—
205C4×D4—671no—
206C4×D4—671no—
207C4×D4—1341no—
208C4×D4—1341no—
209C4×D4C2×D6711yescharacteristic
210#117—671no—
211#117—671no—
212C2×C4⋊C4—671no—
213C2×C22⋊C4—1341no—
214C2×C42—671no—
215#107—671no—
216#107—671no—
217#107—671no—
218#107C2×D6711yescharacteristic, complemented
219#106—671no—
220#106—671no—
221#106—671no—
222#106C2×D6711yescharacteristic, complemented
223C4⋊C8—671no—
224C4⋊C8—671no—
225C4⋊C8—671no—
226C4⋊C8C2×D6711yescharacteristic, complemented
227#102—671no—
228#102—671no—
229#102—671no—
230#102—671no—
231#102—671no—
232#102—671no—
233#102C2×D6711yescharacteristic
234#102C2×D6711yescharacteristic
235C24—1341no—
236C2×Q8—1341no—
237C2×Q8—1341no—
238C2×D4—674no—
239C2×D4—671no—
240C2×D4—1342no—
241C2×D4—1341no—
242C2×D4C22×D6711yescharacteristic
243C22×C4—21no—
244C22×C4—2684no—
245C22×C4—671no—
246C22×C4—671no—
247C22×C4—671no—
248C22×C4—1341no—
249C22×C4—1341no—
250C22×C4—1341no—
251C22×C4—1341no—
252M16—1342no—
253M16—1342no—
254C2×C8—671no—
255C2×C8—671no—
256C2×C8C22×D6711yescharacteristic
257C2×C8C22×D6711yescharacteristic
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—671no—
261C4⋊C4—671no—
262C4⋊C4—671no—
263C4⋊C4—1341no—
264C4⋊C4—1341no—
265C4⋊C4—1341no—
266C4⋊C4—1341no—
267C4⋊C4—1341no—
268C4⋊C4C22×D6711yescharacteristic
269C4⋊C4C22×D6711yescharacteristic
270C4⋊C4C22×D6711yescharacteristic
271C22⋊C4—21no—
272C22⋊C4—1341no—
273C22⋊C4—1341no—
274C22⋊C4—1341no—
275C22⋊C4—1341no—
276C22⋊C4—1341no—
277C42—671no—
278C42—671no—
279C42—671no—
280C42—1341no—
281C42—1341no—
282C42C22×D6711yescharacteristic
283C23—21no—
284C23—2682no—
285C23—2682no—
286C23—671no—
287C23—1344no—
288C23—1341no—
289Q8—2682no—
290Q8—2682no—
291D4—21no—
292D4—672no—
293D4—1344no—
294D4—1341no—
295D4Q8×D6712yes—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—2682no—
301C2×C4—2681no—
302C2×C4—2682no—
303C2×C4—2682no—
304C2×C4—2682no—
305C2×C4—671no—
306C2×C4—671no—
307C2×C4—672no—
308C2×C4—672no—
309C2×C4—671no—
310C2×C4—1342no—
311C2×C4—1341no—
312C2×C4—1341no—
313C2×C4—1341no—
314C2×C4D4×D6711yescharacteristic
315C2×C4D4×D6711yescharacteristic
316C2×C4C23×D6711yescharacteristic
317C8—21no—
318C8—21no—
319C8—1341no—
320C8—1341no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—2688no—
325C22—2681no—
326C22—2681no—
327C22—672no—
328C22—1342no—
329C22—1341no—
330C22—1341no—
331C22C2×D4×D6711yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—2681no—
338C4—2681no—
339C4—2681no—
340C4—672no—
341C4—672no—
342C4—1341no—
343C4C2×Q8×D6711yescharacteristic
344C4#41×D6711yescharacteristic
345C2—42no—
346C2—2682no—
347C2—1342no—
348C2C22⋊Q8×D6711yescharacteristic
349C2AGL(1,ℤ/8)×D6711yescharacteristic
350C2#1686419311yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#478×C67C2
socleC22×C67C22×D4
hypercenter#478D67
hypocenterC67#3077
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #16863952C2semidirect
3. #16863697C2semidirect
4. #16863696C2semidirect
5. #16863693C2semidirect
6. #16863685C2semidirect
7. #16863684C2semidirect
8. #16863675C2semidirect
9. #16863674C2semidirect
10. #16863437C2semidirect
11. #16863435C2semidirect
12. #16863418C2semidirect
15. #478×C67C2semidirect
20. #14524042C22semidirect
21. #14524042C22semidirect
22. #14524035C22semidirect
37. #14523977C22semidirect
38. #14523976C22semidirect
39. #14523976C22semidirect
40. #14523975C22semidirect
41. #14523974C22semidirect
42. #14523974C22semidirect
43. C4⋊(C67⋊C8)C22semidirect
44. Dic67⋊C8C22semidirect
45. Dic67⋊C8C22semidirect
52. #125×C67C22semidirect
54. #107×C67C22semidirect
55. #106×C67C22semidirect
56. C4⋊C8×C67C22semidirect
65. D4×D67Q8semidirect
176. C67#3077semidirect
182. #478D67semidirect
199. #125C2×D67semidirect
218. #107C2×D67semidirect
222. #106C2×D67semidirect
226. C4⋊C8C2×D67semidirect