Finite Group Databasebeta

← #22572350

Subgroups — #22572350

Order 8576.

462 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864092C211yesmaximal, characteristic, complemented
2#16863760C211yesmaximal, characteristic, complemented
3#16863717C211yesmaximal, characteristic, complemented
4#16863712C211yesmaximal, characteristic, complemented
5#16863696C211yesmaximal, characteristic, complemented
6#16863647C211yesmaximal, characteristic, complemented
7#16863630C211yesmaximal, characteristic, complemented
8#16863588C211yesmaximal, characteristic, complemented
9#16863530C211yesmaximal, characteristic, complemented
10#16863530C211yesmaximal, characteristic, complemented
11#16863506C211yesmaximal, characteristic, complemented
12#16863468C211yesmaximal, characteristic, complemented
13#16863401C211yesmaximal, characteristic, complemented
14C2×#14523964C211yesmaximal, characteristic, complemented
15D42×C67C211yesmaximal, characteristic, complemented
16C2×#11761613C2211yescharacteristic
17C2×#11761613C2211yescharacteristic, complemented
18C2×#11761613C2211yescharacteristic, complemented
19C2×#11761612C2211yescharacteristic, complemented
20C22×C67⋊C8C2211yescharacteristic, complemented
21#14524038—21no—
22#14524035C2211yescharacteristic, complemented
23#14524025—21no—
24#14524017C2211yescharacteristic, complemented
25#14524017C2211yescharacteristic, complemented
26#14524015—21no—
27C4×D268C2211yescharacteristic, complemented
28(C22⋊D67)⋊C4—21no—
29#14524002C2211yescharacteristic, complemented
30C2×C4⋊Dic67C2211yescharacteristic, complemented
31#14523976C2211yescharacteristic, complemented
32#14523974C2211yescharacteristic, complemented
33#14523974C2211yescharacteristic, complemented
34C4⋊(C67⋊C8)C2211yescharacteristic, complemented
35#14523964C2211yescharacteristic
36#14523964C2211yescharacteristic
37#14523964C2211yescharacteristic
38#14523964C2211yescharacteristic, complemented
39#14523964C2211yescharacteristic, complemented
40#14523964C2211yescharacteristic, complemented
41#14523964C2211yescharacteristic, complemented
42#14523964C2211yescharacteristic, complemented
43#14523964C2211yescharacteristic, complemented
44#14523963C2211yescharacteristic, complemented
45C22⋊(C67⋊C8)C2211yescharacteristic, complemented
46C22⋊(C67⋊C8)C2211yescharacteristic, complemented
47#14523954C2211yescharacteristic, complemented
48C22×D4×C67—21no—
49C22×D4×C67C2211yescharacteristic
50C22×D4×C67C2211yescharacteristic
51#127×C67C2211yescharacteristic
52#121×C67—21no—
53#121×C67C2211yescharacteristic
54#121×C67C2211yescharacteristic
55#120×C67—21no—
56#120×C67—21no—
57C4×D4×C67C2211yescharacteristic
58C4×D4×C67C2211yescharacteristic
59C2×D268C2311yescharacteristic
60C2×C22⋊D67—21no—
61C2×C22⋊D67—21no—
62C22×Dic67—21no—
63C2×C4×D67—21no—
64#11761613—22no—
65#11761613—41no—
66#11761613—41no—
67#11761612—41no—
68C2×C67⋊C8—41no—
69C2×C67⋊C8C2311yescharacteristic
70C2×C67⋊C8C2311yescharacteristic
71C2×C67⋊C8C2311yescharacteristic
72C2×C67⋊C8C2311yescharacteristic
73C4⋊Dic67—21no—
74C4⋊Dic67C2311yescharacteristic
75C4⋊Dic67C2311yescharacteristic
76C4⋊Dic67C2311yescharacteristic
77Dic67⋊C4—21no—
78Dic67⋊C4—21no—
79C22⋊Dic67—21no—
80(C2×D67)⋊C4—21no—
81(C2×D67)⋊C4—21no—
82(C2×D67)⋊C4—21no—
83C4×Dic67—21no—
84C24×C67—21no—
85C24×C67—21no—
86C2×D4×C67—22no—
87C2×D4×C67—22no—
88C2×D4×C67—21no—
89C2×D4×C67—21no—
90C2×D4×C67—21no—
91C2×D4×C67—21no—
92C2×D4×C67—21no—
93C2×D4×C67—21no—
94C2×D4×C67—41no—
95C2×D4×C67—41no—
96C2×D4×C67—44no—
97C2×D4×C67D411yescharacteristic
98C2×D4×C67C2311yescharacteristic
99C2×D4×C67C2311yescharacteristic
100C2×D4×C67C2311yescharacteristic
101C22×C4×C67—21no—
102C22×C4×C67C2311yescharacteristic
103C22×C4×C67C2311yescharacteristic
104C4⋊C4×C67D411yescharacteristic
105C4⋊C4×C67C2311yescharacteristic
106C22⋊C4×C67—21no—
107C22⋊C4×C67—21no—
108C22⋊C4×C67—21no—
109C22⋊C4×C67D411yescharacteristic
110C22⋊C4×C67D411yescharacteristic
111C42×C67C2311yescharacteristic
112C23×C67—21no—
113C23×C67—21no—
114C23×C67—21no—
115C23×C67—42no—
116C23×C67—42no—
117C23×C67—42no—
118C23×C67—42no—
119C23×C67—81no—
120C23×C67—81no—
121C23×C67C2×D411yescharacteristic
122C23×C67C2×D411yescharacteristic
123C22×D67—21no—
124D4×C67—22no—
125D4×C67—21no—
126D4×C67—21no—
127D4×C67—21no—
128D4×C67—21no—
129D4×C67—21no—
130D4×C67—42no—
131D4×C67—42no—
132D4×C67—42no—
133D4×C67—42no—
134D4×C67—42no—
135D4×C67—81no—
136D4×C67#4112yes—
137C2×C4×C67—21no—
138C2×C4×C67—21no—
139C2×C4×C67—21no—
140C2×C4×C67—21no—
141C2×C4×C67—21no—
142C2×C4×C67—42no—
143C2×C4×C67C2×D411yescharacteristic
144C2×C4×C67C2×D411yescharacteristic
145C2×C4×C67C2×D411yescharacteristic
146C2×C4×C67C2×D411yescharacteristic
147C2×C4×C67C2411yescharacteristic
148C22⋊D67—42no—
149C22⋊D67—81no—
150C2×Dic67—21no—
151C2×Dic67—21no—
152C2×Dic67—21no—
153C2×Dic67—81no—
154D268—21no—
155D268—21no—
156C4×D67—81no—
157C67⋊C8—21no—
158C67⋊C8—21no—
159C67⋊C8—21no—
160C67⋊C8—21no—
161C22×C67—21no—
162C22×C67—21no—
163C22×C67—21no—
164C22×C67—22no—
165C22×C67—21no—
166C22×C67—21no—
167C22×C67—41no—
168C22×C67—41no—
169C22×C67—41no—
170C22×C67—41no—
171C22×C67—41no—
172C22×C67—41no—
173C22×C67—41no—
174C22×C67—41no—
175C22×C67—84no—
176C22×C67—84no—
177C22×C67AGL(1,ℤ/8)12yes—
178C22×C67C22×D411yescharacteristic
179C2×D67—41no—
180C2×D67—41no—
181C2×D67—41no—
182C4×C67—21no—
183C4×C67—21no—
184C4×C67—21no—
185C4×C67—21no—
186C4×C67—41no—
187C4×C67C2×#4111yescharacteristic
188C4×C67H(5,𝔽2)11yescharacteristic
189Dic67—41no—
190Dic67—41no—
191Dic67—41no—
192C2×C67—22no—
193C2×C67—41no—
194C2×C67—42no—
195C2×C67—81no—
196C2×C67—81no—
197C2×C67#54611yescharacteristic
198C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
199C2×C67#57611yescharacteristic
200D67—81no—
201#3318—671nomaximal, Sylow
202C67#331811yescharacteristic, complemented, Sylow
203#553—671no—
204D42D6711yescharacteristic, complemented
205#490—671no—
206#482—671no—
207#481—671no—
208#478—671no—
209#469—671no—
210#466—671no—
211#459—671no—
212#447—671no—
213#447—671no—
214#442—671no—
215#435—671no—
216#418—671no—
217C2×#102—671no—
218C22×D4—21no—
219C22×D4C2×D6711yescharacteristic
220C22×D4C2×D6711yescharacteristic
221C2×D8—671no—
222C2×D8—671no—
223C2×D8—671no—
224C2×M16—671no—
225C22×C8—671no—
226#127C2×D6711yescharacteristic
227#126—1341no—
228#125—671no—
229#123—1341no—
230#121—21no—
231#121—671no—
232#121—671no—
233#121—1341no—
234#121C2×D6711yescharacteristic
235#121C2×D6711yescharacteristic
236#120—21no—
237#120—21no—
238C4×D4—671no—
239C4×D4—1341no—
240C4×D4C2×D6711yescharacteristic
241C4×D4C2×D6711yescharacteristic
242#117—671no—
243C2×C4⋊C4—671no—
244#107—671no—
245#106—671no—
246#106—671no—
247C4⋊C8—671no—
248#102—671no—
249#102—671no—
250#102—671no—
251#102—671no—
252#102—671no—
253#102—671no—
254#102—671no—
255#102—671no—
256#102—671no—
257#102—671no—
258C22⋊C8—671no—
259C22⋊C8—671no—
260#97—671no—
261C24—21no—
262C24—21no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—44no—
274C2×D4—671no—
275C2×D4—1341no—
276C2×D4—1341no—
277C2×D4C22⋊D6711yescharacteristic
278C2×D4C22×D6711yescharacteristic
279C2×D4C22×D6711yescharacteristic
280C2×D4C22×D6711yescharacteristic
281C22×C4—21no—
282C22×C4—1341no—
283C22×C4—1341no—
284C22×C4C22×D6711yescharacteristic
285C22×C4C22×D6711yescharacteristic
286D8—2681no—
287D8—2681no—
288D8—1342no—
289M16—2681no—
290C2×C8—2681no—
291C2×C8—671no—
292C2×C8—671no—
293C2×C8—671no—
294C2×C8—671no—
295C4⋊C4—671no—
296C4⋊C4—671no—
297C4⋊C4—671no—
298C4⋊C4—1341no—
299C4⋊C4—1341no—
300C4⋊C4—1341no—
301C4⋊C4C22⋊D6711yescharacteristic
302C4⋊C4C22×D6711yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—1341no—
307C22⋊C4—1341no—
308C22⋊C4—1341no—
309C22⋊C4—1341no—
310C22⋊C4C22⋊D6711yescharacteristic
311C22⋊C4C22⋊D6711yescharacteristic
312C42—1341no—
313C42C22×D6711yescharacteristic
314C23—21no—
315C23—21no—
316C23—21no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—81no—
322C23—81no—
323C23—1341no—
324C23C2×C22⋊D6711yescharacteristic
325C23C2×C22⋊D6711yescharacteristic
326D4—22no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—42no—
333D4—42no—
334D4—42no—
335D4—42no—
336D4—42no—
337D4—81no—
338D4—2682no—
339D4—5361no—
340D4—1341no—
341D4—1341no—
342D4#1176162712yes—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—5361no—
350C2×C4—5361no—
351C2×C4—1341no—
352C2×C4—1341no—
353C2×C4—1341no—
354C2×C4C2×C22⋊D6711yescharacteristic
355C2×C4C2×C22⋊D6711yescharacteristic
356C2×C4C2×C22⋊D6711yescharacteristic
357C2×C4C2×C22⋊D6711yescharacteristic
358C2×C4C23×D6711yescharacteristic
359C8—1341no—
360C8—1341no—
361C8—1341no—
362C8—1341no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—22no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—84no—
378C22—84no—
379C22—2681no—
380C22—2681no—
381C22—2681no—
382C22#1452406912yes—
383C22C22×C22⋊D6711yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—41no—
389C4—2681no—
390C4—2681no—
391C4—2681no—
392C4C2×#1176162711yescharacteristic
393C4#1452408811yescharacteristic
394C2—22no—
395C2—41no—
396C2—42no—
397C2—81no—
398C2—81no—
399C2—5361no—
400C2#1686404011yescharacteristic
401C2C2×#1452406911yescharacteristic
402C2#1686419911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
FittingD42×C67C2
socleC22×C67C22×D4
hypercenterD42D67
hypocenterC67#3318
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. #16864092C2semidirect
2. #16863760C2semidirect
3. #16863717C2semidirect
4. #16863712C2semidirect
5. #16863696C2semidirect
6. #16863647C2semidirect
7. #16863630C2semidirect
8. #16863588C2semidirect
9. #16863530C2semidirect
10. #16863530C2semidirect
11. #16863506C2semidirect
12. #16863468C2semidirect
13. #16863401C2semidirect
14. C2×#14523964C2semidirect
15. D42×C67C2semidirect
17. C2×#11761613C22semidirect
18. C2×#11761613C22semidirect
19. C2×#11761612C22semidirect
20. C22×C67⋊C8C22semidirect
22. #14524035C22semidirect
24. #14524017C22semidirect
25. #14524017C22semidirect
27. C4×D268C22semidirect
29. #14524002C22semidirect
30. C2×C4⋊Dic67C22semidirect
31. #14523976C22semidirect
32. #14523974C22semidirect
33. #14523974C22semidirect
34. C4⋊(C67⋊C8)C22semidirect
38. #14523964C22semidirect
39. #14523964C22semidirect
40. #14523964C22semidirect
41. #14523964C22semidirect
42. #14523964C22semidirect
43. #14523964C22semidirect
44. #14523963C22semidirect
45. C22⋊(C67⋊C8)C22semidirect
46. C22⋊(C67⋊C8)C22semidirect
47. #14523954C22semidirect
202. C67#3318semidirect
204. D42D67semidirect