Finite Group Databasebeta

← #22571055

Subgroups — #22571055

Order 8576.

422 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16863960C211yesmaximal, characteristic, complemented
2#16863734C211yesmaximal, characteristic, complemented
3#16863717C211yesmaximal, characteristic, complemented
4#16863714C211yesmaximal, characteristic, complemented
5#16863709C211yesmaximal, characteristic, complemented
6#16863647C211yesmaximal, characteristic, complemented
7#16863644C211yesmaximal, characteristic, complemented
8#16863630C211yesmaximal, characteristic, complemented
9#16863627C211yesmaximal, characteristic, complemented
10#16863546C211yesmaximal, characteristic, complemented
11#16863541C211yesmaximal, characteristic, complemented
12#16863541C211yesmaximal, characteristic, complemented
13#16863529C211yesmaximal, characteristic, complemented
14#16863363C211yesmaximal, characteristic, complemented
15#537×C67C211yesmaximal, characteristic, complemented
16C22×D268C2211yescharacteristic
17C2×#11761615C2211yescharacteristic
18C2×#11761615C2211yescharacteristic
19C2×#11761613C2211yescharacteristic
20C2×#11761613C2211yescharacteristic
21C2×#11761612C2211yescharacteristic, complemented
22C22×C67⋊C8C2211yescharacteristic, complemented
23#14524039—21no—
24#14524032—21no—
25#14524017C2211yescharacteristic
26#14524017C2211yescharacteristic
27#14524017C2211yescharacteristic
28#14524017C2211yescharacteristic
29#14524011—21no—
30#14524011—21no—
31#14524002C2211yescharacteristic
32#14523976C2211yescharacteristic, complemented
33#14523976C2211yescharacteristic, complemented
34#14523974C2211yescharacteristic, complemented
35#14523974C2211yescharacteristic, complemented
36#14523967C2211yescharacteristic, complemented
37#14523967C2211yescharacteristic, complemented
38#14523964C2211yescharacteristic, complemented
39#14523964C2211yescharacteristic, complemented
40#14523963C2211yescharacteristic
41#14523963C2211yescharacteristic
42#14523963C2211yescharacteristic
43#14523963C2211yescharacteristic
44C22⋊(C67⋊C8)C2211yescharacteristic
45C22⋊(C67⋊C8)C2211yescharacteristic
46C22⋊(C67⋊C8)C2211yescharacteristic, complemented
47C22⋊(C67⋊C8)C2211yescharacteristic, complemented
48C2×#41×C67C2211yescharacteristic
49C2×#41×C67C2211yescharacteristic, complemented
50C22⋊Q8×C67—21no—
51C22⋊Q8×C67C2211yescharacteristic
52C22⋊Q8×C67C2211yescharacteristic, complemented
53#121×C67—21no—
54#121×C67—21no—
55#121×C67—21no—
56#121×C67C2211yescharacteristic
57#121×C67C2211yescharacteristic, complemented
58C2×C4⋊C4×C67C2211yescharacteristic, complemented
59C23×D67—21no—
60C2×D268—21no—
61C2×D268—41no—
62C2×D268—41no—
63C2×D268C2311yescharacteristic
64C2×D268C2311yescharacteristic
65C2×C22⋊D67—21no—
66C2×C22⋊D67—21no—
67C2×C22⋊D67—21no—
68C2×C22⋊D67—21no—
69#11761615—41no—
70#11761615—41no—
71#11761613—41no—
72#11761613—41no—
73#11761612—41no—
74C2×C67⋊C8—41no—
75C2×C67⋊C8C2311yescharacteristic
76C2×C67⋊C8C2311yescharacteristic
77C2×C67⋊C8C2311yescharacteristic
78C2×C67⋊C8C2311yescharacteristic
79C4⋊Dic67C2311yescharacteristic
80C4⋊Dic67C2311yescharacteristic
81C22⋊Dic67—21no—
82(C2×D67)⋊C4—21no—
83(C2×D67)⋊C4—21no—
84(C2×D67)⋊C4—21no—
85(C2×D67)⋊C4—21no—
86C4×Dic67—21no—
87#41×C67—41no—
88#41×C67—41no—
89#41×C67—41no—
90#41×C67—41no—
91C2×Q8×C67D411yescharacteristic
92C2×Q8×C67C2311yescharacteristic
93C2×D4×C67—21no—
94C2×D4×C67—21no—
95C2×D4×C67—21no—
96C2×D4×C67D411yescharacteristic
97C2×D4×C67D411yescharacteristic
98C2×D4×C67D411yescharacteristic
99C2×D4×C67C2311yescharacteristic
100C22×C4×C67—21no—
101C22×C4×C67—21no—
102C22×C4×C67C2311yescharacteristic
103C22×C4×C67C2311yescharacteristic
104C22×C4×C67C2311yescharacteristic
105C4⋊C4×C67—21no—
106C4⋊C4×C67—21no—
107C4⋊C4×C67—21no—
108C4⋊C4×C67C2311yescharacteristic
109C4⋊C4×C67C2311yescharacteristic
110C22⋊C4×C67—21no—
111C22⋊C4×C67—21no—
112C22⋊C4×C67—21no—
113C22⋊C4×C67—21no—
114C23×C67—21no—
115C23×C67C2×D411yescharacteristic
116C23×C67C2×D411yescharacteristic
117C23×C67C2×D411yescharacteristic
118C22×D67—21no—
119C22×D67—21no—
120C22×D67—42no—
121C22×D67—81no—
122C22×D67—81no—
123Q8×C67—21no—
124Q8×C67—21no—
125Q8×C67—41no—
126D4×C67—21no—
127D4×C67—21no—
128D4×C67—41no—
129D4×C67—41no—
130D4×C67—41no—
131D4×C67—42no—
132D4×C67—81no—
133D4×C67—81no—
134C2×C4×C67—21no—
135C2×C4×C67—21no—
136C2×C4×C67—21no—
137C2×C4×C67—21no—
138C2×C4×C67—21no—
139C2×C4×C67—21no—
140C2×C4×C67—21no—
141C2×C4×C67—21no—
142C2×C4×C67—21no—
143C2×C4×C67—42no—
144C2×C4×C67—81no—
145C2×C4×C67C2×D411yescharacteristic
146C2×C4×C67C2×D411yescharacteristic
147C2×C4×C67C2×D411yescharacteristic
148C2×C4×C67C2411yescharacteristic
149C22⋊D67—81no—
150C22⋊D67—81no—
151C22⋊D67—81no—
152C22⋊D67—81no—
153C2×Dic67—21no—
154C2×Dic67—21no—
155D268—21no—
156D268—21no—
157D268—21no—
158D268—21no—
159D268—42no—
160C67⋊C8—21no—
161C67⋊C8—21no—
162C67⋊C8—21no—
163C67⋊C8—21no—
164C22×C67—21no—
165C22×C67—21no—
166C22×C67—21no—
167C22×C67—21no—
168C22×C67—21no—
169C22×C67—21no—
170C22×C67—21no—
171C22×C67—21no—
172C22×C67—21no—
173C22×C67—41no—
174C22×C67—41no—
175C22×C67—41no—
176C22×C67C22×D411yescharacteristic
177C2×D67—41no—
178C2×D67—41no—
179C2×D67—41no—
180C2×D67—41no—
181C2×D67—41no—
182C2×D67—41no—
183C2×D67—84no—
184C4×C67—21no—
185C4×C67—21no—
186C4×C67—21no—
187C4×C67—41no—
188C4×C67—41no—
189C4×C67—41no—
190C4×C67H(5,𝔽2)11yescharacteristic
191C4×C67H(5,𝔽2)11yescharacteristic
192Dic67—41no—
193Dic67—41no—
194C2×C67—41no—
195C2×C67—41no—
196C2×C67—41no—
197C2×C67—81no—
198C2×C67#53411yescharacteristic
199C2×C67#57611yescharacteristic
200C2×C67#57711yescharacteristic
201D67—81no—
202D67—81no—
203#3198—671nomaximal, Sylow
204C67#319811yescharacteristic, complemented, Sylow
205#537D6711yescharacteristic, complemented
206#535—671no—
207#485—671no—
208#482—671no—
209#481—671no—
210#480—671no—
211#469—671no—
212#468—671no—
213#466—671no—
214#465—671no—
215#450—671no—
216#449—671no—
217#449—671no—
218#447—671no—
219#408—671no—
220C2×#41C2×D6711yescharacteristic
221C2×#41C2×D6711yescharacteristic, complemented
222C22×D4—671no—
223C2×SD16—671no—
224C2×SD16—671no—
225C2×D8—671no—
226C2×D8—671no—
227C2×M16—671no—
228C22×C8—671no—
229#127—1341no—
230#124—1341no—
231C22⋊Q8—21no—
232C22⋊Q8C2×D6711yescharacteristic
233C22⋊Q8C2×D6711yescharacteristic, complemented
234#121—21no—
235#121—21no—
236#121—21no—
237#121—671no—
238#121—671no—
239#121—671no—
240#121—671no—
241#121C2×D6711yescharacteristic
242#121C2×D6711yescharacteristic, complemented
243#120—1341no—
244#120—1341no—
245#117—671no—
246C2×C4⋊C4C2×D6711yescharacteristic, complemented
247#107—671no—
248#107—671no—
249#106—671no—
250#106—671no—
251#103—671no—
252#103—671no—
253#102—671no—
254#102—671no—
255#102—671no—
256#102—671no—
257#102—671no—
258#102—671no—
259C22⋊C8—671no—
260C22⋊C8—671no—
261C22⋊C8—671no—
262C22⋊C8—671no—
263C24—1341no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268C2×Q8C22⋊D6711yescharacteristic
269C2×Q8C22×D6711yescharacteristic
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—2681no—
274C2×D4—2681no—
275C2×D4—671no—
276C2×D4—671no—
277C2×D4—1341no—
278C2×D4—1341no—
279C2×D4—1341no—
280C2×D4—1341no—
281C2×D4—1341no—
282C2×D4C22⋊D6711yescharacteristic
283C2×D4C22⋊D6711yescharacteristic
284C2×D4C22⋊D6711yescharacteristic
285C2×D4C22×D6711yescharacteristic
286C22×C4—21no—
287C22×C4—21no—
288C22×C4C22×D6711yescharacteristic
289C22×C4C22×D6711yescharacteristic
290C22×C4C22×D6711yescharacteristic
291SD16—2681no—
292SD16—2681no—
293D8—2681no—
294D8—2681no—
295M16—2681no—
296C2×C8—2681no—
297C2×C8—671no—
298C2×C8—671no—
299C2×C8—671no—
300C2×C8—671no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—671no—
305C4⋊C4—671no—
306C4⋊C4C22×D6711yescharacteristic
307C4⋊C4C22×D6711yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—1341no—
313C22⋊C4—1341no—
314C22⋊C4—1341no—
315C22⋊C4—1341no—
316C22⋊C4—1341no—
317C42—1341no—
318C23—21no—
319C23—2682no—
320C23—5361no—
321C23—5361no—
322C23—1341no—
323C23—1341no—
324C23C2×C22⋊D6711yescharacteristic
325C23C2×C22⋊D6711yescharacteristic
326C23C2×C22⋊D6711yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—41no—
330D4—21no—
331D4—21no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—42no—
336D4—81no—
337D4—81no—
338D4—2682no—
339D4—5361no—
340D4—5361no—
341D4—5361no—
342D4—5361no—
343D4—1341no—
344D4—1341no—
345D4—1341no—
346D4—1341no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—81no—
358C2×C4—1341no—
359C2×C4—1341no—
360C2×C4C2×C22⋊D6711yescharacteristic
361C2×C4C2×C22⋊D6711yescharacteristic
362C2×C4C2×C22⋊D6711yescharacteristic
363C2×C4C23×D6711yescharacteristic
364C8—1341no—
365C8—1341no—
366C8—1341no—
367C8—1341no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—2681no—
381C22—2681no—
382C22—2681no—
383C22—2681no—
384C22—2681no—
385C22—2681no—
386C22—5364no—
387C22C22×C22⋊D6711yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—2681no—
395C4—2681no—
396C4#1452408811yescharacteristic
397C4#1452408811yescharacteristic
398C2—41no—
399C2—41no—
400C2—41no—
401C2—81no—
402C2—5361no—
403C2—5361no—
404C2#1686395511yescharacteristic
405C2#1686419911yescharacteristic
406C2#1686420411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#537×C67C2
socleC22×C67C22×D4
hypercenter#537D67
hypocenterC67#3198
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×#41 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #16863960C2semidirect
2. #16863734C2semidirect
3. #16863717C2semidirect
4. #16863714C2semidirect
5. #16863709C2semidirect
6. #16863647C2semidirect
7. #16863644C2semidirect
8. #16863630C2semidirect
9. #16863627C2semidirect
10. #16863546C2semidirect
11. #16863541C2semidirect
12. #16863541C2semidirect
13. #16863529C2semidirect
14. #16863363C2semidirect
15. #537×C67C2semidirect
21. C2×#11761612C22semidirect
22. C22×C67⋊C8C22semidirect
32. #14523976C22semidirect
33. #14523976C22semidirect
34. #14523974C22semidirect
35. #14523974C22semidirect
36. #14523967C22semidirect
37. #14523967C22semidirect
38. #14523964C22semidirect
39. #14523964C22semidirect
46. C22⋊(C67⋊C8)C22semidirect
47. C22⋊(C67⋊C8)C22semidirect
49. C2×#41×C67C22semidirect
52. C22⋊Q8×C67C22semidirect
57. #121×C67C22semidirect
58. C2×C4⋊C4×C67C22semidirect
204. C67#3198semidirect
205. #537D67semidirect
221. C2×#41C2×D67semidirect
233. C22⋊Q8C2×D67semidirect
242. #121C2×D67semidirect
246. C2×C4⋊C4C2×D67semidirect