Finite Group Databasebeta

← C67×#3318 #22557749

Subgroups — C67×#3318 #22557749

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#553×C67C211yesmaximal, characteristic, complemented
2D42×C67C211yesmaximal, characteristic, complemented
3#490×C67C211yesmaximal, characteristic, complemented
4#482×C67C211yesmaximal, characteristic, complemented
5#481×C67C211yesmaximal, characteristic, complemented
6#478×C67C211yesmaximal, characteristic, complemented
7#469×C67C211yesmaximal, characteristic, complemented
8#466×C67C211yesmaximal, characteristic, complemented
9#459×C67C211yesmaximal, characteristic, complemented
10#447×C67C211yesmaximal, characteristic, complemented
11#447×C67C211yesmaximal, characteristic, complemented
12#442×C67C211yesmaximal, characteristic, complemented
13#435×C67C211yesmaximal, characteristic, complemented
14#418×C67C211yesmaximal, characteristic, complemented
15C2×#102×C67C211yesmaximal, characteristic, complemented
16C22×D4×C67—21no—
17C22×D4×C67C2211yescharacteristic
18C22×D4×C67C2211yescharacteristic
19C2×D8×C67C2211yescharacteristic
20C2×D8×C67C2211yescharacteristic, complemented
21C2×D8×C67C2211yescharacteristic, complemented
22C2×M16×C67C2211yescharacteristic, complemented
23C22×C8×C67C2211yescharacteristic, complemented
24#127×C67C2211yescharacteristic
25#126×C67—21no—
26#125×C67C2211yescharacteristic, complemented
27#123×C67—21no—
28#121×C67—21no—
29#121×C67—21no—
30#121×C67C2211yescharacteristic
31#121×C67C2211yescharacteristic
32#121×C67C2211yescharacteristic, complemented
33#121×C67C2211yescharacteristic, complemented
34#120×C67—21no—
35#120×C67—21no—
36C4×D4×C67—21no—
37C4×D4×C67C2211yescharacteristic
38C4×D4×C67C2211yescharacteristic
39C4×D4×C67C2211yescharacteristic, complemented
40#117×C67C2211yescharacteristic, complemented
41C2×C4⋊C4×C67C2211yescharacteristic, complemented
42#107×C67C2211yescharacteristic, complemented
43#106×C67C2211yescharacteristic, complemented
44#106×C67C2211yescharacteristic, complemented
45C4⋊C8×C67C2211yescharacteristic, complemented
46#102×C67C2211yescharacteristic
47#102×C67C2211yescharacteristic
48#102×C67C2211yescharacteristic
49#102×C67C2211yescharacteristic, complemented
50#102×C67C2211yescharacteristic, complemented
51#102×C67C2211yescharacteristic, complemented
52#102×C67C2211yescharacteristic, complemented
53#102×C67C2211yescharacteristic, complemented
54#102×C67C2211yescharacteristic, complemented
55#102×C67C2211yescharacteristic, complemented
56C22⋊C8×C67C2211yescharacteristic, complemented
57C22⋊C8×C67C2211yescharacteristic, complemented
58#97×C67C2211yescharacteristic, complemented
59C24×C67—21no—
60C24×C67—21no—
61C2×D4×C67—22no—
62C2×D4×C67—22no—
63C2×D4×C67—21no—
64C2×D4×C67—21no—
65C2×D4×C67—21no—
66C2×D4×C67—21no—
67C2×D4×C67—21no—
68C2×D4×C67—21no—
69C2×D4×C67—21no—
70C2×D4×C67—21no—
71C2×D4×C67—41no—
72C2×D4×C67—41no—
73C2×D4×C67—44no—
74C2×D4×C67D411yescharacteristic
75C2×D4×C67C2311yescharacteristic
76C2×D4×C67C2311yescharacteristic
77C2×D4×C67C2311yescharacteristic
78C2×D4×C67C2311yescharacteristic
79C22×C4×C67—21no—
80C22×C4×C67—21no—
81C22×C4×C67—21no—
82C22×C4×C67C2311yescharacteristic
83C22×C4×C67C2311yescharacteristic
84D8×C67—22no—
85D8×C67—41no—
86D8×C67—41no—
87M16×C67—41no—
88C2×C8×C67—41no—
89C2×C8×C67C2311yescharacteristic
90C2×C8×C67C2311yescharacteristic
91C2×C8×C67C2311yescharacteristic
92C2×C8×C67C2311yescharacteristic
93C4⋊C4×C67—21no—
94C4⋊C4×C67—21no—
95C4⋊C4×C67—21no—
96C4⋊C4×C67D411yescharacteristic
97C4⋊C4×C67C2311yescharacteristic
98C4⋊C4×C67C2311yescharacteristic
99C4⋊C4×C67C2311yescharacteristic
100C4⋊C4×C67C2311yescharacteristic
101C22⋊C4×C67—21no—
102C22⋊C4×C67—21no—
103C22⋊C4×C67—21no—
104C22⋊C4×C67—21no—
105C22⋊C4×C67—21no—
106C22⋊C4×C67—21no—
107C22⋊C4×C67—21no—
108C22⋊C4×C67D411yescharacteristic
109C22⋊C4×C67D411yescharacteristic
110C42×C67—21no—
111C42×C67C2311yescharacteristic
112C23×C67—21no—
113C23×C67—21no—
114C23×C67—21no—
115C23×C67—21no—
116C23×C67—42no—
117C23×C67—42no—
118C23×C67—42no—
119C23×C67—42no—
120C23×C67—81no—
121C23×C67—81no—
122C23×C67C2×D411yescharacteristic
123C23×C67C2×D411yescharacteristic
124D4×C67—22no—
125D4×C67—21no—
126D4×C67—21no—
127D4×C67—21no—
128D4×C67—21no—
129D4×C67—21no—
130D4×C67—21no—
131D4×C67—21no—
132D4×C67—42no—
133D4×C67—42no—
134D4×C67—42no—
135D4×C67—42no—
136D4×C67—42no—
137D4×C67—42no—
138D4×C67—81no—
139D4×C67—81no—
140D4×C67#4112yes—
141C2×C4×C67—21no—
142C2×C4×C67—21no—
143C2×C4×C67—21no—
144C2×C4×C67—21no—
145C2×C4×C67—21no—
146C2×C4×C67—21no—
147C2×C4×C67—21no—
148C2×C4×C67—21no—
149C2×C4×C67—42no—
150C2×C4×C67—81no—
151C2×C4×C67—81no—
152C2×C4×C67C2×D411yescharacteristic
153C2×C4×C67C2×D411yescharacteristic
154C2×C4×C67C2×D411yescharacteristic
155C2×C4×C67C2×D411yescharacteristic
156C2×C4×C67C2411yescharacteristic
157C8×C67—21no—
158C8×C67—21no—
159C8×C67—21no—
160C8×C67—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—41no—
176C22×C67—41no—
177C22×C67—41no—
178C22×C67—84no—
179C22×C67—84no—
180C22×C67AGL(1,ℤ/8)12yes—
181C22×C67C22×D411yescharacteristic
182C4×C67—21no—
183C4×C67—21no—
184C4×C67—21no—
185C4×C67—21no—
186C4×C67—41no—
187C4×C67—41no—
188C4×C67—41no—
189C4×C67—41no—
190C4×C67C2×#4111yescharacteristic
191C4×C67H(5,𝔽2)11yescharacteristic
192C2×C67—22no—
193C2×C67—41no—
194C2×C67—42no—
195C2×C67—81no—
196C2×C67—81no—
197C2×C67—81no—
198C2×C67#54611yescharacteristic
199C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
200C2×C67#57611yescharacteristic
201#3318C6711yesmaximal, characteristic, direct product factor, Sylow
202C67#331811yescharacteristic, direct product factor, Sylow
203#553C2×C6711yescharacteristic, complemented
204D42C2×C6711yescharacteristic, complemented
205#490C2×C6711yescharacteristic, complemented
206#482C2×C6711yescharacteristic, complemented
207#481C2×C6711yescharacteristic, complemented
208#478C2×C6711yescharacteristic, complemented
209#469C2×C6711yescharacteristic, complemented
210#466C2×C6711yescharacteristic, complemented
211#459C2×C6711yescharacteristic, complemented
212#447C2×C6711yescharacteristic, complemented
213#447C2×C6711yescharacteristic, complemented
214#442C2×C6711yescharacteristic, complemented
215#435C2×C6711yescharacteristic, complemented
216#418C2×C6711yescharacteristic, complemented
217C2×#102C2×C6711yescharacteristic, complemented
218C22×D4—21no—
219C22×D4C22×C6711yescharacteristic
220C22×D4C22×C6711yescharacteristic
221C2×D8C22×C6711yescharacteristic
222C2×D8C22×C6711yescharacteristic, complemented
223C2×D8C22×C6711yescharacteristic, complemented
224C2×M16C22×C6711yescharacteristic, complemented
225C22×C8C22×C6711yescharacteristic, complemented
226#127C22×C6711yescharacteristic
227#126—21no—
228#125C22×C6711yescharacteristic, complemented
229#123—21no—
230#121—21no—
231#121—21no—
232#121C22×C6711yescharacteristic
233#121C22×C6711yescharacteristic
234#121C22×C6711yescharacteristic, complemented
235#121C22×C6711yescharacteristic, complemented
236#120—21no—
237#120—21no—
238C4×D4—21no—
239C4×D4C22×C6711yescharacteristic
240C4×D4C22×C6711yescharacteristic
241C4×D4C22×C6711yescharacteristic, complemented
242#117C22×C6711yescharacteristic, complemented
243C2×C4⋊C4C22×C6711yescharacteristic, complemented
244#107C22×C6711yescharacteristic, complemented
245#106C22×C6711yescharacteristic, complemented
246#106C22×C6711yescharacteristic, complemented
247C4⋊C8C22×C6711yescharacteristic, complemented
248#102C22×C6711yescharacteristic
249#102C22×C6711yescharacteristic
250#102C22×C6711yescharacteristic
251#102C22×C6711yescharacteristic, complemented
252#102C22×C6711yescharacteristic, complemented
253#102C22×C6711yescharacteristic, complemented
254#102C22×C6711yescharacteristic, complemented
255#102C22×C6711yescharacteristic, complemented
256#102C22×C6711yescharacteristic, complemented
257#102C22×C6711yescharacteristic, complemented
258C22⋊C8C22×C6711yescharacteristic, complemented
259C22⋊C8C22×C6711yescharacteristic, complemented
260#97C22×C6711yescharacteristic, complemented
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—21no—
272C2×D4—21no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—44no—
276C2×D4D4×C6711yescharacteristic
277C2×D4C23×C6711yescharacteristic
278C2×D4C23×C6711yescharacteristic
279C2×D4C23×C6711yescharacteristic
280C2×D4C23×C6711yescharacteristic
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4C23×C6711yescharacteristic
285C22×C4C23×C6711yescharacteristic
286D8—22no—
287D8—41no—
288D8—41no—
289M16—41no—
290C2×C8—41no—
291C2×C8C23×C6711yescharacteristic
292C2×C8C23×C6711yescharacteristic
293C2×C8C23×C6711yescharacteristic
294C2×C8C23×C6711yescharacteristic
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4D4×C6711yescharacteristic
299C4⋊C4C23×C6711yescharacteristic
300C4⋊C4C23×C6711yescharacteristic
301C4⋊C4C23×C6711yescharacteristic
302C4⋊C4C23×C6711yescharacteristic
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4D4×C6711yescharacteristic
311C22⋊C4D4×C6711yescharacteristic
312C42—21no—
313C42C23×C6711yescharacteristic
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—81no—
323C23—81no—
324C23C2×D4×C6711yescharacteristic
325C23C2×D4×C6711yescharacteristic
326D4—22no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—42no—
335D4—42no—
336D4—42no—
337D4—42no—
338D4—42no—
339D4—42no—
340D4—81no—
341D4—81no—
342D4#41×C6712yes—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—42no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4C2×D4×C6711yescharacteristic
355C2×C4C2×D4×C6711yescharacteristic
356C2×C4C2×D4×C6711yescharacteristic
357C2×C4C2×D4×C6711yescharacteristic
358C2×C4C24×C6711yescharacteristic
359C8—21no—
360C8—21no—
361C8—21no—
362C8—21no—
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—41no—
378C22—41no—
379C22—41no—
380C22—84no—
381C22—84no—
382C22AGL(1,ℤ/8)×C6712yes—
383C22C22×D4×C6711yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4C2×#41×C6711yescharacteristic
393C4H(5,𝔽2)×C6711yescharacteristic
394C2—22no—
395C2—41no—
396C2—42no—
397C2—81no—
398C2—81no—
399C2—81no—
400C2#546×C6711yescharacteristic
401C2C2×AGL(1,ℤ/8)×C6711yescharacteristic
402C2#576×C6711yescharacteristic
subgroupG/H
center, socleC22×C67C22×D4
commutator, Frattini, cosocleC2×C4C24×C67
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C67 ◁ C4×D4×C67 ◁ G

Complements

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

NH ≅ G/Ntype
1. #553×C67C2semidirect
2. D42×C67C2semidirect
3. #490×C67C2semidirect
4. #482×C67C2semidirect
5. #481×C67C2semidirect
6. #478×C67C2semidirect
7. #469×C67C2semidirect
8. #466×C67C2semidirect
9. #459×C67C2semidirect
10. #447×C67C2semidirect
11. #447×C67C2semidirect
12. #442×C67C2semidirect
13. #435×C67C2semidirect
14. #418×C67C2semidirect
15. C2×#102×C67C2semidirect
20. C2×D8×C67C22semidirect
21. C2×D8×C67C22semidirect
22. C2×M16×C67C22semidirect
23. C22×C8×C67C22semidirect
26. #125×C67C22semidirect
32. #121×C67C22semidirect
33. #121×C67C22semidirect
39. C4×D4×C67C22semidirect
40. #117×C67C22semidirect
41. C2×C4⋊C4×C67C22semidirect
42. #107×C67C22semidirect
43. #106×C67C22semidirect
44. #106×C67C22semidirect
45. C4⋊C8×C67C22semidirect
49. #102×C67C22semidirect
50. #102×C67C22semidirect
51. #102×C67C22semidirect
52. #102×C67C22semidirect
53. #102×C67C22semidirect
54. #102×C67C22semidirect
55. #102×C67C22semidirect
56. C22⋊C8×C67C22semidirect
57. C22⋊C8×C67C22semidirect
58. #97×C67C22semidirect
201. #3318C67direct (N × H)
202. C67#3318direct (N × H)
203. #553C2×C67semidirect
204. D42C2×C67semidirect
205. #490C2×C67semidirect
206. #482C2×C67semidirect
207. #481C2×C67semidirect
208. #478C2×C67semidirect
209. #469C2×C67semidirect
210. #466C2×C67semidirect
211. #459C2×C67semidirect
212. #447C2×C67semidirect
213. #447C2×C67semidirect
214. #442C2×C67semidirect
215. #435C2×C67semidirect
216. #418C2×C67semidirect
217. C2×#102C2×C67semidirect
222. C2×D8C22×C67semidirect
223. C2×D8C22×C67semidirect
224. C2×M16C22×C67semidirect
225. C22×C8C22×C67semidirect
228. #125C22×C67semidirect
234. #121C22×C67semidirect
235. #121C22×C67semidirect
241. C4×D4C22×C67semidirect
242. #117C22×C67semidirect
243. C2×C4⋊C4C22×C67semidirect
244. #107C22×C67semidirect
245. #106C22×C67semidirect
246. #106C22×C67semidirect
247. C4⋊C8C22×C67semidirect
251. #102C22×C67semidirect
252. #102C22×C67semidirect
253. #102C22×C67semidirect
254. #102C22×C67semidirect
255. #102C22×C67semidirect
256. #102C22×C67semidirect
257. #102C22×C67semidirect
258. C22⋊C8C22×C67semidirect
259. C22⋊C8C22×C67semidirect
260. #97C22×C67semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#553×C672
D42×C672
#490×C672
#482×C672
#481×C672
#478×C672
#469×C672
#466×C672
#459×C672
#447×C672
#442×C672
#435×C672
#418×C672
C2×#102×C672
#331867