Finite Group Databasebeta

← #22571297

Subgroups — #22571297

Order 8576.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16863965C211yesmaximal, characteristic, complemented
2#16863786C211yesmaximal, characteristic, complemented
3#16863778C211yesmaximal, characteristic, complemented
4#16863758C211yesmaximal, characteristic, complemented
5#16863744C211yesmaximal, characteristic, complemented
6#16863618C211yesmaximal, characteristic, complemented
7#16863611C211yesmaximal, characteristic, complemented
8#16863596C211yesmaximal, characteristic, complemented
9#16863591C211yesmaximal, characteristic, complemented
10#16863556C211yesmaximal, characteristic, complemented
11#16863546C211yesmaximal, characteristic, complemented
12#16863539C211yesmaximal, characteristic, complemented
13#16863529C211yesmaximal, characteristic, complemented
14#16863457C211yesmaximal, characteristic, complemented
15#542×C67C211yesmaximal, characteristic, complemented
16C2×#11761628C2211yescharacteristic
17C22×D268C2211yescharacteristic
18C2×#11761618C2211yescharacteristic, complemented
19C2×#11761616C2211yescharacteristic, complemented
20C2×#11761615C2211yescharacteristic
21C2×#11761615C2211yescharacteristic
22C2×#11761615C2211yescharacteristic, complemented
23C2×#11761613C2211yescharacteristic
24C2×#11761613C2211yescharacteristic
25C2×#11761613C2211yescharacteristic, complemented
26#14524040C2211yescharacteristic
27#14524040C2211yescharacteristic
28#14524031C2211yescharacteristic
29#14524031C2211yescharacteristic
30#14524018—21no—
31#14524018—21no—
32#14524013—21no—
33#14524013—21no—
34C4⋊(C67⋊C8)C2211yescharacteristic, complemented
35C4⋊(C67⋊C8)C2211yescharacteristic, complemented
36#14523966C2211yescharacteristic, complemented
37#14523966C2211yescharacteristic, complemented
38#14523963C2211yescharacteristic
39#14523963C2211yescharacteristic
40#14523963C2211yescharacteristic
41#14523963C2211yescharacteristic
42#14523963C2211yescharacteristic, complemented
43#14523963C2211yescharacteristic, complemented
44C22⋊(C67⋊C8)C2211yescharacteristic
45C22⋊(C67⋊C8)C2211yescharacteristic
46#14523954C2211yescharacteristic, complemented
47C4×C67⋊C8C2211yescharacteristic, complemented
48C4⋊Q8×C67C2211yescharacteristic, complemented
49#126×C67—21no—
50#124×C67—21no—
51#124×C67C2211yescharacteristic, complemented
52#123×C67—21no—
53C22⋊Q8×C67—21no—
54C22⋊Q8×C67C2211yescharacteristic
55#121×C67C2211yescharacteristic
56C4×Q8×C67C2211yescharacteristic, complemented
57C4×D4×C67C2211yescharacteristic, complemented
58#117×C67C2211yescharacteristic
59C23×D67—21no—
60#11761628—41no—
61#11761628—41no—
62C2×Dic134C2311yescharacteristic
63C2×D268—21no—
64C2×D268—21no—
65C2×D268—21no—
66C2×D268—21no—
67C2×D268—21no—
68C2×D268—41no—
69C2×D268—41no—
70C2×D268C2311yescharacteristic
71C2×D268C2311yescharacteristic
72C2×D268C2311yescharacteristic
73C2×C22⋊D67—21no—
74C2×C4×D67—21no—
75#11761618—41no—
76#11761616—41no—
77#11761615—41no—
78#11761615—41no—
79#11761615—41no—
80#11761613—41no—
81#11761613—41no—
82#11761613—41no—
83C2×C67⋊C8C2311yescharacteristic
84C2×C67⋊C8C2311yescharacteristic
85C2×C67⋊C8C2311yescharacteristic
86C2×C67⋊C8C2311yescharacteristic
87(C2×D67)⋊C4—21no—
88(C2×D67)⋊C4—21no—
89(C2×D67)⋊C4—21no—
90(C2×D67)⋊C4—21no—
91C2×Q8×C67—21no—
92C2×Q8×C67C2311yescharacteristic
93C2×D4×C67—21no—
94C2×D4×C67C2311yescharacteristic
95C22×C4×C67—21no—
96C22×C4×C67C2311yescharacteristic
97C4⋊C4×C67—21no—
98C4⋊C4×C67—21no—
99C4⋊C4×C67—21no—
100C4⋊C4×C67D411yescharacteristic
101C4⋊C4×C67D411yescharacteristic
102C4⋊C4×C67C2311yescharacteristic
103C4⋊C4×C67C2311yescharacteristic
104C22⋊C4×C67—21no—
105C22⋊C4×C67—21no—
106C22⋊C4×C67—21no—
107C22⋊C4×C67—21no—
108C22⋊C4×C67—21no—
109C22⋊C4×C67D411yescharacteristic
110C22⋊C4×C67D411yescharacteristic
111C42×C67—21no—
112C42×C67C2311yescharacteristic
113C42×C67C2311yescharacteristic
114C23×C67—21no—
115C23×C67C2×D411yescharacteristic
116C22×D67—21no—
117C22×D67—21no—
118C22×D67—21no—
119C22×D67—42no—
120C22×D67—81no—
121C22×D67—81no—
122Q8×C67—21no—
123Q8×C67—21no—
124Q8×C67—81no—
125D4×C67—21no—
126D4×C67—21no—
127D4×C67—81no—
128C2×C4×C67—21no—
129C2×C4×C67—21no—
130C2×C4×C67—21no—
131C2×C4×C67—21no—
132C2×C4×C67—21no—
133C2×C4×C67—81no—
134C2×C4×C67C2×D411yescharacteristic
135C2×C4×C67C2×D411yescharacteristic
136C2×C4×C67C2×D411yescharacteristic
137C2×C4×C67C2×D411yescharacteristic
138C2×C4×C67C2×D411yescharacteristic
139C2×C4×C67C2411yescharacteristic
140C22⋊D67—42no—
141C2×Dic67—21no—
142D268—21no—
143D268—21no—
144D268—21no—
145D268—21no—
146D268—21no—
147D268—21no—
148D268—42no—
149D268—81no—
150D268—81no—
151D268—81no—
152D268—81no—
153C4×D67—42no—
154Dic134—21no—
155Dic134—21no—
156C67⋊C8—21no—
157C67⋊C8—21no—
158C67⋊C8—21no—
159C67⋊C8—21no—
160C22×C67—21no—
161C22×C67—21no—
162C22×C67—21no—
163C22×C67—41no—
164C22×C67—41no—
165C22×C67—41no—
166C22×C67C22×D411yescharacteristic
167C2×D67—41no—
168C2×D67—41no—
169C2×D67—41no—
170C2×D67—41no—
171C2×D67—41no—
172C2×D67—41no—
173C2×D67—41no—
174C2×D67—41no—
175C2×D67—41no—
176C2×D67—84no—
177C4×C67—21no—
178C4×C67—21no—
179C4×C67—21no—
180C4×C67—21no—
181C4×C67—21no—
182C4×C67—41no—
183C4×C67—41no—
184C4×C67—41no—
185C4×C67H(5,𝔽2)11yescharacteristic
186C4×C67H(5,𝔽2)11yescharacteristic
187Dic67—41no—
188C2×C67—41no—
189C2×C67—81no—
190C2×C67#53511yescharacteristic
191C2×C67#57611yescharacteristic
192C2×C67#57711yescharacteristic
193D67—81no—
194D67—81no—
195D67—81no—
196#3224—671nomaximal, Sylow
197C67#322411yescharacteristic, complemented, Sylow
198#542D6711yescharacteristic, complemented
199#535—671no—
200#496—671no—
201#495—671no—
202#489—671no—
203#486—671no—
204#464—671no—
205#463—671no—
206#460—671no—
207#459—671no—
208#452—671no—
209#450—671no—
210#449—671no—
211#447—671no—
212#433—671no—
213C2×#41—671no—
214C22×D4—671no—
215C2×Q16—671no—
216C2×SD16—671no—
217C2×SD16—671no—
218C2×SD16—671no—
219C2×SD16—671no—
220C2×D8—671no—
221C2×D8—671no—
222C2×D8—671no—
223C4⋊Q8C2×D6711yescharacteristic, complemented
224#127—671no—
225#127—671no—
226#126—21no—
227#124—21no—
228#124—671no—
229#124—671no—
230#124C2×D6711yescharacteristic, complemented
231#123—21no—
232C22⋊Q8—21no—
233C22⋊Q8C2×D6711yescharacteristic
234#121—1341no—
235#121—1341no—
236#121C2×D6711yescharacteristic
237#120—1341no—
238#120—1341no—
239C4×Q8C2×D6711yescharacteristic, complemented
240C4×D4C2×D6711yescharacteristic, complemented
241#117C2×D6711yescharacteristic
242C4⋊C8—671no—
243C4⋊C8—671no—
244#103—671no—
245#103—671no—
246#102—671no—
247#102—671no—
248#102—671no—
249#102—671no—
250#102—671no—
251#102—671no—
252C22⋊C8—671no—
253C22⋊C8—671no—
254#97—671no—
255C4×C8—671no—
256C24—1341no—
257#41—2681no—
258#41—2681no—
259C2×Q8—21no—
260C2×Q8—671no—
261C2×Q8C22×D6711yescharacteristic
262C2×D4—21no—
263C2×D4—2681no—
264C2×D4—2681no—
265C2×D4—671no—
266C2×D4—671no—
267C2×D4—671no—
268C2×D4—1341no—
269C2×D4—1341no—
270C2×D4—1341no—
271C2×D4—1341no—
272C2×D4—1341no—
273C2×D4—1341no—
274C2×D4C22×D6711yescharacteristic
275C22×C4—21no—
276C22×C4—1341no—
277C22×C4C22×D6711yescharacteristic
278Q16—2681no—
279SD16—2681no—
280SD16—2681no—
281SD16—2681no—
282SD16—2681no—
283D8—2681no—
284D8—2681no—
285D8—2681no—
286C2×C8—671no—
287C2×C8—671no—
288C2×C8—671no—
289C2×C8—671no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4C22⋊D6711yescharacteristic
294C4⋊C4C22⋊D6711yescharacteristic
295C4⋊C4C22×D6711yescharacteristic
296C4⋊C4C22×D6711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—1341no—
303C22⋊C4—1341no—
304C22⋊C4—1341no—
305C22⋊C4—1341no—
306C22⋊C4C22⋊D6711yescharacteristic
307C22⋊C4C22⋊D6711yescharacteristic
308C42—21no—
309C42C22×D6711yescharacteristic
310C42C22×D6711yescharacteristic
311C23—21no—
312C23—2682no—
313C23—5361no—
314C23—5361no—
315C23—1341no—
316C23—1341no—
317C23—1341no—
318C23C2×C22⋊D6711yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—81no—
322Q8—1341no—
323Q8—1341no—
324D4—21no—
325D4—21no—
326D4—81no—
327D4—2682no—
328D4—2682no—
329D4—5361no—
330D4—5361no—
331D4—5361no—
332D4—5361no—
333D4—1341no—
334D4—1341no—
335D4—1341no—
336D4—1341no—
337D4—1341no—
338D4—1341no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—81no—
345C2×C4—2682no—
346C2×C4—1341no—
347C2×C4C2×C22⋊D6711yescharacteristic
348C2×C4C2×C22⋊D6711yescharacteristic
349C2×C4C2×C22⋊D6711yescharacteristic
350C2×C4C2×C22⋊D6711yescharacteristic
351C2×C4C2×C22⋊D6711yescharacteristic
352C2×C4C23×D6711yescharacteristic
353C8—1341no—
354C8—1341no—
355C8—1341no—
356C8—1341no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—2681no—
364C22—2681no—
365C22—2681no—
366C22—2681no—
367C22—2681no—
368C22—2681no—
369C22—2681no—
370C22—2681no—
371C22—2681no—
372C22—5364no—
373C22C22×C22⋊D6711yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—2681no—
383C4#1452408911yescharacteristic
384C4#1452408911yescharacteristic
385C2—41no—
386C2—81no—
387C2—5361no—
388C2—5361no—
389C2—5361no—
390C2#1686396011yescharacteristic
391C2#1686419911yescharacteristic
392C2#1686420411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#542×C67C2
socleC22×C67C22×D4
hypercenter#542D67
hypocenterC67#3224
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #16863965C2semidirect
2. #16863786C2semidirect
3. #16863778C2semidirect
4. #16863758C2semidirect
5. #16863744C2semidirect
6. #16863618C2semidirect
7. #16863611C2semidirect
8. #16863596C2semidirect
9. #16863591C2semidirect
10. #16863556C2semidirect
11. #16863546C2semidirect
12. #16863539C2semidirect
13. #16863529C2semidirect
14. #16863457C2semidirect
15. #542×C67C2semidirect
18. C2×#11761618C22semidirect
19. C2×#11761616C22semidirect
22. C2×#11761615C22semidirect
25. C2×#11761613C22semidirect
34. C4⋊(C67⋊C8)C22semidirect
35. C4⋊(C67⋊C8)C22semidirect
36. #14523966C22semidirect
37. #14523966C22semidirect
42. #14523963C22semidirect
43. #14523963C22semidirect
46. #14523954C22semidirect
47. C4×C67⋊C8C22semidirect
48. C4⋊Q8×C67C22semidirect
51. #124×C67C22semidirect
56. C4×Q8×C67C22semidirect
57. C4×D4×C67C22semidirect
197. C67#3224semidirect
198. #542D67semidirect
223. C4⋊Q8C2×D67semidirect
230. #124C2×D67semidirect
239. C4×Q8C2×D67semidirect
240. C4×D4C2×D67semidirect