Finite Group Databasebeta

← #22570396

Subgroups — #22570396

Order 8576.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864019C211yesmaximal, characteristic, complemented
2#16863870C211yesmaximal, characteristic, complemented
3#16863720C211yesmaximal, characteristic, complemented
4#16863685C211yesmaximal, characteristic, complemented
5#16863675C211yesmaximal, characteristic, complemented
6#16863614C211yesmaximal, characteristic, complemented
7#16863614C211yesmaximal, characteristic, complemented
8#16863546C211yesmaximal, characteristic, complemented
9#16863532C211yesmaximal, characteristic
10#16863507C211yesmaximal, characteristic, complemented
11#16863499C211yesmaximal, characteristic, complemented
12#16863481C211yesmaximal, characteristic, complemented
13#16863478C211yesmaximal, characteristic, complemented
14#16863454C211yesmaximal, characteristic, complemented
15#485×C67C211yesmaximal, characteristic, complemented
16C22×D268C2211yescharacteristic
17C2×#11761615C2211yescharacteristic
18C2×#11761615C2211yescharacteristic
19C2×D536C2211yescharacteristic
20C2×D536C2211yescharacteristic
21#14524042C2211yescharacteristic, complemented
22#14524038—21no—
23#14524032C2211yescharacteristic, complemented
24#14524029—21no—
25#14524025—21no—
26#14524023—21no—
27#14524017C2211yescharacteristic
28Q8×Dic67C2211yescharacteristic, complemented
29C4×D268—21no—
30C4×D268—21no—
31#14524005C2211yescharacteristic
32#14524005C2211yescharacteristic
33#14524005C2211yescharacteristic
34#14524005C2211yescharacteristic
35#14524005C2211yescharacteristic, complemented
36#14524002C2211yescharacteristic
37C22⋊C4×D67—21no—
38C22⋊C4×D67—21no—
39#14523976C2211yescharacteristic, complemented
40#14523974C2211yescharacteristic, complemented
41Dic67⋊C8C2211yescharacteristic, complemented
42Dic67⋊C8C2211yescharacteristic, complemented
43#14523965C2211yescharacteristic
44#14523965C2211yescharacteristic
45#14523963C2211yescharacteristic
46#14523963C2211yescharacteristic
47#14523963C2211yescharacteristic, complemented
48#14523963C2211yescharacteristic, complemented
49C22⋊(C67⋊C8)C2211yescharacteristic
50#14523953C2211yescharacteristic, complemented
51C8×Dic67C2211yescharacteristic, complemented
52C22⋊Q8×C67C2211yescharacteristic
53#117×C67C2211yescharacteristic
54#107×C67C2211yescharacteristic, complemented
55#106×C67C2211yescharacteristic, complemented
56#103×C67C2211yescharacteristic, complemented
57#103×C67C2211yescharacteristic, complemented
58C22⋊C8×C67C2211yescharacteristic
59C23×D67—21no—
60C2×Dic134—21no—
61C2×D268—21no—
62C2×D268—22no—
63C2×D268—22no—
64C2×D268C2311yescharacteristic
65C2×D268C2311yescharacteristic
66C2×D268C2311yescharacteristic
67C2×C22⋊D67—21no—
68C2×C4×D67—21no—
69C2×C4×D67—21no—
70C2×C4×D67—21no—
71C2×C4×D67—21no—
72C2×C4×D67—21no—
73#11761615—22no—
74#11761615—22no—
75D536—22no—
76D536—22no—
77C2×C67⋊C8C2311yescharacteristic
78C2×C67⋊C8C2311yescharacteristic
79C4⋊Dic67—21no—
80C4⋊Dic67D411yescharacteristic
81C4⋊Dic67D411yescharacteristic
82C4⋊Dic67C2311yescharacteristic
83Dic67⋊C4—21no—
84C22⋊Dic67D411yescharacteristic
85C22⋊Dic67D411yescharacteristic
86(C2×D67)⋊C4—21no—
87(C2×D67)⋊C4—21no—
88(C2×D67)⋊C4—21no—
89(C2×D67)⋊C4—21no—
90(C2×D67)⋊C4—21no—
91(C2×D67)⋊C4—21no—
92(C2×D67)⋊C4—21no—
93(C2×D67)⋊C4—21no—
94C4×Dic67—21no—
95C4×Dic67C2311yescharacteristic
96C4×Dic67C2311yescharacteristic
97C2×Q8×C67C2311yescharacteristic
98C22×C4×C67C2311yescharacteristic
99C2×C8×C67C2311yescharacteristic
100C2×C8×C67C2311yescharacteristic
101C4⋊C4×C67—21no—
102C4⋊C4×C67C2311yescharacteristic
103C4⋊C4×C67C2311yescharacteristic
104C4⋊C4×C67C2311yescharacteristic
105C22⋊C4×C67—21no—
106C22⋊C4×C67—21no—
107C42×C67—21no—
108C23×C67C2×D411yescharacteristic
109C22×D67—21no—
110C22×D67—21no—
111C22×D67—24no—
112C22×D67—21no—
113C22×D67—81no—
114C22×D67—81no—
115Q8×C67—21no—
116Q8×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67C2×D411yescharacteristic
124C2×C4×C67C2411yescharacteristic
125C22⋊D67—81no—
126C2×Dic67—21no—
127C2×Dic67C2×D411yescharacteristic
128C2×Dic67C2×D411yescharacteristic
129C2×Dic67C2×D411yescharacteristic
130C2×Dic67C2×D411yescharacteristic
131D268—24no—
132D268—21no—
133D268—21no—
134D268—21no—
135D268—21no—
136D268#4112yes—
137D268#4112yes—
138C4×D67—81no—
139C4×D67—81no—
140C4×D67—81no—
141C4×D67—81no—
142C4×D67—81no—
143Dic134—81no—
144C8×C67—21no—
145C8×C67—21no—
146C67⋊C8—21no—
147C67⋊C8—21no—
148C22×C67—21no—
149C22×C67—21no—
150C22×C67—21no—
151C22×C67C22×D411yescharacteristic
152C2×D67—22no—
153C2×D67—22no—
154C2×D67—41no—
155C2×D67—41no—
156C2×D67—41no—
157C2×D67—41no—
158C2×D67—41no—
159C2×D67—41no—
160C2×D67—41no—
161C2×D67—84no—
162C4×C67—21no—
163C4×C67—41no—
164C4×C67—41no—
165C4×C67—41no—
166C4×C67—41no—
167C4×C67C2×#4111yescharacteristic
168C4×C67C2×#4111yescharacteristic
169Dic67—21no—
170Dic67—21no—
171Dic67—21no—
172Dic67—21no—
173Dic67—41no—
174C2×C67—41no—
175C2×C67#52511yescharacteristic
176C2×C67#57611yescharacteristic
177C2×C67#57711yescharacteristic
178D67—42no—
179D67—42no—
180D67—81no—
181#3118—671nomaximal, Sylow
182C67#311811yescharacteristic, complemented, Sylow
183#542—671no—
184#518—671no—
185#485D6711yescharacteristic, complemented
186#482—671no—
187#476—671no—
188#474—671no—
189#463—671no—
190#463—671no—
191#450—671no—
192#447—671no—
193#442—671no—
194#440—671no—
195C4×SD16—671no—
196C4×D8—671no—
197#433—671no—
198C22×D4—671no—
199C2×SD16—671no—
200C2×SD16—671no—
201C2×D8—671no—
202C2×D8—671no—
203C4⋊Q8—671no—
204#126—1341no—
205#124—671no—
206#124—1341no—
207#123—1341no—
208C22⋊Q8—1341no—
209C22⋊Q8C2×D6711yescharacteristic
210#121—671no—
211C4×Q8—671no—
212C4×D4—671no—
213C4×D4—671no—
214C4×D4—671no—
215C4×D4—671no—
216C4×D4—671no—
217C4×D4—1341no—
218C4×D4—1341no—
219#117—671no—
220#117C2×D6711yescharacteristic
221C2×C22⋊C4—1341no—
222C2×C22⋊C4—1341no—
223#107—671no—
224#107C2×D6711yescharacteristic, complemented
225#106—671no—
226#106C2×D6711yescharacteristic, complemented
227C4⋊C8—671no—
228C4⋊C8—671no—
229#103C2×D6711yescharacteristic, complemented
230#103C2×D6711yescharacteristic, complemented
231#102—671no—
232#102—671no—
233#102—671no—
234#102—671no—
235#102—671no—
236#102—671no—
237C22⋊C8—671no—
238C22⋊C8C2×D6711yescharacteristic
239#97—671no—
240C4×C8—671no—
241C24—1341no—
242C2×Q8—1341no—
243C2×Q8C22×D6711yescharacteristic
244C2×D4—671no—
245C2×D4—671no—
246C2×D4—671no—
247C2×D4—1342no—
248C2×D4—1342no—
249C2×D4—1341no—
250C2×D4—1341no—
251C22×C4—1341no—
252C22×C4—1341no—
253C22×C4—1341no—
254C22×C4—1341no—
255C22×C4—1341no—
256C22×C4C22×D6711yescharacteristic
257SD16—1342no—
258SD16—1342no—
259D8—1342no—
260D8—1342no—
261C2×C8—671no—
262C2×C8—671no—
263C2×C8C22×D6711yescharacteristic
264C2×C8C22×D6711yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—671no—
267C4⋊C4—671no—
268C4⋊C4—671no—
269C4⋊C4—1341no—
270C4⋊C4—1341no—
271C4⋊C4C22×D6711yescharacteristic
272C4⋊C4C22×D6711yescharacteristic
273C4⋊C4C22×D6711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—671no—
277C22⋊C4—671no—
278C22⋊C4—1341no—
279C22⋊C4—1341no—
280C22⋊C4—1341no—
281C22⋊C4—1341no—
282C22⋊C4—1341no—
283C22⋊C4—1341no—
284C22⋊C4—1341no—
285C22⋊C4—1341no—
286C42—21no—
287C42—671no—
288C42—671no—
289C42—1341no—
290C23—5361no—
291C23—5361no—
292C23—1344no—
293C23—1341no—
294C23—1341no—
295C23—1341no—
296C23D4×D6711yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—5361no—
300D4—5361no—
301D4—672no—
302D4—672no—
303D4—1344no—
304D4—1341no—
305D4—1341no—
306D4—1341no—
307D4—1341no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—5361no—
315C2×C4—5361no—
316C2×C4—5361no—
317C2×C4—5361no—
318C2×C4—5361no—
319C2×C4—671no—
320C2×C4—671no—
321C2×C4—671no—
322C2×C4—671no—
323C2×C4—1341no—
324C2×C4D4×D6711yescharacteristic
325C2×C4C23×D6711yescharacteristic
326C8—21no—
327C8—21no—
328C8—1341no—
329C8—1341no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—2681no—
334C22—2681no—
335C22—2681no—
336C22—2681no—
337C22—2681no—
338C22—2681no—
339C22—2681no—
340C22—5364no—
341C22—1342no—
342C22—1342no—
343C22C2×D4×D6711yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—2681no—
350C4—1341no—
351C4—1341no—
352C4—1341no—
353C4—1341no—
354C4#41×D6711yescharacteristic
355C4#41×D6711yescharacteristic
356C2—41no—
357C2—2682no—
358C2—2682no—
359C2—5361no—
360C2#1686390411yescharacteristic
361C2#1686419811yescharacteristic
362C2#1686420511yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#485×C67C2
socleC22×C67C22×D4
hypercenter#485D67
hypocenterC67#3118
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #16864019C2semidirect
2. #16863870C2semidirect
3. #16863720C2semidirect
4. #16863685C2semidirect
5. #16863675C2semidirect
6. #16863614C2semidirect
7. #16863614C2semidirect
8. #16863546C2semidirect
10. #16863507C2semidirect
11. #16863499C2semidirect
12. #16863481C2semidirect
13. #16863478C2semidirect
14. #16863454C2semidirect
15. #485×C67C2semidirect
21. #14524042C22semidirect
23. #14524032C22semidirect
28. Q8×Dic67C22semidirect
35. #14524005C22semidirect
39. #14523976C22semidirect
40. #14523974C22semidirect
41. Dic67⋊C8C22semidirect
42. Dic67⋊C8C22semidirect
47. #14523963C22semidirect
48. #14523963C22semidirect
50. #14523953C22semidirect
51. C8×Dic67C22semidirect
54. #107×C67C22semidirect
55. #106×C67C22semidirect
56. #103×C67C22semidirect
57. #103×C67C22semidirect
182. C67#3118semidirect
185. #485D67semidirect
224. #107C2×D67semidirect
226. #106C2×D67semidirect
229. #103C2×D67semidirect
230. #103C2×D67semidirect