Finite Group Databasebeta

← #22571063

Subgroups — #22571063

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#16863974C211yesmaximal, characteristic, complemented
2#16863960C211yesmaximal, characteristic, complemented
3#16863738C211yesmaximal, characteristic, complemented
4#16863721C211yesmaximal, characteristic, complemented
5#16863715C211yesmaximal, characteristic, complemented
6#16863710C211yesmaximal, characteristic, complemented
7#16863649C211yesmaximal, characteristic, complemented
8#16863642C211yesmaximal, characteristic, complemented
9#16863632C211yesmaximal, characteristic, complemented
10#16863625C211yesmaximal, characteristic, complemented
11#16863549C211yesmaximal, characteristic, complemented
12#16863543C211yesmaximal, characteristic, complemented
13#16863543C211yesmaximal, characteristic, complemented
14#16863532C211yesmaximal, characteristic, complemented
15#408×C67C211yesmaximal, characteristic, complemented
16C2×#11761628C2211yescharacteristic, complemented
17C22×D268C2211yescharacteristic
18C2×#11761617C2211yescharacteristic
19C2×#11761617C2211yescharacteristic
20C2×D536C2211yescharacteristic
21C2×D536C2211yescharacteristic
22#14524039—21no—
23#14524032—21no—
24#14524022C2211yescharacteristic
25#14524022C2211yescharacteristic, complemented
26#14524019—21no—
27#14524017C2211yescharacteristic
28#14524017C2211yescharacteristic
29#14524017C2211yescharacteristic
30#14524017C2211yescharacteristic
31#14524017C2211yescharacteristic
32#14524017C2211yescharacteristic, complemented
33#14524015—21no—
34#14524015—21no—
35#14524014—21no—
36#14524011—21no—
37#14524011—21no—
38#14524002C2211yescharacteristic
39C2×C4⋊Dic67C2211yescharacteristic, complemented
40#14523977C2211yescharacteristic, complemented
41#14523977C2211yescharacteristic, complemented
42#14523975C2211yescharacteristic, complemented
43#14523975C2211yescharacteristic, complemented
44#14523968C2211yescharacteristic, complemented
45#14523968C2211yescharacteristic, complemented
46#14523965C2211yescharacteristic
47#14523965C2211yescharacteristic
48#14523965C2211yescharacteristic
49#14523965C2211yescharacteristic
50#14523965C2211yescharacteristic, complemented
51#14523965C2211yescharacteristic, complemented
52C2×#41×C67C2211yescharacteristic
53C2×M16×C67C2211yescharacteristic, complemented
54C22×C8×C67C2211yescharacteristic, complemented
55C22⋊C8×C67C2211yescharacteristic
56C22⋊C8×C67C2211yescharacteristic
57C22⋊C8×C67C2211yescharacteristic, complemented
58C22⋊C8×C67C2211yescharacteristic, complemented
59C23×D67—21no—
60#11761628—41no—
61#11761628—41no—
62C2×Dic134C2311yescharacteristic
63C2×D268—21no—
64C2×D268—41no—
65C2×D268—41no—
66C2×D268C2311yescharacteristic
67C2×D268C2311yescharacteristic
68C2×D268C2311yescharacteristic
69C2×C22⋊D67—21no—
70C2×C22⋊D67—21no—
71C2×C22⋊D67—21no—
72C2×C22⋊D67—21no—
73C2×C22⋊D67—21no—
74C2×C22⋊D67—21no—
75C2×C22⋊D67—21no—
76C22×Dic67—21no—
77C2×C4×D67—21no—
78#11761617—41no—
79#11761617—41no—
80D536—41no—
81D536—41no—
82C4⋊Dic67—21no—
83C4⋊Dic67C2311yescharacteristic
84C4⋊Dic67C2311yescharacteristic
85C4⋊Dic67C2311yescharacteristic
86C4⋊Dic67C2311yescharacteristic
87Dic67⋊C4—21no—
88Dic67⋊C4—21no—
89C22⋊Dic67—21no—
90C22⋊Dic67—21no—
91C22⋊Dic67—21no—
92(C2×D67)⋊C4—21no—
93(C2×D67)⋊C4—21no—
94(C2×D67)⋊C4—21no—
95(C2×D67)⋊C4—21no—
96(C2×D67)⋊C4—21no—
97(C2×D67)⋊C4—21no—
98C4×Dic67—21no—
99#41×C67—41no—
100#41×C67—41no—
101C2×Q8×C67D411yescharacteristic
102C2×D4×C67D411yescharacteristic
103C2×D4×C67D411yescharacteristic
104C2×D4×C67D411yescharacteristic
105C22×C4×C67C2311yescharacteristic
106C22×C4×C67C2311yescharacteristic
107C22×C4×C67C2311yescharacteristic
108M16×C67—41no—
109C2×C8×C67—41no—
110C2×C8×C67C2311yescharacteristic
111C2×C8×C67C2311yescharacteristic
112C2×C8×C67C2311yescharacteristic
113C2×C8×C67C2311yescharacteristic
114C23×C67C2×D411yescharacteristic
115C23×C67C2×D411yescharacteristic
116C23×C67C2×D411yescharacteristic
117C22×D67—21no—
118C22×D67—21no—
119C22×D67—21no—
120C22×D67—42no—
121C22×D67—81no—
122C22×D67—81no—
123Q8×C67—41no—
124D4×C67—41no—
125D4×C67—41no—
126D4×C67—41no—
127C2×C4×C67—21no—
128C2×C4×C67—21no—
129C2×C4×C67—21no—
130C2×C4×C67—21no—
131C2×C4×C67—21no—
132C2×C4×C67—21no—
133C2×C4×C67C2×D411yescharacteristic
134C2×C4×C67C2×D411yescharacteristic
135C2×C4×C67C2×D411yescharacteristic
136C2×C4×C67C2411yescharacteristic
137C22⋊D67—42no—
138C22⋊D67—81no—
139C22⋊D67—81no—
140C22⋊D67—81no—
141C22⋊D67—81no—
142C22⋊D67—81no—
143C22⋊D67—81no—
144C2×Dic67—21no—
145C2×Dic67—21no—
146C2×Dic67—21no—
147C2×Dic67—21no—
148C2×Dic67—21no—
149C2×Dic67—81no—
150D268—21no—
151D268—21no—
152D268—21no—
153D268—21no—
154D268—21no—
155D268—21no—
156D268—42no—
157C4×D67—42no—
158Dic134—21no—
159Dic134—21no—
160C8×C67—21no—
161C8×C67—21no—
162C8×C67—21no—
163C8×C67—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×C67C22×D411yescharacteristic
174C2×D67—41no—
175C2×D67—41no—
176C2×D67—41no—
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×C67H(5,𝔽2)11yescharacteristic
188C4×C67H(5,𝔽2)11yescharacteristic
189Dic67—41no—
190Dic67—41no—
191Dic67—41no—
192Dic67—41no—
193Dic67—41no—
194C2×C67—41no—
195C2×C67—41no—
196C2×C67—41no—
197C2×C67#53411yescharacteristic
198C2×C67#57611yescharacteristic
199C2×C67#57711yescharacteristic
200D67—81no—
201D67—81no—
202D67—81no—
203#3198—671nomaximal, Sylow
204C67#319811yescharacteristic, complemented, Sylow
205#537—671no—
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#408D6711yescharacteristic, complemented
220C2×#41—671no—
221C2×#41C2×D6711yescharacteristic
222C22×D4—671no—
223C2×SD16—671no—
224C2×SD16—671no—
225C2×D8—671no—
226C2×D8—671no—
227C2×M16C2×D6711yescharacteristic, complemented
228C22×C8C2×D6711yescharacteristic, complemented
229#127—1341no—
230#124—1341no—
231C22⋊Q8—671no—
232C22⋊Q8—671no—
233C22⋊Q8—1341no—
234#121—671no—
235#121—671no—
236#121—671no—
237#121—671no—
238#121—671no—
239#121—671no—
240#121—1341no—
241#121—1341no—
242#121—1341no—
243#120—1341no—
244#120—1341no—
245#117—671no—
246C2×C4⋊C4—671no—
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⋊C8C2×D6711yescharacteristic
260C22⋊C8C2×D6711yescharacteristic
261C22⋊C8C2×D6711yescharacteristic, complemented
262C22⋊C8C2×D6711yescharacteristic, complemented
263C24—1341no—
264#41—41no—
265#41—41no—
266#41—2681no—
267#41—2681no—
268C2×Q8—671no—
269C2×Q8D26811yescharacteristic
270C2×D4—2681no—
271C2×D4—2681no—
272C2×D4—671no—
273C2×D4—671no—
274C2×D4—671no—
275C2×D4—1341no—
276C2×D4—1341no—
277C2×D4—1341no—
278C2×D4—1341no—
279C2×D4—1341no—
280C2×D4—1341no—
281C2×D4—1341no—
282C2×D4—1341no—
283C2×D4D26811yescharacteristic
284C2×D4D26811yescharacteristic
285C2×D4D26811yescharacteristic
286C22×C4—1341no—
287C22×C4—1341no—
288C22×C4C22×D6711yescharacteristic
289C22×C4C22×D6711yescharacteristic
290C22×C4C22×D6711yescharacteristic
291SD16—2681no—
292SD16—2681no—
293D8—2681no—
294D8—2681no—
295M16—41no—
296C2×C8—41no—
297C2×C8C22×D6711yescharacteristic
298C2×C8C22×D6711yescharacteristic
299C2×C8C22×D6711yescharacteristic
300C2×C8C22×D6711yescharacteristic
301C4⋊C4—671no—
302C4⋊C4—671no—
303C4⋊C4—671no—
304C4⋊C4—671no—
305C4⋊C4—1341no—
306C4⋊C4—1341no—
307C4⋊C4—1341no—
308C22⋊C4—1341no—
309C22⋊C4—1341no—
310C22⋊C4—1341no—
311C22⋊C4—1341no—
312C22⋊C4—1341no—
313C22⋊C4—1341no—
314C22⋊C4—1341no—
315C22⋊C4—1341no—
316C22⋊C4—1341no—
317C42—1341no—
318C23—2682no—
319C23—5361no—
320C23—5361no—
321C23—1341no—
322C23—1341no—
323C23—1341no—
324C23C2×D26811yescharacteristic
325C23C2×D26811yescharacteristic
326C23C2×D26811yescharacteristic
327Q8—41no—
328Q8—1341no—
329Q8—1341no—
330D4—41no—
331D4—41no—
332D4—41no—
333D4—2682no—
334D4—2682no—
335D4—5361no—
336D4—5361no—
337D4—5361no—
338D4—5361no—
339D4—5361no—
340D4—5361no—
341D4—1341no—
342D4—1341no—
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—2682no—
354C2×C4—5361no—
355C2×C4—1341no—
356C2×C4—1341no—
357C2×C4—1341no—
358C2×C4—1341no—
359C2×C4—1341no—
360C2×C4C2×D26811yescharacteristic
361C2×C4C2×D26811yescharacteristic
362C2×C4C2×D26811yescharacteristic
363C2×C4C23×D6711yescharacteristic
364C8—21no—
365C8—21no—
366C8—21no—
367C8—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—2681no—
378C22—2681no—
379C22—2681no—
380C22—2681no—
381C22—2681no—
382C22—2681no—
383C22—2681no—
384C22—2681no—
385C22—2681no—
386C22—5364no—
387C22C22×D26811yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—2681no—
392C4—2681no—
393C4—2681no—
394C4—2681no—
395C4—2681no—
396C4#1452408811yescharacteristic
397C4#1452408811yescharacteristic
398C2—41no—
399C2—41no—
400C2—41no—
401C2—5361no—
402C2—5361no—
403C2—5361no—
404C2#1686395811yescharacteristic
405C2#1686420111yescharacteristic
406C2#1686420811yescharacteristic
subgroupG/H
centerC22C22×D268
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#408×C67C2
socleC22×C67C22×D4
hypercenter#408D67
hypocenterC67#3198
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #16863974C2semidirect
2. #16863960C2semidirect
3. #16863738C2semidirect
4. #16863721C2semidirect
5. #16863715C2semidirect
6. #16863710C2semidirect
7. #16863649C2semidirect
8. #16863642C2semidirect
9. #16863632C2semidirect
10. #16863625C2semidirect
11. #16863549C2semidirect
12. #16863543C2semidirect
13. #16863543C2semidirect
14. #16863532C2semidirect
15. #408×C67C2semidirect
16. C2×#11761628C22semidirect
25. #14524022C22semidirect
32. #14524017C22semidirect
39. C2×C4⋊Dic67C22semidirect
40. #14523977C22semidirect
41. #14523977C22semidirect
42. #14523975C22semidirect
43. #14523975C22semidirect
44. #14523968C22semidirect
45. #14523968C22semidirect
50. #14523965C22semidirect
51. #14523965C22semidirect
53. C2×M16×C67C22semidirect
54. C22×C8×C67C22semidirect
57. C22⋊C8×C67C22semidirect
58. C22⋊C8×C67C22semidirect
204. C67#3198semidirect
219. #408D67semidirect
227. C2×M16C2×D67semidirect
228. C22×C8C2×D67semidirect
261. C22⋊C8C2×D67semidirect
262. C22⋊C8C2×D67semidirect