Finite Group Databasebeta

← #36335949

Subgroups — #36335949

Order 21376.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923920C211yesmaximal, characteristic, complemented
2#24923903C211yesmaximal, characteristic, complemented
3#24923614C211yesmaximal, characteristic, complemented
4#24923584C211yesmaximal, characteristic, complemented
5#24923548C211yesmaximal, characteristic, complemented
6#24923506C211yesmaximal, characteristic, complemented
7#24923503C211yesmaximal, characteristic, complemented
8#24923463C211yesmaximal, characteristic, complemented
9#24923405C211yesmaximal, characteristic, complemented
10#24923404C211yesmaximal, characteristic, complemented
11#24923351C211yesmaximal, characteristic, complemented
12(C22⋊D167)⋊C8C211yesmaximal, characteristic, complemented
13C2×#19128752C211yesmaximal, characteristic, complemented
14#102×D167C211yesmaximal, characteristic, complemented
15#480×C167C211yesmaximal, characteristic, complemented
16C2×#14581950—21no—
17C22×D668C2211yescharacteristic
18C2×D4×D167—21no—
19C2×D4×D167C2211yescharacteristic
20C2×D1336C2211yescharacteristic
21C2×D1336C2211yescharacteristic, complemented
22C2×#14581935C2211yescharacteristic, complemented
23C22×C167⋊C8C2211yescharacteristic, complemented
24C2×C8×D167C2211yescharacteristic, complemented
25#19128831C2211yescharacteristic, complemented
26#19128828C2211yescharacteristic
27#19128817—21no—
28#19128815—21no—
29#19128810—21no—
30#19128807C2211yescharacteristic
31#19128807C2211yescharacteristic, complemented
32#19128806C2211yescharacteristic, complemented
33#19128805—21no—
34#19128802—21no—
35#19128800—21no—
36#19128794C2211yescharacteristic
37#19128794C2211yescharacteristic, complemented
38C4×C22⋊D167C2211yescharacteristic
39C4⋊C4×D167C2211yescharacteristic, complemented
40#19128765C2211yescharacteristic, complemented
41Dic167⋊C8C2211yescharacteristic, complemented
42#19128754C2211yescharacteristic
43#19128754C2211yescharacteristic, complemented
44#19128753C2211yescharacteristic, complemented
45#19128752C2211yescharacteristic
46#19128752C2211yescharacteristic
47#19128752C2211yescharacteristic, complemented
48#19128752C2211yescharacteristic, complemented
49#19128752C2211yescharacteristic, complemented
50(C2×D167)⋊C8C2211yescharacteristic, complemented
51C8×Dic167C2211yescharacteristic, complemented
52#121×C167C2211yescharacteristic
53C2×C4⋊C4×C167C2211yescharacteristic, complemented
54#107×C167C2211yescharacteristic, complemented
55#107×C167C2211yescharacteristic, complemented
56#102×C167C2211yescharacteristic, complemented
57#102×C167C2211yescharacteristic, complemented
58C22⋊C8×C167C2211yescharacteristic, complemented
59C23×D167—21no—
60C23×D167—21no—
61#14581950—44no—
62C2×Dic334—21no—
63C2×D668—22no—
64C2×D668—21no—
65C2×D668—21no—
66C2×D668—21no—
67C2×D668—41no—
68C2×D668C2311yescharacteristic
69C2×D668C2311yescharacteristic
70C2×D668C2311yescharacteristic
71C2×C22⋊D167—21no—
72C2×C22⋊D167—21no—
73C2×C22⋊D167—21no—
74C2×C22⋊D167—21no—
75C2×C22⋊D167D411yescharacteristic
76D4×D167—22no—
77D4×D167—41no—
78D4×D167—44no—
79C2×C4×D167—21no—
80C2×C4×D167—21no—
81C2×C4×D167—21no—
82C2×C4×D167C2311yescharacteristic
83D1336—22no—
84D1336—41no—
85#14581935—41no—
86C2×C167⋊C8—41no—
87C2×C167⋊C8C2311yescharacteristic
88C2×C167⋊C8C2311yescharacteristic
89C8×D167—41no—
90C4⋊Dic167C2311yescharacteristic
91Dic167⋊C4—21no—
92Dic167⋊C4—21no—
93Dic167⋊C4D411yescharacteristic
94C22⋊Dic167D411yescharacteristic
95(C2×D167)⋊C4—21no—
96(C2×D167)⋊C4—21no—
97(C2×D167)⋊C4—21no—
98(C2×D167)⋊C4—21no—
99(C2×D167)⋊C4—21no—
100(C2×D167)⋊C4D411yescharacteristic
101C4×Dic167C2311yescharacteristic
102C2×D4×C167—21no—
103C2×D4×C167C2311yescharacteristic
104C22×C4×C167—21no—
105C22×C4×C167C2311yescharacteristic
106C2×C8×C167C2311yescharacteristic
107C2×C8×C167C2311yescharacteristic
108C4⋊C4×C167—21no—
109C4⋊C4×C167C2311yescharacteristic
110C4⋊C4×C167C2311yescharacteristic
111C4⋊C4×C167C2311yescharacteristic
112C22⋊C4×C167—21no—
113C22×D167—21no—
114C22×D167—21no—
115C22×D167—21no—
116C22×D167—42no—
117C22×D167—42no—
118C22×D167—42no—
119C22×D167—42no—
120C22×D167—81no—
121C22×D167—81no—
122C22×D167C2×D411yescharacteristic
123Dic334—42no—
124D668—21no—
125D668—21no—
126D668—21no—
127D668—21no—
128D668—21no—
129D668—42no—
130D668—42no—
131D668—42no—
132D668#4112yescomplemented
133C22⋊D167—42no—
134C22⋊D167—42no—
135C22⋊D167—42no—
136C22⋊D167—81no—
137C22⋊D167D814yescomplemented
138C2×Dic167—21no—
139C2×Dic167C2×D411yescharacteristic
140C2×Dic167C2×D411yescharacteristic
141C2×Dic167C2×D411yescharacteristic
142C4×D167—21no—
143C4×D167—21no—
144C4×D167—42no—
145C4×D167—42no—
146C4×D167—42no—
147C167⋊C8—21no—
148C167⋊C8—21no—
149C23×C167—21no—
150C23×C167C2×D411yescharacteristic
151D4×C167—21no—
152D4×C167—21no—
153D4×C167—42no—
154C2×C4×C167—21no—
155C2×C4×C167—21no—
156C2×C4×C167—21no—
157C2×C4×C167—21no—
158C2×C4×C167—21no—
159C2×C4×C167—42no—
160C2×C4×C167C2×D411yescharacteristic
161C2×C4×C167C2411yescharacteristic
162C8×C167—21no—
163C8×C167—21no—
164C22×C167—21no—
165C22×C167—21no—
166C22×C167—41no—
167C22×C167—41no—
168C22×C167—41no—
169C22×C167C2×D812yes—
170C22×C167C22×D411yescharacteristic
171C2×D167—21no—
172C2×D167—21no—
173C2×D167—22no—
174C2×D167—41no—
175C2×D167—41no—
176C2×D167—41no—
177C2×D167—41no—
178C2×D167—41no—
179C2×D167—41no—
180C2×D167—41no—
181C2×D167—41no—
182C2×D167—84no—
183C2×D167—84no—
184C2×D167C2×D812yes—
185C4×C167—21no—
186C4×C167—41no—
187C4×C167—41no—
188C4×C167—41no—
189C4×C167C2×#4111yescharacteristic
190C4×C167H(5,𝔽2)11yescharacteristic
191Dic167—21no—
192Dic167—21no—
193Dic167—41no—
194Dic167C2×D812yes—
195C2×C167—22no—
196C2×C167—81no—
197C2×C167#54611yescharacteristic
198C2×C167C22×D811yescharacteristic
199C2×C167#57711yescharacteristic
200D167—22no—
201D167—42no—
202D167—81no—
203D167—81no—
204C167#329411yescharacteristic, complemented, Sylow
205#3294—1671nomaximal, Sylow
206#547—1671no—
207D42—1671no—
208#486—1671no—
209#480—1671no—
210#480D16711yescharacteristic, complemented
211#474—1671no—
212#466—1671no—
213#466—1671no—
214#459—1671no—
215#447—1671no—
216#447—1671no—
217C4×D8—1671no—
218C8×D4—1671no—
219C2×#102—1671no—
220C2×#102—1671no—
221C2×#41—3341no—
222C22×D4—3341no—
223C22×D4—1671no—
224C22×D4—1671no—
225C2×D8—1671no—
226C2×D8—1671no—
227C2×D8—1671no—
228C22×C8—1671no—
229C22×C8—1671no—
230C4⋊Q8—1671no—
231#127—1671no—
232#123—3341no—
233#123—3341no—
234C22⋊Q8—3341no—
235#121—3341no—
236#121—1671no—
237#121—1671no—
238#121—1671no—
239#121C2×D16711yescharacteristic
240#120—3341no—
241#120—3341no—
242C4×D4—1671no—
243C4×D4—1671no—
244C4×D4—1671no—
245C2×C4⋊C4—1671no—
246C2×C4⋊C4C2×D16711yescharacteristic, complemented
247#107—1671no—
248#107C2×D16711yescharacteristic, complemented
249#107C2×D16711yescharacteristic, complemented
250C4⋊C8—1671no—
251#102—1671no—
252#102—1671no—
253#102—1671no—
254#102—1671no—
255#102—1671no—
256#102—1671no—
257#102—1671no—
258#102—1671no—
259#102C2×D16711yescharacteristic, complemented
260#102C2×D16711yescharacteristic, complemented
261C22⋊C8—1671no—
262C22⋊C8C2×D16711yescharacteristic, complemented
263C4×C8—1671no—
264C24—3341no—
265C24—3341no—
266#41—6684no—
267C2×Q8—3341no—
268C2×D4—21no—
269C2×D4—3342no—
270C2×D4—3341no—
271C2×D4—3341no—
272C2×D4—3342no—
273C2×D4—3341no—
274C2×D4—3341no—
275C2×D4—3341no—
276C2×D4—3341no—
277C2×D4—3341no—
278C2×D4—6681no—
279C2×D4—6681no—
280C2×D4—6684no—
281C2×D4—1671no—
282C2×D4—1671no—
283C2×D4—1671no—
284C2×D4—1671no—
285C2×D4C22×D16711yescharacteristic
286C22×C4—21no—
287C22×C4—3341no—
288C22×C4—3341no—
289C22×C4—3341no—
290C22×C4—1671no—
291C22×C4C22×D16711yescharacteristic
292D8—3342no—
293D8—6681no—
294D8—6681no—
295C2×C8—6681no—
296C2×C8—6681no—
297C2×C8—1671no—
298C2×C8—1671no—
299C2×C8C22×D16711yescharacteristic
300C2×C8C22×D16711yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—3341no—
303C4⋊C4—3341no—
304C4⋊C4—1671no—
305C4⋊C4—1671no—
306C4⋊C4C22×D16711yescharacteristic
307C4⋊C4C22×D16711yescharacteristic
308C4⋊C4C22×D16711yescharacteristic
309C22⋊C4—21no—
310C22⋊C4—3341no—
311C22⋊C4—3341no—
312C22⋊C4—3341no—
313C22⋊C4—3341no—
314C22⋊C4—3341no—
315C22⋊C4—1671no—
316C22⋊C4—1671no—
317C42—1671no—
318C23—21no—
319C23—13361no—
320C23—13361no—
321C23—3341no—
322C23—3341no—
323C23—3341no—
324C23—6682no—
325C23—6682no—
326C23—6682no—
327C23—6682no—
328C23—1671no—
329C23D4×D16711yescharacteristic
330Q8—6682no—
331D4—21no—
332D4—21no—
333D4—42no—
334D4—13361no—
335D4—3341no—
336D4—3341no—
337D4—3341no—
338D4—3341no—
339D4—3341no—
340D4—6682no—
341D4—6682no—
342D4—6682no—
343D4—6682no—
344D4—6682no—
345D4—6682no—
346D4—1674no—
347D4—1672no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—42no—
354C2×C4—3341no—
355C2×C4—3341no—
356C2×C4—3341no—
357C2×C4—6682no—
358C2×C4—6682no—
359C2×C4—6682no—
360C2×C4—1671no—
361C2×C4—1671no—
362C2×C4—1671no—
363C2×C4D4×D16711yescharacteristic
364C2×C4C23×D16711yescharacteristic
365C8—21no—
366C8—21no—
367C8—3341no—
368C8—3341no—
369C22—21no—
370C22—21no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—13364no—
375C22—13364no—
376C22—3341no—
377C22—3341no—
378C22—3342no—
379C22—6681no—
380C22—6681no—
381C22—6681no—
382C22—6681no—
383C22—6681no—
384C22—6681no—
385C22—6681no—
386C22—6681no—
387C22—1672no—
388C22D8×D16712yes—
389C22C2×D4×D16711yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—3341no—
395C4—3341no—
396C4—6681no—
397C4—1672no—
398C4#41×D16711yescharacteristic
399C4#1912887811yescharacteristic
400C2—22no—
401C2—81no—
402C2—13361no—
403C2—13361no—
404C2—3342no—
405C2—6682no—
406C2#2492391111yescharacteristic
407C2C2×D8×D16711yescharacteristic
408C2#2492407811yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#480×C167C2
socleC22×C167C22×D4
hypercenter#480D167
hypocenterC167#3294
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #24923920C2semidirect
2. #24923903C2semidirect
3. #24923614C2semidirect
4. #24923584C2semidirect
5. #24923548C2semidirect
6. #24923506C2semidirect
7. #24923503C2semidirect
8. #24923463C2semidirect
9. #24923405C2semidirect
10. #24923404C2semidirect
11. #24923351C2semidirect
12. (C22⋊D167)⋊C8C2semidirect
13. C2×#19128752C2semidirect
14. #102×D167C2semidirect
15. #480×C167C2semidirect
21. C2×D1336C22semidirect
22. C2×#14581935C22semidirect
23. C22×C167⋊C8C22semidirect
24. C2×C8×D167C22semidirect
25. #19128831C22semidirect
31. #19128807C22semidirect
32. #19128806C22semidirect
37. #19128794C22semidirect
39. C4⋊C4×D167C22semidirect
40. #19128765C22semidirect
41. Dic167⋊C8C22semidirect
43. #19128754C22semidirect
44. #19128753C22semidirect
47. #19128752C22semidirect
48. #19128752C22semidirect
49. #19128752C22semidirect
50. (C2×D167)⋊C8C22semidirect
51. C8×Dic167C22semidirect
53. C2×C4⋊C4×C167C22semidirect
54. #107×C167C22semidirect
55. #107×C167C22semidirect
56. #102×C167C22semidirect
57. #102×C167C22semidirect
58. C22⋊C8×C167C22semidirect
132. D668#41semidirect
137. C22⋊D167D8semidirect
204. C167#3294semidirect
210. #480D167semidirect
246. C2×C4⋊C4C2×D167semidirect
248. #107C2×D167semidirect
249. #107C2×D167semidirect
259. #102C2×D167semidirect
260. #102C2×D167semidirect
262. C22⋊C8C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249239202
#249239032
#249236142
#249235842
#249235482
#249235062
#249235032
#249234632
#249234052
#249234042
#249233512
(C22⋊D167)⋊C82
C2×#191287522
#102×D1672
#480×C1672
#3294167