Finite Group Databasebeta

← #36336092

Subgroups — #36336092

Order 21376.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923919C211yesmaximal, characteristic, complemented
2#24923907C211yesmaximal, characteristic, complemented
3#24923630C211yesmaximal, characteristic, complemented
4#24923598C211yesmaximal, characteristic, complemented
5#24923588C211yesmaximal, characteristic, complemented
6#24923557C211yesmaximal, characteristic, complemented
7#24923520C211yesmaximal, characteristic, complemented
8#24923495C211yesmaximal, characteristic, complemented
9#24923486C211yesmaximal, characteristic, complemented
10#24923401C211yesmaximal, characteristic, complemented
11#24923381C211yesmaximal, characteristic, complemented
12#24923343C211yesmaximal, characteristic, complemented
13#24923270C211yesmaximal, characteristic, complemented
14#102×D167C211yesmaximal, characteristic, complemented
15#449×C167C211yesmaximal, characteristic, complemented
16C2×#14581949—21no—
17C2×D4×D167C2211yescharacteristic
18C2×#14581935C2211yescharacteristic
19C2×#14581934C2211yescharacteristic, complemented
20C2×C8×D167C2211yescharacteristic
21#19128831C2211yescharacteristic, complemented
22#19128821C2211yescharacteristic
23#19128815—21no—
24#19128814—21no—
25#19128813—21no—
26#19128809—21no—
27#19128808C2211yescharacteristic
28#19128806C2211yescharacteristic
29#19128804—21no—
30#19128803C2211yescharacteristic
31#19128800—21no—
32D4×Dic167C2211yescharacteristic
33D4×Dic167C2211yescharacteristic, complemented
34C4×C22⋊D167C2211yescharacteristic
35C2×C4⋊Dic167C2211yescharacteristic, complemented
36C4⋊C4×D167C2211yescharacteristic
37C2×(C2×D167)⋊C4—21no—
38#19128764C2211yescharacteristic, complemented
39#19128764C2211yescharacteristic, complemented
40#19128763C2211yescharacteristic, complemented
41Dic167⋊C8C2211yescharacteristic, complemented
42#19128756C2211yescharacteristic, complemented
43#19128756C2211yescharacteristic, complemented
44#19128754C2211yescharacteristic
45#19128754C2211yescharacteristic
46#19128753C2211yescharacteristic
47#19128753C2211yescharacteristic
48#19128753C2211yescharacteristic, complemented
49#19128752C2211yescharacteristic
50(C2×D167)⋊C8C2211yescharacteristic
51#19128742C2211yescharacteristic, complemented
52C2×#41×C167C2211yescharacteristic
53C2×SD16×C167C2211yescharacteristic, complemented
54C2×D8×C167C2211yescharacteristic
55#121×C167C2211yescharacteristic, complemented
56#103×C167C2211yescharacteristic, complemented
57#102×C167C2211yescharacteristic
58C22⋊C8×C167C2211yescharacteristic, complemented
59C23×D167—21no—
60#14581949—44no—
61C2×Dic334—21no—
62C2×D668C2311yescharacteristic
63C2×C22⋊D167—21no—
64C2×C22⋊D167—21no—
65C2×C22⋊D167—21no—
66C2×C22⋊D167D411yescharacteristic
67D4×D167—22no—
68D4×D167—41no—
69C22×Dic167—21no—
70C22×Dic167—21no—
71C22×Dic167—21no—
72C2×C4×D167—21no—
73C2×C4×D167C2311yescharacteristic
74#14581935—41no—
75#14581934—41no—
76C2×C167⋊C8C2311yescharacteristic
77C2×C167⋊C8C2311yescharacteristic
78C8×D167—41no—
79C4⋊Dic167—21no—
80C4⋊Dic167C2311yescharacteristic
81C4⋊Dic167C2311yescharacteristic
82C4⋊Dic167C2311yescharacteristic
83Dic167⋊C4—21no—
84Dic167⋊C4—21no—
85Dic167⋊C4—21no—
86Dic167⋊C4D411yescharacteristic
87C22⋊Dic167—21no—
88C22⋊Dic167—21no—
89C22⋊Dic167—21no—
90C22⋊Dic167D411yescharacteristic
91(C2×D167)⋊C4—21no—
92(C2×D167)⋊C4—21no—
93(C2×D167)⋊C4—21no—
94(C2×D167)⋊C4—21no—
95(C2×D167)⋊C4D411yescharacteristic
96C4×Dic167C2311yescharacteristic
97#41×C167—22no—
98#41×C167—41no—
99C2×Q8×C167C2311yescharacteristic
100C2×D4×C167—21no—
101C2×D4×C167—21no—
102C2×D4×C167C2311yescharacteristic
103C2×D4×C167C2311yescharacteristic
104C22×C4×C167—21no—
105C22×C4×C167C2311yescharacteristic
106SD16×C167—41no—
107D8×C167—22no—
108C2×C8×C167C2311yescharacteristic
109C2×C8×C167C2311yescharacteristic
110C4⋊C4×C167C2311yescharacteristic
111C22⋊C4×C167—21no—
112C22×D167—21no—
113C22×D167—42no—
114C22×D167—42no—
115C22×D167—81no—
116C22×D167C2×D411yescharacteristic
117Dic334—42no—
118D668—21no—
119D668—21no—
120C22⋊D167—22no—
121C22⋊D167—42no—
122C22⋊D167—42no—
123C22⋊D167—81no—
124C2×Dic167—21no—
125C2×Dic167—21no—
126C2×Dic167—21no—
127C2×Dic167—42no—
128C2×Dic167—42no—
129C2×Dic167—81no—
130C2×Dic167C2×D411yescharacteristic
131C2×Dic167C2×D411yescharacteristic
132C2×Dic167C2×D411yescharacteristic
133C4×D167—21no—
134C4×D167—21no—
135C4×D167—42no—
136C167⋊C8—21no—
137C167⋊C8—21no—
138C23×C167—21no—
139C23×C167—21no—
140C23×C167C2×D411yescharacteristic
141Q8×C167—21no—
142Q8×C167—21no—
143D4×C167—21no—
144D4×C167—21no—
145D4×C167—21no—
146D4×C167—42no—
147D4×C167—42no—
148D4×C167#4112yescomplemented
149C2×C4×C167—21no—
150C2×C4×C167—21no—
151C2×C4×C167—21no—
152C2×C4×C167—21no—
153C2×C4×C167—42no—
154C2×C4×C167C2×D411yescharacteristic
155C2×C4×C167C2411yescharacteristic
156C8×C167—21no—
157C8×C167—21no—
158C22×C167—21no—
159C22×C167—22no—
160C22×C167—21no—
161C22×C167—21no—
162C22×C167—41no—
163C22×C167—41no—
164C22×C167—41no—
165C22×C167—41no—
166C22×C167—41no—
167C22×C167C22×D411yescharacteristic
168C2×D167—21no—
169C2×D167—21no—
170C2×D167—41no—
171C2×D167—41no—
172C2×D167—41no—
173C2×D167—84no—
174C2×D167AGL(1,ℤ/8)12yes—
175C4×C167—21no—
176C4×C167—41no—
177C4×C167—41no—
178C4×C167C2×#4111yescharacteristic
179C4×C167H(5,𝔽2)11yescharacteristic
180Dic167—21no—
181Dic167—21no—
182Dic167—21no—
183Dic167—41no—
184Dic167—41no—
185Dic167—41no—
186C2×C167—42no—
187C2×C167—41no—
188C2×C167—81no—
189C2×C167#54611yescharacteristic
190C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C167#57711yescharacteristic
192D167—22no—
193D167—81no—
194C167#330611yescharacteristic, complemented, Sylow
195#3306—1671nomaximal, Sylow
196#547—1671no—
197#546—1671no—
198#489—1671no—
199#483—1671no—
200#481—1671no—
201#476—1671no—
202#469—1671no—
203#465—1671no—
204#463—1671no—
205#449D16711yescharacteristic, complemented
206#447—1671no—
207#442—1671no—
208#435—1671no—
209#417—1671no—
210C2×#102—1671no—
211C2×#41—3341no—
212C2×#41C2×D16711yescharacteristic
213C22×D4—1671no—
214C2×SD16C2×D16711yescharacteristic, complemented
215C2×D8—1671no—
216C2×D8C2×D16711yescharacteristic
217C2×M16—1671no—
218C22×C8—1671no—
219C4⋊Q8—1671no—
220#124—1671no—
221#123—3341no—
222#123—3341no—
223#123—3341no—
224C22⋊Q8—3341no—
225C22⋊Q8—1671no—
226#121—3341no—
227#121—1671no—
228#121—1671no—
229#121C2×D16711yescharacteristic, complemented
230#120—3341no—
231C4×D4—1671no—
232C4×D4—1671no—
233C4×D4—1671no—
234C2×C4⋊C4—1671no—
235C2×C4⋊C4—1671no—
236C2×C22⋊C4—3341no—
237#106—1671no—
238#106—1671no—
239#106—1671no—
240C4⋊C8—1671no—
241#103—1671no—
242#103—1671no—
243#103C2×D16711yescharacteristic, complemented
244#102—1671no—
245#102—1671no—
246#102—1671no—
247#102—1671no—
248#102—1671no—
249#102—1671no—
250#102C2×D16711yescharacteristic
251C22⋊C8—1671no—
252C22⋊C8C2×D16711yescharacteristic, complemented
253#97—1671no—
254C24—3341no—
255#41—22no—
256#41—41no—
257#41—6684no—
258C2×Q8—3341no—
259C2×Q8C22×D16711yescharacteristic
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—3341no—
263C2×D4—3342no—
264C2×D4—3341no—
265C2×D4—3341no—
266C2×D4—6681no—
267C2×D4—1671no—
268C2×D4—1671no—
269C2×D4C22×D16711yescharacteristic
270C2×D4C22×D16711yescharacteristic
271C22×C4—21no—
272C22×C4—3341no—
273C22×C4—3341no—
274C22×C4—3341no—
275C22×C4—3341no—
276C22×C4—1671no—
277C22×C4C22×D16711yescharacteristic
278SD16—41no—
279D8—22no—
280D8—6681no—
281M16—6681no—
282C2×C8—6681no—
283C2×C8—1671no—
284C2×C8—1671no—
285C2×C8C22×D16711yescharacteristic
286C2×C8C22×D16711yescharacteristic
287C4⋊C4—3341no—
288C4⋊C4—3341no—
289C4⋊C4—3341no—
290C4⋊C4—3341no—
291C4⋊C4—1671no—
292C4⋊C4—1671no—
293C4⋊C4—1671no—
294C4⋊C4—1671no—
295C4⋊C4C22×D16711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—3341no—
298C22⋊C4—3341no—
299C22⋊C4—3341no—
300C22⋊C4—3341no—
301C22⋊C4—3341no—
302C22⋊C4—3341no—
303C22⋊C4—3341no—
304C22⋊C4—1671no—
305C22⋊C4—1671no—
306C42—1671no—
307C23—21no—
308C23—21no—
309C23—13361no—
310C23—3341no—
311C23—6682no—
312C23—6682no—
313C23—1671no—
314C23D4×D16711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—6682no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—42no—
323D4—13361no—
324D4—3342no—
325D4—3341no—
326D4—3341no—
327D4—6682no—
328D4—6682no—
329D4#1458194912yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—13361no—
336C2×C4—3341no—
337C2×C4—3341no—
338C2×C4—3341no—
339C2×C4—3341no—
340C2×C4—3341no—
341C2×C4—6682no—
342C2×C4—6682no—
343C2×C4—6682no—
344C2×C4—1671no—
345C2×C4—1671no—
346C2×C4—1671no—
347C2×C4D4×D16711yescharacteristic
348C2×C4C23×D16711yescharacteristic
349C8—21no—
350C8—21no—
351C8—3341no—
352C8—3341no—
353C22—21no—
354C22—22no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—13364no—
363C22—3341no—
364C22—3341no—
365C22—6681no—
366C22—6681no—
367C22—6681no—
368C22—1672no—
369C22C2×D4×D16711yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—3341no—
374C4—3341no—
375C4—3341no—
376C4—6681no—
377C4—6681no—
378C4—6681no—
379C4C2×#1458194911yescharacteristic
380C4#1912887711yescharacteristic
381C2—41no—
382C2—42no—
383C2—81no—
384C2—13361no—
385C2—3342no—
386C2#2492390611yescharacteristic
387C2AGL(1,ℤ/8)×D16711yescharacteristic
388C2#2492407611yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#449×C167C2
socleC22×C167C22×D4
hypercenter#449D167
hypocenterC167#3306
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #24923919C2semidirect
2. #24923907C2semidirect
3. #24923630C2semidirect
4. #24923598C2semidirect
5. #24923588C2semidirect
6. #24923557C2semidirect
7. #24923520C2semidirect
8. #24923495C2semidirect
9. #24923486C2semidirect
10. #24923401C2semidirect
11. #24923381C2semidirect
12. #24923343C2semidirect
13. #24923270C2semidirect
14. #102×D167C2semidirect
15. #449×C167C2semidirect
19. C2×#14581934C22semidirect
21. #19128831C22semidirect
33. D4×Dic167C22semidirect
35. C2×C4⋊Dic167C22semidirect
38. #19128764C22semidirect
39. #19128764C22semidirect
40. #19128763C22semidirect
41. Dic167⋊C8C22semidirect
42. #19128756C22semidirect
43. #19128756C22semidirect
48. #19128753C22semidirect
51. #19128742C22semidirect
53. C2×SD16×C167C22semidirect
55. #121×C167C22semidirect
56. #103×C167C22semidirect
58. C22⋊C8×C167C22semidirect
148. D4×C167#41semidirect
194. C167#3306semidirect
205. #449D167semidirect
214. C2×SD16C2×D167semidirect
229. #121C2×D167semidirect
243. #103C2×D167semidirect
252. C22⋊C8C2×D167semidirect
329. D4#14581949semidirect

Maximal subgroups

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

subgroupindex
#249239192
#249239072
#249236302
#249235982
#249235882
#249235572
#249235202
#249234952
#249234862
#249234012
#249233812
#249233432
#249232702
#102×D1672
#449×C1672
#3306167