Finite Group Databasebeta

← #36336099

Subgroups — #36336099

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#24923923C211yesmaximal, characteristic, complemented
2#24923628C211yesmaximal, characteristic, complemented
3#24923597C211yesmaximal, characteristic, complemented
4#24923585C211yesmaximal, characteristic, complemented
5#24923559C211yesmaximal, characteristic, complemented
6#24923520C211yesmaximal, characteristic, complemented
7#24923500C211yesmaximal, characteristic, complemented
8#24923481C211yesmaximal, characteristic, complemented
9#24923414C211yesmaximal, characteristic, complemented
10#24923403C211yesmaximal, characteristic, complemented
11#24923379C211yesmaximal, characteristic, complemented
12#24923341C211yesmaximal, characteristic, complemented
13#24923270C211yesmaximal, characteristic, complemented
14C2×#19128753C211yesmaximal, characteristic, complemented
15#546×C167C211yesmaximal, characteristic, complemented
16C2×#14581949—21no—
17C2×#14581937C2211yescharacteristic, complemented
18C2×#14581935C2211yescharacteristic
19C2×#14581935C2211yescharacteristic
20C2×#14581934C2211yescharacteristic, complemented
21C22×C167⋊C8C2211yescharacteristic
22#19128832C2211yescharacteristic, complemented
23#19128816—21no—
24#19128816—21no—
25#19128812—21no—
26#19128806C2211yescharacteristic
27#19128806C2211yescharacteristic, complemented
28C4×D668C2211yescharacteristic, complemented
29C2×C4⋊Dic167C2211yescharacteristic
30C2×C4⋊Dic167C2211yescharacteristic, complemented
31#19128763C2211yescharacteristic, complemented
32#19128763C2211yescharacteristic, complemented
33#19128763C2211yescharacteristic, complemented
34C4⋊(C167⋊C8)C2211yescharacteristic, complemented
35#19128756C2211yescharacteristic, complemented
36#19128756C2211yescharacteristic, complemented
37#19128756C2211yescharacteristic, complemented
38#19128753C2211yescharacteristic
39#19128753C2211yescharacteristic
40#19128753C2211yescharacteristic
41#19128753C2211yescharacteristic
42#19128753C2211yescharacteristic
43#19128753C2211yescharacteristic
44#19128752C2211yescharacteristic, complemented
45C22⋊(C167⋊C8)C2211yescharacteristic
46C22⋊(C167⋊C8)C2211yescharacteristic, complemented
47#19128743C2211yescharacteristic, complemented
48C2×#41×C167C2211yescharacteristic
49C22×D4×C167C2211yescharacteristic
50#124×C167C2211yescharacteristic
51#123×C167—21no—
52C22⋊Q8×C167C2211yescharacteristic
53#121×C167—21no—
54#121×C167C2211yescharacteristic
55#120×C167—21no—
56C4×D4×C167C2211yescharacteristic
57C4×D4×C167C2211yescharacteristic
58C2×C22⋊C4×C167—21no—
59#14581949—44no—
60C2×Dic334—21no—
61C2×D668C2311yescharacteristic
62C2×C22⋊D167—21no—
63C2×C22⋊D167—21no—
64C22×Dic167—21no—
65C22×Dic167—21no—
66C2×C4×D167—21no—
67#14581937—41no—
68#14581935—22no—
69#14581935—41no—
70#14581934—41no—
71C2×C167⋊C8—41no—
72C2×C167⋊C8C2311yescharacteristic
73C2×C167⋊C8C2311yescharacteristic
74C2×C167⋊C8C2311yescharacteristic
75C2×C167⋊C8C2311yescharacteristic
76C4⋊Dic167—21no—
77C4⋊Dic167—21no—
78C4⋊Dic167—21no—
79C4⋊Dic167C2311yescharacteristic
80C4⋊Dic167C2311yescharacteristic
81C4⋊Dic167C2311yescharacteristic
82(C2×D167)⋊C4—21no—
83(C2×D167)⋊C4—21no—
84(C2×D167)⋊C4—21no—
85C24×C167—21no—
86#41×C167—22no—
87#41×C167—41no—
88C2×Q8×C167C2311yescharacteristic
89C2×D4×C167—22no—
90C2×D4×C167—21no—
91C2×D4×C167—21no—
92C2×D4×C167—21no—
93C2×D4×C167—41no—
94C2×D4×C167D411yescharacteristic
95C2×D4×C167C2311yescharacteristic
96C2×D4×C167C2311yescharacteristic
97C22×C4×C167—21no—
98C22×C4×C167—21no—
99C22×C4×C167C2311yescharacteristic
100C22×C4×C167C2311yescharacteristic
101C4⋊C4×C167—21no—
102C4⋊C4×C167D411yescharacteristic
103C4⋊C4×C167C2311yescharacteristic
104C22⋊C4×C167—21no—
105C22⋊C4×C167—21no—
106C22⋊C4×C167—21no—
107C22⋊C4×C167—21no—
108C22⋊C4×C167—21no—
109C22⋊C4×C167D411yescharacteristic
110C22⋊C4×C167D411yescharacteristic
111C42×C167C2311yescharacteristic
112C22×D167—21no—
113Dic334—42no—
114D668—21no—
115D668—21no—
116C22⋊D167—42no—
117C22⋊D167—42no—
118C2×Dic167—21no—
119C2×Dic167—21no—
120C2×Dic167—21no—
121C2×Dic167—42no—
122C2×Dic167—42no—
123C4×D167—42no—
124C167⋊C8—21no—
125C167⋊C8—21no—
126C167⋊C8—21no—
127C167⋊C8—21no—
128C23×C167—21no—
129C23×C167—21no—
130C23×C167—42no—
131C23×C167—42no—
132C23×C167—81no—
133C23×C167C2×D411yescharacteristic
134C23×C167C2×D411yescharacteristic
135Q8×C167—21no—
136Q8×C167—21no—
137D4×C167—22no—
138D4×C167—21no—
139D4×C167—21no—
140D4×C167—21no—
141D4×C167—42no—
142D4×C167—42no—
143D4×C167—81no—
144D4×C167#4112yescomplemented
145C2×C4×C167—21no—
146C2×C4×C167—21no—
147C2×C4×C167—21no—
148C2×C4×C167—21no—
149C2×C4×C167—21no—
150C2×C4×C167—21no—
151C2×C4×C167—42no—
152C2×C4×C167—81no—
153C2×C4×C167C2×D411yescharacteristic
154C2×C4×C167C2×D411yescharacteristic
155C2×C4×C167C2×D411yescharacteristic
156C2×C4×C167C2×D411yescharacteristic
157C2×C4×C167C2411yescharacteristic
158C22×C167—21no—
159C22×C167—21no—
160C22×C167—21no—
161C22×C167—22no—
162C22×C167—21no—
163C22×C167—21no—
164C22×C167—41no—
165C22×C167—41no—
166C22×C167—41no—
167C22×C167—41no—
168C22×C167—41no—
169C22×C167—84no—
170C22×C167AGL(1,ℤ/8)12yes—
171C22×C167C22×D411yescharacteristic
172C2×D167—41no—
173C2×D167—41no—
174C2×D167—41no—
175C4×C167—21no—
176C4×C167—21no—
177C4×C167—21no—
178C4×C167—21no—
179C4×C167—41no—
180C4×C167—41no—
181C4×C167C2×#4111yescharacteristic
182C4×C167H(5,𝔽2)11yescharacteristic
183Dic167—41no—
184Dic167—41no—
185Dic167—41no—
186C2×C167—22no—
187C2×C167—41no—
188C2×C167—42no—
189C2×C167—81no—
190C2×C167#54611yescharacteristic
191C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C167#57711yescharacteristic
193D167—81no—
194C167#330611yescharacteristic, complemented, Sylow
195#3306—1671nomaximal, Sylow
196#547—1671no—
197#546D16711yescharacteristic, complemented
198#489—1671no—
199#483—1671no—
200#481—1671no—
201#476—1671no—
202#469—1671no—
203#465—1671no—
204#463—1671no—
205#449—1671no—
206#447—1671no—
207#442—1671no—
208#435—1671no—
209#417—1671no—
210C2×#102—1671no—
211C2×#41—3341no—
212C2×#41C2×D16711yescharacteristic
213C22×D4C2×D16711yescharacteristic
214C2×SD16—1671no—
215C2×D8—1671no—
216C2×D8—1671no—
217C2×M16—1671no—
218C22×C8—1671no—
219C4⋊Q8—1671no—
220#124C2×D16711yescharacteristic
221#123—21no—
222#123—3341no—
223#123—3341no—
224C22⋊Q8—3341no—
225C22⋊Q8C2×D16711yescharacteristic
226#121—21no—
227#121—1671no—
228#121—1671no—
229#121C2×D16711yescharacteristic
230#120—21no—
231C4×D4—1671no—
232C4×D4C2×D16711yescharacteristic
233C4×D4C2×D16711yescharacteristic
234C2×C4⋊C4—1671no—
235C2×C4⋊C4—1671no—
236C2×C22⋊C4—21no—
237#106—1671no—
238#106—1671no—
239#106—1671no—
240C4⋊C8—1671no—
241#103—1671no—
242#103—1671no—
243#103—1671no—
244#102—1671no—
245#102—1671no—
246#102—1671no—
247#102—1671no—
248#102—1671no—
249#102—1671no—
250#102—1671no—
251C22⋊C8—1671no—
252C22⋊C8—1671no—
253#97—1671no—
254C24—21no—
255#41—22no—
256#41—41no—
257#41—6684no—
258C2×Q8—3341no—
259C2×Q8C22×D16711yescharacteristic
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—3341no—
266C2×D4—3341no—
267C2×D4—1671no—
268C2×D4C22⋊D16711yescharacteristic
269C2×D4C22×D16711yescharacteristic
270C2×D4C22×D16711yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—3341no—
274C22×C4—3341no—
275C22×C4—3341no—
276C22×C4C22×D16711yescharacteristic
277C22×C4C22×D16711yescharacteristic
278SD16—6681no—
279D8—3342no—
280D8—6681no—
281M16—6681no—
282C2×C8—6681no—
283C2×C8—1671no—
284C2×C8—1671no—
285C2×C8—1671no—
286C2×C8—1671no—
287C4⋊C4—21no—
288C4⋊C4—3341no—
289C4⋊C4—3341no—
290C4⋊C4—3341no—
291C4⋊C4—1671no—
292C4⋊C4—1671no—
293C4⋊C4—1671no—
294C4⋊C4C22⋊D16711yescharacteristic
295C4⋊C4C22×D16711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—3341no—
302C22⋊C4—3341no—
303C22⋊C4—3341no—
304C22⋊C4C22⋊D16711yescharacteristic
305C22⋊C4C22⋊D16711yescharacteristic
306C42C22×D16711yescharacteristic
307C23—21no—
308C23—21no—
309C23—42no—
310C23—42no—
311C23—81no—
312C23—3341no—
313C23C2×C22⋊D16711yescharacteristic
314C23C2×C22⋊D16711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—6682no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—42no—
324D4—81no—
325D4—3341no—
326D4—3341no—
327D4—6682no—
328D4—6682no—
329D4#1458194912yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—81no—
338C2×C4—3341no—
339C2×C4—3341no—
340C2×C4—3341no—
341C2×C4—6682no—
342C2×C4—6682no—
343C2×C4—6682no—
344C2×C4C2×C22⋊D16711yescharacteristic
345C2×C4C2×C22⋊D16711yescharacteristic
346C2×C4C2×C22⋊D16711yescharacteristic
347C2×C4C2×C22⋊D16711yescharacteristic
348C2×C4C23×D16711yescharacteristic
349C8—3341no—
350C8—3341no—
351C8—3341no—
352C8—3341no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—21no—
358C22—21no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—84no—
365C22—6681no—
366C22—6681no—
367C22—6681no—
368C22#1912885712yes—
369C22C22×C22⋊D16711yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—6681no—
377C4—6681no—
378C4—6681no—
379C4C2×#1458194911yescharacteristic
380C4#1912887711yescharacteristic
381C2—22no—
382C2—41no—
383C2—42no—
384C2—81no—
385C2—13361no—
386C2#2492391311yescharacteristic
387C2C2×#1912885711yescharacteristic
388C2#2492408011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#546×C167C2
socleC22×C167C22×D4
hypercenter#546D167
hypocenterC167#3306
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #24923923C2semidirect
2. #24923628C2semidirect
3. #24923597C2semidirect
4. #24923585C2semidirect
5. #24923559C2semidirect
6. #24923520C2semidirect
7. #24923500C2semidirect
8. #24923481C2semidirect
9. #24923414C2semidirect
10. #24923403C2semidirect
11. #24923379C2semidirect
12. #24923341C2semidirect
13. #24923270C2semidirect
14. C2×#19128753C2semidirect
15. #546×C167C2semidirect
17. C2×#14581937C22semidirect
20. C2×#14581934C22semidirect
22. #19128832C22semidirect
27. #19128806C22semidirect
28. C4×D668C22semidirect
30. C2×C4⋊Dic167C22semidirect
31. #19128763C22semidirect
32. #19128763C22semidirect
33. #19128763C22semidirect
34. C4⋊(C167⋊C8)C22semidirect
35. #19128756C22semidirect
36. #19128756C22semidirect
37. #19128756C22semidirect
44. #19128752C22semidirect
46. C22⋊(C167⋊C8)C22semidirect
47. #19128743C22semidirect
144. D4×C167#41semidirect
194. C167#3306semidirect
197. #546D167semidirect
329. D4#14581949semidirect

Maximal subgroups

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

subgroupindex
#249239232
#249236282
#249235972
#249235852
#249235592
#249235202
#249235002
#249234812
#249234142
#249234032
#249233792
#249233412
#249232702
C2×#191287532
#546×C1672
#3306167