Finite Group Databasebeta

← #36334275

Subgroups — #36334275

Order 21376.

450 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923798C211yesmaximal, characteristic, complemented
2C4×D4×D167C211yesmaximal, characteristic, complemented
3#24923598C211yesmaximal, characteristic, complemented
4#24923581C211yesmaximal, characteristic, complemented
5#24923570C211yesmaximal, characteristic, complemented
6#24923569C211yesmaximal, characteristic, complemented
7#24923481C211yesmaximal, characteristic, complemented
8#24923418C211yesmaximal, characteristic, complemented
9#24923401C211yesmaximal, characteristic
10#24923381C211yesmaximal, characteristic, complemented
11#24923378C211yesmaximal, characteristic, complemented
12#24923372C211yesmaximal, characteristic, complemented
13#24923370C211yesmaximal, characteristic, complemented
14#24923325C211yesmaximal, characteristic, complemented
15#463×C167C211yesmaximal, characteristic, complemented
16C2×D4×D167C2211yescharacteristic
17C22×C4×D167—21no—
18C2×#14581937C2211yescharacteristic
19C2×#14581935C2211yescharacteristic
20#19128825—21no—
21#19128824C2211yescharacteristic, complemented
22#19128814—21no—
23#19128808C2211yescharacteristic
24#19128803C2211yescharacteristic
25Q8×Dic167C2211yescharacteristic, complemented
26C4×D668C2211yescharacteristic
27D4×Dic167C2211yescharacteristic
28D4×Dic167C2211yescharacteristic, complemented
29#19128794C2211yescharacteristic
30C4×C22⋊D167—21no—
31(C22⋊D167)⋊C4—21no—
32(C22⋊D167)⋊C4—21no—
33#19128788—21no—
34C4⋊C4×D167C2211yescharacteristic
35C22⋊C4×D167—21no—
36C42×D167C2211yescharacteristic
37#19128766C2211yescharacteristic, complemented
38#19128765C2211yescharacteristic, complemented
39#19128764C2211yescharacteristic, complemented
40#19128763C2211yescharacteristic, complemented
41C4⋊(C167⋊C8)C2211yescharacteristic, complemented
42#19128756C2211yescharacteristic, complemented
43#19128754C2211yescharacteristic
44#19128754C2211yescharacteristic
45#19128753C2211yescharacteristic
46#19128753C2211yescharacteristic
47#19128753C2211yescharacteristic, complemented
48(C2×D167)⋊C8C2211yescharacteristic
49(C2×D167)⋊C8C2211yescharacteristic
50#19128742C2211yescharacteristic, complemented
51#19128742C2211yescharacteristic, complemented
52C2×SD16×C167C2211yescharacteristic
53C2×D8×C167C2211yescharacteristic
54#124×C167C2211yescharacteristic, complemented
55C4×D4×C167C2211yescharacteristic
56C4⋊C8×C167C2211yescharacteristic, complemented
57#103×C167C2211yescharacteristic, complemented
58#102×C167C2211yescharacteristic, complemented
59C23×D167—21no—
60C2×D668C2311yescharacteristic
61C2×C22⋊D167—21no—
62C2×C22⋊D167—21no—
63D4×D167—22no—
64D4×D167—22no—
65C22×Dic167—21no—
66C22×Dic167—21no—
67C2×C4×D167—21no—
68C2×C4×D167—21no—
69C2×C4×D167—21no—
70C2×C4×D167—44no—
71C2×C4×D167D411yescharacteristic
72C2×C4×D167D411yescharacteristic
73C2×C4×D167C2311yescharacteristic
74#14581937—22no—
75#14581935—22no—
76C2×C167⋊C8C2311yescharacteristic
77C2×C167⋊C8C2311yescharacteristic
78C4⋊Dic167—21no—
79C4⋊Dic167C2311yescharacteristic
80C4⋊Dic167C2311yescharacteristic
81C4⋊Dic167C2311yescharacteristic
82Dic167⋊C4—21no—
83Dic167⋊C4—21no—
84Dic167⋊C4—21no—
85Dic167⋊C4—21no—
86C22⋊Dic167—21no—
87C22⋊Dic167—21no—
88C22⋊Dic167—21no—
89(C2×D167)⋊C4—21no—
90(C2×D167)⋊C4—21no—
91(C2×D167)⋊C4—21no—
92C4×Dic167—21no—
93C4×Dic167D411yescharacteristic
94C4×Dic167D411yescharacteristic
95C4×Dic167C2311yescharacteristic
96C2×Q8×C167C2311yescharacteristic
97C2×D4×C167C2311yescharacteristic
98C2×D4×C167C2311yescharacteristic
99C22×C4×C167—21no—
100SD16×C167—22no—
101D8×C167—22no—
102C2×C8×C167C2311yescharacteristic
103C2×C8×C167C2311yescharacteristic
104C4⋊C4×C167C2311yescharacteristic
105C22⋊C4×C167—21no—
106C22⋊C4×C167—21no—
107C22⋊C4×C167—21no—
108C42×C167C2311yescharacteristic
109C22×D167—24no—
110C22×D167—21no—
111C22×D167—42no—
112C22×D167—42no—
113C22×D167C2×D411yescharacteristic
114D668—21no—
115D668#4112yes—
116C22⋊D167—24no—
117C22⋊D167—42no—
118C2×Dic167—21no—
119C2×Dic167—21no—
120C2×Dic167—21no—
121C2×Dic167—42no—
122C2×Dic167—42no—
123C2×Dic167C2×D411yescharacteristic
124C2×Dic167C2×D411yescharacteristic
125C2×Dic167C2×D411yescharacteristic
126C4×D167—22no—
127C4×D167—21no—
128C4×D167—21no—
129C4×D167—41no—
130C4×D167—42no—
131C4×D167—42no—
132C4×D167—42no—
133C167⋊C8—21no—
134C167⋊C8—21no—
135C23×C167—21no—
136C23×C167—21no—
137Q8×C167—21no—
138Q8×C167—21no—
139D4×C167—21no—
140D4×C167—21no—
141D4×C167—21no—
142D4×C167#4112yes—
143C2×C4×C167—21no—
144C2×C4×C167—21no—
145C2×C4×C167—42no—
146C2×C4×C167C2×D411yescharacteristic
147C2×C4×C167C2×D411yescharacteristic
148C2×C4×C167C2411yescharacteristic
149C8×C167—21no—
150C8×C167—21no—
151C22×C167—22no—
152C22×C167—41no—
153C22×C167—41no—
154C22×C167—41no—
155C22×C167—41no—
156C22×C167—41no—
157C22×C167C22×D411yescharacteristic
158C2×D167—21no—
159C2×D167—21no—
160C2×D167—22no—
161C2×D167—48no—
162C2×D167—41no—
163C2×D167—41no—
164C2×D167AGL(1,ℤ/8)12yes—
165C4×C167—21no—
166C4×C167—21no—
167C4×C167—41no—
168C4×C167—41no—
169C4×C167C2×#4111yescharacteristic
170C4×C167C2×#4111yescharacteristic
171Dic167—21no—
172Dic167—21no—
173Dic167—21no—
174Dic167—41no—
175Dic167—41no—
176Dic167—41no—
177C2×C167—42no—
178C2×C167—81no—
179C2×C167#52511yescharacteristic
180C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
181C2×C167#57511yescharacteristic
182D167—22no—
183D167—42no—
184C167#311411yescharacteristic, complemented, Sylow
185#3114—1671nomaximal, Sylow
186#529—1671no—
187C2×C4×D4—1671no—
188#483—1671no—
189#480—1671no—
190#478—1671no—
191#478—1671no—
192#463—1671no—
193#463D16711yescharacteristic, complemented
194#450—1671no—
195#447—1671no—
196#442—1671no—
197#442—1671no—
198#440—1671no—
199#440—1671no—
200#432—1671no—
201C22×D4—1671no—
202C23×C4—3341no—
203C2×SD16—1671no—
204C2×SD16C2×D16711yescharacteristic
205C2×D8—1671no—
206C2×D8C2×D16711yescharacteristic
207#126—3341no—
208#125—1671no—
209#124C2×D16711yescharacteristic, complemented
210#123—3341no—
211C22⋊Q8—1671no—
212#121—1671no—
213C4×Q8—1671no—
214C4×D4—3341no—
215C4×D4—3341no—
216C4×D4—3341no—
217C4×D4—1671no—
218C4×D4—1671no—
219C4×D4—1671no—
220C4×D4—1671no—
221C4×D4C2×D16711yescharacteristic
222#117—3341no—
223C2×C4⋊C4—1671no—
224C2×C22⋊C4—3341no—
225C2×C42—1671no—
226#107—1671no—
227#107—1671no—
228#106—1671no—
229#106—1671no—
230C4⋊C8—1671no—
231C4⋊C8C2×D16711yescharacteristic, complemented
232#103—1671no—
233#103C2×D16711yescharacteristic, complemented
234#102—1671no—
235#102—1671no—
236#102—1671no—
237#102—1671no—
238#102—1671no—
239#102C2×D16711yescharacteristic, complemented
240C22⋊C8—1671no—
241C22⋊C8—1671no—
242#97—1671no—
243#97—1671no—
244C24—3341no—
245C2×Q8C22×D16711yescharacteristic
246C2×D4—3342no—
247C2×D4—3342no—
248C2×D4—3341no—
249C2×D4—3341no—
250C2×D4—1671no—
251C2×D4C22×D16711yescharacteristic
252C2×D4C22×D16711yescharacteristic
253C22×C4—21no—
254C22×C4—3341no—
255C22×C4—3341no—
256C22×C4—3341no—
257C22×C4—3341no—
258C22×C4—3341no—
259C22×C4—6684no—
260C22×C4—1671no—
261C22×C4—1671no—
262C22×C4—1671no—
263SD16—22no—
264SD16—3342no—
265D8—22no—
266D8—3342no—
267C2×C8—1671no—
268C2×C8—1671no—
269C2×C8C22×D16711yescharacteristic
270C2×C8C22×D16711yescharacteristic
271C4⋊C4—3341no—
272C4⋊C4—3341no—
273C4⋊C4—3341no—
274C4⋊C4—3341no—
275C4⋊C4—3341no—
276C4⋊C4—1671no—
277C4⋊C4—1671no—
278C4⋊C4—1671no—
279C4⋊C4C22×D16711yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—3341no—
284C22⋊C4—3341no—
285C22⋊C4—3341no—
286C22⋊C4—3341no—
287C22⋊C4—3341no—
288C22⋊C4—3341no—
289C42—3341no—
290C42—1671no—
291C42—1671no—
292C42—1671no—
293C42C22×D16711yescharacteristic
294C23—21no—
295C23—21no—
296C23—3344no—
297C23—3341no—
298C23—6682no—
299C23—6682no—
300C23—1671no—
301Q8—21no—
302Q8—21no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—3344no—
307D4—3341no—
308D4—6682no—
309D4—1672no—
310D4#1458194912yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—3342no—
315C2×C4—3341no—
316C2×C4—3341no—
317C2×C4—3341no—
318C2×C4—3341no—
319C2×C4—3341no—
320C2×C4—6681no—
321C2×C4—6682no—
322C2×C4—6682no—
323C2×C4—6682no—
324C2×C4—6682no—
325C2×C4—6682no—
326C2×C4—1671no—
327C2×C4—1671no—
328C2×C4—1671no—
329C2×C4D4×D16711yescharacteristic
330C2×C4D4×D16711yescharacteristic
331C2×C4C23×D16711yescharacteristic
332C8—21no—
333C8—21no—
334C8—3341no—
335C8—3341no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—3341no—
343C22—3341no—
344C22—3342no—
345C22—6688no—
346C22—6681no—
347C22—6681no—
348C22—1672no—
349C22C2×D4×D16711yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—3341no—
355C4—3341no—
356C4—3341no—
357C4—6681no—
358C4—6681no—
359C4—6681no—
360C4#41×D16711yescharacteristic
361C4C2×#1458194911yescharacteristic
362C2—42no—
363C2—81no—
364C2—3342no—
365C2—6682no—
366C2#2492377911yescharacteristic
367C2AGL(1,ℤ/8)×D16711yescharacteristic
368C2#2492406611yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#463×C167C2
socleC22×C167C22×D4
prime radical#24923401C2
hypercenter#463D167
hypocenterC167#3114
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #24923798C2semidirect
2. C4×D4×D167C2semidirect
3. #24923598C2semidirect
4. #24923581C2semidirect
5. #24923570C2semidirect
6. #24923569C2semidirect
7. #24923481C2semidirect
8. #24923418C2semidirect
10. #24923381C2semidirect
11. #24923378C2semidirect
12. #24923372C2semidirect
13. #24923370C2semidirect
14. #24923325C2semidirect
15. #463×C167C2semidirect
21. #19128824C22semidirect
25. Q8×Dic167C22semidirect
28. D4×Dic167C22semidirect
37. #19128766C22semidirect
38. #19128765C22semidirect
39. #19128764C22semidirect
40. #19128763C22semidirect
41. C4⋊(C167⋊C8)C22semidirect
42. #19128756C22semidirect
47. #19128753C22semidirect
50. #19128742C22semidirect
51. #19128742C22semidirect
54. #124×C167C22semidirect
56. C4⋊C8×C167C22semidirect
57. #103×C167C22semidirect
58. #102×C167C22semidirect
184. C167#3114semidirect
193. #463D167semidirect
209. #124C2×D167semidirect
231. C4⋊C8C2×D167semidirect
233. #103C2×D167semidirect
239. #102C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249237982
C4×D4×D1672
#249235982
#249235812
#249235702
#249235692
#249234812
#249234182
#249234012
#249233812
#249233782
#249233722
#249233702
#249233252
#463×C1672
#3114167