Finite Group Databasebeta

← #36336159

Subgroups — #36336159

Order 21376.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923959C211yesmaximal, characteristic, complemented
2#24923633C211yesmaximal, characteristic, complemented
3#24923590C211yesmaximal, characteristic, complemented
4#24923580C211yesmaximal, characteristic, complemented
5#24923569C211yesmaximal, characteristic, complemented
6#24923513C211yesmaximal, characteristic, complemented
7#24923496C211yesmaximal, characteristic, complemented
8#24923466C211yesmaximal, characteristic, complemented
9#24923420C211yesmaximal, characteristic, complemented
10#24923420C211yesmaximal, characteristic, complemented
11#24923369C211yesmaximal, characteristic, complemented
12#24923341C211yesmaximal, characteristic, complemented
13#24923274C211yesmaximal, characteristic, complemented
14C2×#19128753C211yesmaximal, characteristic, complemented
15D42×C167C211yesmaximal, characteristic
16C2×#14581938C2211yescharacteristic
17C2×#14581938C2211yescharacteristic, complemented
18C2×#14581938C2211yescharacteristic, complemented
19C2×#14581934C2211yescharacteristic, complemented
20C22×C167⋊C8C2211yescharacteristic, complemented
21#19128825—21no—
22#19128824C2211yescharacteristic, complemented
23#19128822—21no—
24#19128813—21no—
25#19128811C2211yescharacteristic, complemented
26#19128811C2211yescharacteristic, complemented
27C4×Dic334C2211yescharacteristic, complemented
28#19128791C2211yescharacteristic, complemented
29#19128788—21no—
30C2×C4⋊Dic167C2211yescharacteristic, complemented
31#19128765C2211yescharacteristic, complemented
32#19128765C2211yescharacteristic, complemented
33#19128763C2211yescharacteristic, complemented
34C4⋊(C167⋊C8)C2211yescharacteristic, complemented
35#19128755C2211yescharacteristic, complemented
36#19128753C2211yescharacteristic
37#19128753C2211yescharacteristic
38#19128753C2211yescharacteristic
39#19128753C2211yescharacteristic, complemented
40#19128753C2211yescharacteristic, complemented
41#19128753C2211yescharacteristic, complemented
42#19128753C2211yescharacteristic, complemented
43#19128753C2211yescharacteristic, complemented
44#19128753C2211yescharacteristic, complemented
45C22⋊(C167⋊C8)C2211yescharacteristic, complemented
46C22⋊(C167⋊C8)C2211yescharacteristic, complemented
47#19128743C2211yescharacteristic, complemented
48C22×D4×C167—21no—
49C22×D4×C167C2211yescharacteristic
50C22×D4×C167C2211yescharacteristic
51#127×C167C2211yescharacteristic
52#121×C167—21no—
53#121×C167C2211yescharacteristic
54#121×C167C2211yescharacteristic
55#120×C167—21no—
56#120×C167—21no—
57C4×D4×C167C2211yescharacteristic
58C4×D4×C167C2211yescharacteristic
59C2×Dic334C2311yescharacteristic
60C22×Dic167—21no—
61#14581938—22no—
62#14581938—41no—
63#14581938—41no—
64#14581934—41no—
65C2×C167⋊C8—41no—
66C2×C167⋊C8C2311yescharacteristic
67C2×C167⋊C8C2311yescharacteristic
68C2×C167⋊C8C2311yescharacteristic
69C2×C167⋊C8C2311yescharacteristic
70C4⋊Dic167—21no—
71C4⋊Dic167C2311yescharacteristic
72C4⋊Dic167C2311yescharacteristic
73C4⋊Dic167C2311yescharacteristic
74Dic167⋊C4—21no—
75Dic167⋊C4—21no—
76Dic167⋊C4—21no—
77Dic167⋊C4—21no—
78Dic167⋊C4—21no—
79C22⋊Dic167—21no—
80C22⋊Dic167—21no—
81C22⋊Dic167—21no—
82C4×Dic167—21no—
83C4×Dic167—21no—
84C24×C167—21no—
85C24×C167—21no—
86C2×D4×C167—22no—
87C2×D4×C167—22no—
88C2×D4×C167—21no—
89C2×D4×C167—21no—
90C2×D4×C167—21no—
91C2×D4×C167—21no—
92C2×D4×C167—21no—
93C2×D4×C167—21no—
94C2×D4×C167—41no—
95C2×D4×C167—41no—
96C2×D4×C167—44no—
97C2×D4×C167D411yescharacteristic
98C2×D4×C167C2311yescharacteristic
99C2×D4×C167C2311yescharacteristic
100C2×D4×C167C2311yescharacteristic
101C22×C4×C167—21no—
102C22×C4×C167C2311yescharacteristic
103C22×C4×C167C2311yescharacteristic
104C4⋊C4×C167D411yescharacteristic
105C4⋊C4×C167C2311yescharacteristic
106C22⋊C4×C167—21no—
107C22⋊C4×C167—21no—
108C22⋊C4×C167—21no—
109C22⋊C4×C167D411yescharacteristic
110C22⋊C4×C167D411yescharacteristic
111C42×C167C2311yescharacteristic
112Dic334—21no—
113Dic334—21no—
114C2×Dic167—21no—
115C2×Dic167—21no—
116C2×Dic167—21no—
117C2×Dic167—21no—
118C2×Dic167—42no—
119C167⋊C8—21no—
120C167⋊C8—21no—
121C167⋊C8—21no—
122C167⋊C8—21no—
123C23×C167—21no—
124C23×C167—21no—
125C23×C167—21no—
126C23×C167—42no—
127C23×C167—42no—
128C23×C167—42no—
129C23×C167—42no—
130C23×C167—81no—
131C23×C167—81no—
132C23×C167C2×D411yescharacteristic
133C23×C167C2×D411yescharacteristic
134D4×C167—22no—
135D4×C167—21no—
136D4×C167—21no—
137D4×C167—21no—
138D4×C167—21no—
139D4×C167—21no—
140D4×C167—42no—
141D4×C167—42no—
142D4×C167—42no—
143D4×C167—42no—
144D4×C167—42no—
145D4×C167—81no—
146D4×C167#4112yes—
147C2×C4×C167—21no—
148C2×C4×C167—21no—
149C2×C4×C167—21no—
150C2×C4×C167—21no—
151C2×C4×C167—21no—
152C2×C4×C167—42no—
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—41no—
170C22×C167—41no—
171C22×C167—41no—
172C22×C167—84no—
173C22×C167—84no—
174C22×C167AGL(1,ℤ/8)12yes—
175C22×C167C22×D411yescharacteristic
176C4×C167—21no—
177C4×C167—21no—
178C4×C167—21no—
179C4×C167—21no—
180C4×C167—41no—
181C4×C167C2×#4111yescharacteristic
182C4×C167H(5,𝔽2)11yescharacteristic
183Dic167—41no—
184Dic167—41no—
185Dic167—41no—
186Dic167—41no—
187C2×C167—22no—
188C2×C167—41no—
189C2×C167—42no—
190C2×C167—81no—
191C2×C167—81no—
192C2×C167#54611yescharacteristic
193C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
194C2×C167#57711yescharacteristic
195C167#331011yescharacteristic, complemented, Sylow
196#3310—1671nomaximal, Sylow
197#552—1671no—
198D42D16711yescharacteristic
199#490—1671no—
200#482—1671no—
201#480—1671no—
202#478—1671no—
203#468—1671no—
204#465—1671no—
205#460—1671no—
206#450—1671no—
207#450—1671no—
208#440—1671no—
209#435—1671no—
210#418—1671no—
211C2×#102—1671no—
212C22×D4—21no—
213C22×D4C2×D16711yescharacteristic
214C22×D4C2×D16711yescharacteristic
215C2×SD16—1671no—
216C2×SD16—1671no—
217C2×SD16—1671no—
218C2×M16—1671no—
219C22×C8—1671no—
220#127C2×D16711yescharacteristic
221#126—3341no—
222#125—3341no—
223#125—1671no—
224#123—3341no—
225C22⋊Q8—1671no—
226C22⋊Q8—1671no—
227#121—21no—
228#121C2×D16711yescharacteristic
229#121C2×D16711yescharacteristic
230#120—21no—
231#120—21no—
232C4×Q8—1671no—
233C4×D4C2×D16711yescharacteristic
234C4×D4C2×D16711yescharacteristic
235#117—3341no—
236#117—1671no—
237C2×C4⋊C4—1671no—
238#107—1671no—
239#107—1671no—
240#106—1671no—
241C4⋊C8—1671no—
242#103—1671no—
243#102—1671no—
244#102—1671no—
245#102—1671no—
246#102—1671no—
247#102—1671no—
248#102—1671no—
249#102—1671no—
250#102—1671no—
251#102—1671no—
252C22⋊C8—1671no—
253C22⋊C8—1671no—
254#97—1671no—
255C24—21no—
256C24—21no—
257C2×Q8—1671no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—44no—
269C2×D4C22⋊D16711yescharacteristic
270C2×D4C22×D16711yescharacteristic
271C2×D4C22×D16711yescharacteristic
272C2×D4C22×D16711yescharacteristic
273C22×C4—21no—
274C22×C4—3341no—
275C22×C4C22×D16711yescharacteristic
276C22×C4C22×D16711yescharacteristic
277SD16—3342no—
278SD16—6681no—
279SD16—6681no—
280M16—6681no—
281C2×C8—6681no—
282C2×C8—1671no—
283C2×C8—1671no—
284C2×C8—1671no—
285C2×C8—1671no—
286C4⋊C4—3341no—
287C4⋊C4—3341no—
288C4⋊C4—3341no—
289C4⋊C4—3341no—
290C4⋊C4—3341no—
291C4⋊C4—3341no—
292C4⋊C4—1671no—
293C4⋊C4—1671no—
294C4⋊C4—1671no—
295C4⋊C4C22⋊D16711yescharacteristic
296C4⋊C4C22×D16711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—3341no—
301C22⋊C4—3341no—
302C22⋊C4—3341no—
303C22⋊C4C22⋊D16711yescharacteristic
304C22⋊C4C22⋊D16711yescharacteristic
305C42—3341no—
306C42—3341no—
307C42C22×D16711yescharacteristic
308C23—21no—
309C23—21no—
310C23—21no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23C2×C22⋊D16711yescharacteristic
318C23C2×C22⋊D16711yescharacteristic
319Q8—3341no—
320Q8—3341no—
321D4—22no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—42no—
329D4—42no—
330D4—42no—
331D4—42no—
332D4—81no—
333D4#1458194912yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—3341no—
341C2×C4—3341no—
342C2×C4—3341no—
343C2×C4—3341no—
344C2×C4—6682no—
345C2×C4C2×C22⋊D16711yescharacteristic
346C2×C4C2×C22⋊D16711yescharacteristic
347C2×C4C2×C22⋊D16711yescharacteristic
348C2×C4C2×C22⋊D16711yescharacteristic
349C2×C4C23×D16711yescharacteristic
350C8—3341no—
351C8—3341no—
352C8—3341no—
353C8—3341no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—84no—
369C22—84no—
370C22#1912885812yes—
371C22C22×C22⋊D16711yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—6681no—
378C4—6681no—
379C4—6681no—
380C4—6681no—
381C4C2×#1458194911yescharacteristic
382C4#1912887711yescharacteristic
383C2—22no—
384C2—41no—
385C2—42no—
386C2—81no—
387C2—81no—
388C2#2492391311yescharacteristic
389C2C2×#1912885811yescharacteristic
390C2#2492408011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting, prime radicalD42×C167C2
socleC22×C167C22×D4
hypercenterD42D167
hypocenterC167#3310
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #24923959C2semidirect
2. #24923633C2semidirect
3. #24923590C2semidirect
4. #24923580C2semidirect
5. #24923569C2semidirect
6. #24923513C2semidirect
7. #24923496C2semidirect
8. #24923466C2semidirect
9. #24923420C2semidirect
10. #24923420C2semidirect
11. #24923369C2semidirect
12. #24923341C2semidirect
13. #24923274C2semidirect
14. C2×#19128753C2semidirect
17. C2×#14581938C22semidirect
18. C2×#14581938C22semidirect
19. C2×#14581934C22semidirect
20. C22×C167⋊C8C22semidirect
22. #19128824C22semidirect
25. #19128811C22semidirect
26. #19128811C22semidirect
27. C4×Dic334C22semidirect
28. #19128791C22semidirect
30. C2×C4⋊Dic167C22semidirect
31. #19128765C22semidirect
32. #19128765C22semidirect
33. #19128763C22semidirect
34. C4⋊(C167⋊C8)C22semidirect
35. #19128755C22semidirect
39. #19128753C22semidirect
40. #19128753C22semidirect
41. #19128753C22semidirect
42. #19128753C22semidirect
43. #19128753C22semidirect
44. #19128753C22semidirect
45. C22⋊(C167⋊C8)C22semidirect
46. C22⋊(C167⋊C8)C22semidirect
47. #19128743C22semidirect
195. C167#3310semidirect

Maximal subgroups

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

subgroupindex
#249239592
#249236332
#249235902
#249235802
#249235692
#249235132
#249234962
#249234662
#249234202
#249233692
#249233412
#249232742
C2×#191287532
D42×C1672
#3310167