Finite Group Databasebeta

← #36336101

Subgroups — #36336101

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#24923915C211yesmaximal, characteristic, complemented
3#24923629C211yesmaximal, characteristic, complemented
4#24923600C211yesmaximal, characteristic, complemented
5#24923588C211yesmaximal, characteristic, complemented
6#24923560C211yesmaximal, characteristic, complemented
7#24923522C211yesmaximal, characteristic, complemented
8#24923498C211yesmaximal, characteristic, complemented
9#24923485C211yesmaximal, characteristic, complemented
10#24923416C211yesmaximal, characteristic, complemented
11#24923405C211yesmaximal, characteristic, complemented
12#24923382C211yesmaximal, characteristic, complemented
13#24923271C211yesmaximal, characteristic, complemented
14C2×#19128754C211yesmaximal, characteristic, complemented
15#435×C167C211yesmaximal, characteristic, complemented
16C2×#14581950C2211yescharacteristic
17C2×#14581949—21no—
18C22×D668C2211yescharacteristic
19C2×#14581939C2211yescharacteristic, complemented
20C2×D1336C2211yescharacteristic
21C2×D1336C2211yescharacteristic
22#19128832C2211yescharacteristic, complemented
23#19128820C2211yescharacteristic
24#19128817—21no—
25#19128816—21no—
26#19128816—21no—
27#19128812—21no—
28#19128811C2211yescharacteristic
29#19128806C2211yescharacteristic
30#19128806C2211yescharacteristic
31#19128806C2211yescharacteristic, complemented
32#19128805—21no—
33#19128800—21no—
34C4×D668C2211yescharacteristic
35C4×D668C2211yescharacteristic, complemented
36C2×C4⋊Dic167C2211yescharacteristic
37C2×C4⋊Dic167C2211yescharacteristic, complemented
38C22⋊C4×D167—21no—
39#19128764C2211yescharacteristic, complemented
40#19128764C2211yescharacteristic, complemented
41#19128764C2211yescharacteristic, complemented
42#19128757C2211yescharacteristic, complemented
43#19128757C2211yescharacteristic, complemented
44#19128757C2211yescharacteristic, complemented
45#19128754C2211yescharacteristic
46#19128754C2211yescharacteristic
47#19128754C2211yescharacteristic
48#19128754C2211yescharacteristic
49#19128754C2211yescharacteristic
50#19128754C2211yescharacteristic
51#19128754C2211yescharacteristic, complemented
52C2×M16×C167C2211yescharacteristic, complemented
53C22×C8×C167C2211yescharacteristic
54C4×D4×C167C2211yescharacteristic
55C4⋊C8×C167C2211yescharacteristic, complemented
56C22⋊C8×C167C2211yescharacteristic
57C22⋊C8×C167C2211yescharacteristic, complemented
58#97×C167C2211yescharacteristic, complemented
59C23×D167—21no—
60#14581950—22no—
61#14581950—41no—
62#14581949—44no—
63C2×Dic334—21no—
64C2×Dic334C2311yescharacteristic
65C2×D668—22no—
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—
75C22×Dic167—21no—
76C22×Dic167—21no—
77C2×C4×D167—21no—
78C2×C4×D167—21no—
79C2×C4×D167—21no—
80#14581939—41no—
81D1336—22no—
82D1336—41no—
83C4⋊Dic167—21no—
84C4⋊Dic167—21no—
85C4⋊Dic167—21no—
86C4⋊Dic167C2311yescharacteristic
87C4⋊Dic167C2311yescharacteristic
88C4⋊Dic167C2311yescharacteristic
89C4⋊Dic167C2311yescharacteristic
90Dic167⋊C4—21no—
91C22⋊Dic167—21no—
92(C2×D167)⋊C4—21no—
93(C2×D167)⋊C4—21no—
94(C2×D167)⋊C4—21no—
95(C2×D167)⋊C4—21no—
96(C2×D167)⋊C4—21no—
97(C2×D167)⋊C4—21no—
98(C2×D167)⋊C4—21no—
99C2×D4×C167D411yescharacteristic
100C22×C4×C167C2311yescharacteristic
101C22×C4×C167C2311yescharacteristic
102M16×C167—41no—
103C2×C8×C167—41no—
104C2×C8×C167C2311yescharacteristic
105C2×C8×C167C2311yescharacteristic
106C2×C8×C167C2311yescharacteristic
107C2×C8×C167C2311yescharacteristic
108C4⋊C4×C167D411yescharacteristic
109C22⋊C4×C167D411yescharacteristic
110C22⋊C4×C167D411yescharacteristic
111C42×C167C2311yescharacteristic
112C22×D167—21no—
113C22×D167—21no—
114C22×D167—21no—
115C22×D167—42no—
116C22×D167—42no—
117C22×D167—81no—
118Dic334—21no—
119Dic334—21no—
120Dic334—42no—
121D668—21no—
122D668—21no—
123D668—21no—
124D668—21no—
125D668—21no—
126D668—42no—
127D668#4112yescomplemented
128C22⋊D167—42no—
129C22⋊D167—42no—
130C22⋊D167—42no—
131C22⋊D167—81no—
132C2×Dic167—21no—
133C2×Dic167—21no—
134C2×Dic167—21no—
135C2×Dic167—21no—
136C2×Dic167—21no—
137C2×Dic167—42no—
138C2×Dic167—42no—
139C4×D167—42no—
140C4×D167—42no—
141C4×D167—81no—
142C23×C167C2×D411yescharacteristic
143C23×C167C2×D411yescharacteristic
144D4×C167—22no—
145C2×C4×C167—21no—
146C2×C4×C167—21no—
147C2×C4×C167—21no—
148C2×C4×C167—21no—
149C2×C4×C167C2×D411yescharacteristic
150C2×C4×C167C2×D411yescharacteristic
151C2×C4×C167C2×D411yescharacteristic
152C2×C4×C167C2×D411yescharacteristic
153C2×C4×C167C2411yescharacteristic
154C8×C167—21no—
155C8×C167—21no—
156C8×C167—21no—
157C8×C167—21no—
158C22×C167—21no—
159C22×C167—21no—
160C22×C167—21no—
161C22×C167—21no—
162C22×C167—21no—
163C22×C167AGL(1,ℤ/8)12yes—
164C22×C167C22×D411yescharacteristic
165C2×D167—22no—
166C2×D167—41no—
167C2×D167—41no—
168C2×D167—41no—
169C2×D167—41no—
170C2×D167—41no—
171C2×D167—41no—
172C2×D167—41no—
173C2×D167—41no—
174C2×D167—84no—
175C4×C167—21no—
176C4×C167—21no—
177C4×C167—21no—
178C4×C167—21no—
179C4×C167C2×#4111yescharacteristic
180C4×C167H(5,𝔽2)11yescharacteristic
181Dic167—41no—
182Dic167—41no—
183Dic167—41no—
184Dic167—41no—
185Dic167—41no—
186C2×C167—22no—
187C2×C167—41no—
188C2×C167#54611yescharacteristic
189C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C167#57711yescharacteristic
191D167—42no—
192D167—81no—
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#449—1671no—
206#447—1671no—
207#442—1671no—
208#435D16711yescharacteristic, complemented
209#417—1671no—
210C2×#102—1671no—
211C2×#41—3341no—
212C2×#41—1671no—
213C22×D4—1671no—
214C2×SD16—1671no—
215C2×D8—1671no—
216C2×D8—1671no—
217C2×M16C2×D16711yescharacteristic, complemented
218C22×C8C2×D16711yescharacteristic
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#121—1671no—
230#120—3341no—
231C4×D4—1671no—
232C4×D4—1671no—
233C4×D4C2×D16711yescharacteristic
234C2×C4⋊C4—1671no—
235C2×C4⋊C4—1671no—
236C2×C22⋊C4—3341no—
237#106—1671no—
238#106—1671no—
239#106—1671no—
240C4⋊C8C2×D16711yescharacteristic, complemented
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⋊C8C2×D16711yescharacteristic
252C22⋊C8C2×D16711yescharacteristic, complemented
253#97C2×D16711yescharacteristic, complemented
254C24—3341no—
255#41—3342no—
256#41—6681no—
257#41—6684no—
258C2×Q8—3341no—
259C2×Q8—1671no—
260C2×D4—3342no—
261C2×D4—3341no—
262C2×D4—3341no—
263C2×D4—3341no—
264C2×D4—3341no—
265C2×D4—3341no—
266C2×D4—6681no—
267C2×D4—1671no—
268C2×D4—1671no—
269C2×D4—1671no—
270C2×D4D66811yescharacteristic
271C22×C4—3341no—
272C22×C4—3341no—
273C22×C4—3341no—
274C22×C4—3341no—
275C22×C4—3341no—
276C22×C4C22×D16711yescharacteristic
277C22×C4C22×D16711yescharacteristic
278SD16—6681no—
279D8—3342no—
280D8—6681no—
281M16—41no—
282C2×C8—41no—
283C2×C8C22×D16711yescharacteristic
284C2×C8C22×D16711yescharacteristic
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⋊C4D66811yescharacteristic
296C22⋊C4—3341no—
297C22⋊C4—3341no—
298C22⋊C4—3341no—
299C22⋊C4—3341no—
300C22⋊C4—3341no—
301C22⋊C4—3341no—
302C22⋊C4—3341no—
303C22⋊C4—3341no—
304C22⋊C4D66811yescharacteristic
305C22⋊C4D66811yescharacteristic
306C42C22×D16711yescharacteristic
307C23—13361no—
308C23—3341no—
309C23—3341no—
310C23—3341no—
311C23—6682no—
312C23—6682no—
313C23C2×D66811yescharacteristic
314C23C2×D66811yescharacteristic
315Q8—3341no—
316Q8—3341no—
317Q8—6682no—
318D4—22no—
319D4—13361no—
320D4—3341no—
321D4—3341no—
322D4—3341no—
323D4—3341no—
324D4—3341no—
325D4—6682no—
326D4—6682no—
327D4—6682no—
328D4—6682no—
329D4—1672no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—13361no—
335C2×C4—3341no—
336C2×C4—3341no—
337C2×C4—3341no—
338C2×C4—3341no—
339C2×C4—3341no—
340C2×C4—6682no—
341C2×C4—6682no—
342C2×C4—6682no—
343C2×C4—6682no—
344C2×C4C2×D66811yescharacteristic
345C2×C4C2×D66811yescharacteristic
346C2×C4C2×D66811yescharacteristic
347C2×C4C2×D66811yescharacteristic
348C2×C4C23×D16711yescharacteristic
349C8—21no—
350C8—21no—
351C8—21no—
352C8—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—13364no—
359C22—3342no—
360C22—6681no—
361C22—6681no—
362C22—6681no—
363C22—6681no—
364C22—6681no—
365C22—6681no—
366C22—6681no—
367C22—6681no—
368C22#1912886012yes—
369C22C22×D66811yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—6681no—
375C4—6681no—
376C4—6681no—
377C4—6681no—
378C4—6681no—
379C4#41×D16711yescharacteristic
380C4#1912887711yescharacteristic
381C2—22no—
382C2—41no—
383C2—13361no—
384C2—13361no—
385C2—6682no—
386C2#2492391411yescharacteristic
387C2C2×#1912886011yescharacteristic
388C2#2492408111yescharacteristic
subgroupG/H
centerC22C22×D668
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#435×C167C2
socleC22×C167C22×D4
hypercenter#435D167
hypocenterC167#3306
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #24923923C2semidirect
2. #24923915C2semidirect
3. #24923629C2semidirect
4. #24923600C2semidirect
5. #24923588C2semidirect
6. #24923560C2semidirect
7. #24923522C2semidirect
8. #24923498C2semidirect
9. #24923485C2semidirect
10. #24923416C2semidirect
11. #24923405C2semidirect
12. #24923382C2semidirect
13. #24923271C2semidirect
14. C2×#19128754C2semidirect
15. #435×C167C2semidirect
19. C2×#14581939C22semidirect
22. #19128832C22semidirect
31. #19128806C22semidirect
35. C4×D668C22semidirect
37. C2×C4⋊Dic167C22semidirect
39. #19128764C22semidirect
40. #19128764C22semidirect
41. #19128764C22semidirect
42. #19128757C22semidirect
43. #19128757C22semidirect
44. #19128757C22semidirect
51. #19128754C22semidirect
52. C2×M16×C167C22semidirect
55. C4⋊C8×C167C22semidirect
57. C22⋊C8×C167C22semidirect
58. #97×C167C22semidirect
127. D668#41semidirect
194. C167#3306semidirect
208. #435D167semidirect
217. C2×M16C2×D167semidirect
240. C4⋊C8C2×D167semidirect
252. C22⋊C8C2×D167semidirect
253. #97C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249239232
#249239152
#249236292
#249236002
#249235882
#249235602
#249235222
#249234982
#249234852
#249234162
#249234052
#249233822
#249232712
C2×#191287542
#435×C1672
#3306167