Finite Group Databasebeta

← #36336160

Subgroups — #36336160

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#24923958C211yesmaximal, characteristic, complemented
2D4×C22⋊D167C211yesmaximal, characteristic
3#24923632C211yesmaximal, characteristic, complemented
4#24923593C211yesmaximal, characteristic, complemented
5#24923581C211yesmaximal, characteristic, complemented
6#24923567C211yesmaximal, characteristic, complemented
7#24923515C211yesmaximal, characteristic, complemented
8#24923497C211yesmaximal, characteristic, complemented
9#24923468C211yesmaximal, characteristic, complemented
10#24923421C211yesmaximal, characteristic, complemented
11#24923420C211yesmaximal, characteristic, complemented
12#24923366C211yesmaximal, characteristic, complemented
13#24923338C211yesmaximal, characteristic, complemented
14#102×D167C211yesmaximal, characteristic, complemented
15#418×C167C211yesmaximal, characteristic, complemented
16C22×C22⋊D167—21no—
17C2×D4×D167C2211yescharacteristic
18C2×#14581939C2211yescharacteristic, complemented
19C2×#14581938C2211yescharacteristic
20C2×#14581938C2211yescharacteristic, complemented
21C2×C8×D167C2211yescharacteristic, complemented
22#19128828C2211yescharacteristic
23#19128825—21no—
24#19128824—21no—
25#19128823C2211yescharacteristic, complemented
26#19128814—21no—
27#19128812C2211yescharacteristic, complemented
28#19128811C2211yescharacteristic, complemented
29#19128806C2211yescharacteristic
30#19128804—21no—
31#19128803C2211yescharacteristic
32#19128801—21no—
33#19128800—21no—
34#19128797C2211yescharacteristic, complemented
35D4×Dic167C2211yescharacteristic
36C4×C22⋊D167C2211yescharacteristic
37#19128789—21no—
38C4⋊C4×D167C2211yescharacteristic, complemented
39#19128766C2211yescharacteristic, complemented
40#19128765C2211yescharacteristic, complemented
41#19128763C2211yescharacteristic, complemented
42Dic167⋊C8C2211yescharacteristic, complemented
43#19128757C2211yescharacteristic, complemented
44#19128754C2211yescharacteristic, complemented
45#19128753C2211yescharacteristic
46#19128753C2211yescharacteristic, complemented
47#19128752C2211yescharacteristic, complemented
48#19128752C2211yescharacteristic, complemented
49C22⋊(C167⋊C8)C2211yescharacteristic, complemented
50(C2×D167)⋊C8C2211yescharacteristic, complemented
51#19128742C2211yescharacteristic, complemented
52C22×D4×C167C2211yescharacteristic
53C2×M16×C167C2211yescharacteristic, complemented
54#117×C167C2211yescharacteristic, complemented
55#102×C167C2211yescharacteristic
56#102×C167C2211yescharacteristic
57#102×C167C2211yescharacteristic, complemented
58#102×C167C2211yescharacteristic, complemented
59C23×D167—21no—
60C2×Dic334C2311yescharacteristic
61C2×D668C2311yescharacteristic
62C2×C22⋊D167—21no—
63C2×C22⋊D167—21no—
64C2×C22⋊D167—21no—
65C2×C22⋊D167—21no—
66C2×C22⋊D167—21no—
67C2×C22⋊D167—44no—
68C2×C22⋊D167D411yescharacteristic
69D4×D167—22no—
70D4×D167—41no—
71C22×Dic167—21no—
72C2×C4×D167—21no—
73C2×C4×D167C2311yescharacteristic
74#14581939—41no—
75#14581938—22no—
76#14581938—41no—
77C2×C167⋊C8C2311yescharacteristic
78C2×C167⋊C8C2311yescharacteristic
79C8×D167—41no—
80C4⋊Dic167—21no—
81C4⋊Dic167—21no—
82C4⋊Dic167C2311yescharacteristic
83C4⋊Dic167C2311yescharacteristic
84Dic167⋊C4—21no—
85Dic167⋊C4—21no—
86Dic167⋊C4—21no—
87Dic167⋊C4—21no—
88Dic167⋊C4D411yescharacteristic
89C22⋊Dic167—21no—
90C22⋊Dic167—21no—
91C22⋊Dic167—21no—
92C22⋊Dic167D411yescharacteristic
93(C2×D167)⋊C4—21no—
94(C2×D167)⋊C4—21no—
95(C2×D167)⋊C4D411yescharacteristic
96C4×Dic167—21no—
97C4×Dic167C2311yescharacteristic
98C24×C167—21no—
99C2×D4×C167—22no—
100C2×D4×C167—21no—
101C2×D4×C167—41no—
102C2×D4×C167C2311yescharacteristic
103C2×D4×C167C2311yescharacteristic
104C22×C4×C167C2311yescharacteristic
105M16×C167—41no—
106C2×C8×C167C2311yescharacteristic
107C2×C8×C167C2311yescharacteristic
108C4⋊C4×C167C2311yescharacteristic
109C4⋊C4×C167C2311yescharacteristic
110C22⋊C4×C167—21no—
111C42×C167—21no—
112C22×D167—21no—
113C22×D167—42no—
114C22×D167—42no—
115C22×D167—81no—
116C22×D167C2×D411yescharacteristic
117Dic334—21no—
118Dic334—21no—
119D668—21no—
120D668—21no—
121C22⋊D167—22no—
122C22⋊D167—42no—
123C22⋊D167—42no—
124C22⋊D167—42no—
125C22⋊D167—42no—
126C22⋊D167—81no—
127C2×Dic167—21no—
128C2×Dic167—21no—
129C2×Dic167—21no—
130C2×Dic167—42no—
131C2×Dic167C2×D411yescharacteristic
132C2×Dic167C2×D411yescharacteristic
133C2×Dic167C2×D411yescharacteristic
134C4×D167—21no—
135C4×D167—21no—
136C4×D167—42no—
137C167⋊C8—21no—
138C167⋊C8—21no—
139C23×C167—21no—
140C23×C167—21no—
141C23×C167—42no—
142C23×C167—42no—
143C23×C167—81no—
144C23×C167C2×D411yescharacteristic
145D4×C167—21no—
146D4×C167—21no—
147D4×C167—21no—
148D4×C167—42no—
149D4×C167#4112yes—
150C2×C4×C167—21no—
151C2×C4×C167—21no—
152C2×C4×C167—21no—
153C2×C4×C167—21no—
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×C167—84no—
168C22×C167C22×D411yescharacteristic
169C2×D167—21no—
170C2×D167—21no—
171C2×D167—41no—
172C2×D167—41no—
173C2×D167—41no—
174C2×D167—84no—
175C2×D167AGL(1,ℤ/8)12yes—
176C4×C167—21no—
177C4×C167—41no—
178C4×C167—41no—
179C4×C167C2×#4111yescharacteristic
180C4×C167H(5,𝔽2)11yescharacteristic
181Dic167—21no—
182Dic167—21no—
183Dic167—21no—
184Dic167—41no—
185Dic167—41no—
186Dic167—41no—
187C2×C167—42no—
188C2×C167—41no—
189C2×C167—81no—
190C2×C167#54611yescharacteristic
191C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C167#57711yescharacteristic
193D167—22no—
194D167—81no—
195C167#331011yescharacteristic, complemented, Sylow
196#3310—1671nomaximal, Sylow
197#552—1671no—
198D42—1671no—
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#418D16711yescharacteristic, complemented
211C2×#102—1671no—
212C22×D4—3341no—
213C22×D4—1671no—
214C22×D4C2×D16711yescharacteristic
215C2×SD16—1671no—
216C2×SD16—1671no—
217C2×SD16—1671no—
218C2×M16C2×D16711yescharacteristic, complemented
219C22×C8—1671no—
220#127—1671no—
221#126—3341no—
222#125—3341no—
223#125—1671no—
224#123—3341no—
225C22⋊Q8—1671no—
226C22⋊Q8—1671no—
227#121—3341no—
228#121—1671no—
229#121—1671no—
230#120—3341no—
231#120—3341no—
232C4×Q8—1671no—
233C4×D4—1671no—
234C4×D4—1671no—
235#117—3341no—
236#117C2×D16711yescharacteristic, complemented
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#102C2×D16711yescharacteristic
249#102C2×D16711yescharacteristic
250#102C2×D16711yescharacteristic, complemented
251#102C2×D16711yescharacteristic, complemented
252C22⋊C8—1671no—
253C22⋊C8—1671no—
254#97—1671no—
255C24—21no—
256C24—3341no—
257C2×Q8—1671no—
258C2×D4—22no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—3342no—
262C2×D4—3341no—
263C2×D4—3341no—
264C2×D4—3341no—
265C2×D4—3341no—
266C2×D4—3341no—
267C2×D4—6681no—
268C2×D4—6684no—
269C2×D4—1671no—
270C2×D4—1671no—
271C2×D4C22×D16711yescharacteristic
272C2×D4C22×D16711yescharacteristic
273C22×C4—3341no—
274C22×C4—3341no—
275C22×C4—1671no—
276C22×C4C22×D16711yescharacteristic
277SD16—3342no—
278SD16—6681no—
279SD16—6681no—
280M16—41no—
281C2×C8—6681no—
282C2×C8—1671no—
283C2×C8—1671no—
284C2×C8C22×D16711yescharacteristic
285C2×C8C22×D16711yescharacteristic
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—3341no—
299C22⋊C4—3341no—
300C22⋊C4—3341no—
301C22⋊C4—3341no—
302C22⋊C4—3341no—
303C22⋊C4—1671no—
304C22⋊C4—1671no—
305C42—21no—
306C42—3341no—
307C42—1671no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23—13361no—
314C23—3341no—
315C23—6682no—
316C23—6682no—
317C23—1671no—
318C23D4×D16711yescharacteristic
319Q8—3341no—
320Q8—3341no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—13361no—
326D4—3342no—
327D4—3341no—
328D4—3341no—
329D4—6682no—
330D4—6682no—
331D4—6682no—
332D4—6682no—
333D4#1458195012yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—3341no—
339C2×C4—3341no—
340C2×C4—3341no—
341C2×C4—3341no—
342C2×C4—3341no—
343C2×C4—6682no—
344C2×C4—6682no—
345C2×C4—1671no—
346C2×C4—1671no—
347C2×C4—1671no—
348C2×C4D4×D16711yescharacteristic
349C2×C4C23×D16711yescharacteristic
350C8—21no—
351C8—21no—
352C8—3341no—
353C8—3341no—
354C22—21no—
355C22—22no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—84no—
364C22—13364no—
365C22—3341no—
366C22—3341no—
367C22—6681no—
368C22—6681no—
369C22—6681no—
370C22—1672no—
371C22C2×D4×D16711yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—3341no—
376C4—3341no—
377C4—3341no—
378C4—6681no—
379C4—6681no—
380C4—6681no—
381C4C2×#1458195011yescharacteristic
382C4#1912887711yescharacteristic
383C2—41no—
384C2—42no—
385C2—81no—
386C2—13361no—
387C2—3342no—
388C2#2492390911yescharacteristic
389C2AGL(1,ℤ/8)×D16711yescharacteristic
390C2#2492407911yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#418×C167C2
socleC22×C167C22×D4
prime radicalD4×C22⋊D167C2
hypercenter#418D167
hypocenterC167#3310
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #24923958C2semidirect
3. #24923632C2semidirect
4. #24923593C2semidirect
5. #24923581C2semidirect
6. #24923567C2semidirect
7. #24923515C2semidirect
8. #24923497C2semidirect
9. #24923468C2semidirect
10. #24923421C2semidirect
11. #24923420C2semidirect
12. #24923366C2semidirect
13. #24923338C2semidirect
14. #102×D167C2semidirect
15. #418×C167C2semidirect
18. C2×#14581939C22semidirect
20. C2×#14581938C22semidirect
21. C2×C8×D167C22semidirect
25. #19128823C22semidirect
27. #19128812C22semidirect
28. #19128811C22semidirect
34. #19128797C22semidirect
38. C4⋊C4×D167C22semidirect
39. #19128766C22semidirect
40. #19128765C22semidirect
41. #19128763C22semidirect
42. Dic167⋊C8C22semidirect
43. #19128757C22semidirect
44. #19128754C22semidirect
46. #19128753C22semidirect
47. #19128752C22semidirect
48. #19128752C22semidirect
49. C22⋊(C167⋊C8)C22semidirect
50. (C2×D167)⋊C8C22semidirect
51. #19128742C22semidirect
53. C2×M16×C167C22semidirect
54. #117×C167C22semidirect
57. #102×C167C22semidirect
58. #102×C167C22semidirect
195. C167#3310semidirect
210. #418D167semidirect
218. C2×M16C2×D167semidirect
236. #117C2×D167semidirect
250. #102C2×D167semidirect
251. #102C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249239582
D4×C22⋊D1672
#249236322
#249235932
#249235812
#249235672
#249235152
#249234972
#249234682
#249234212
#249234202
#249233662
#249233382
#102×D1672
#418×C1672
#3310167