Finite Group Databasebeta

← #36335955

Subgroups — #36335955

Order 21376.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923977C211yesmaximal, characteristic, complemented
2D4×C22⋊D167C211yesmaximal, characteristic
3#24923614C211yesmaximal, characteristic, complemented
4#24923588C211yesmaximal, characteristic, complemented
5#24923586C211yesmaximal, characteristic, complemented
6#24923557C211yesmaximal, characteristic, complemented
7#24923500C211yesmaximal, characteristic, complemented
8#24923495C211yesmaximal, characteristic, complemented
9#24923467C211yesmaximal, characteristic, complemented
10#24923418C211yesmaximal, characteristic, complemented
11SD16×Dic167C211yesmaximal, characteristic, complemented
12(C22⋊D167)⋊C8C211yesmaximal, characteristic, complemented
13C2×#19128753C211yesmaximal, characteristic, complemented
14#102×D167C211yesmaximal, characteristic, complemented
15#450×C167C211yesmaximal, characteristic, complemented
16C22×C22⋊D167—21no—
17C2×D4×D167C2211yescharacteristic
18C2×#14581937C2211yescharacteristic, complemented
19C22×C167⋊C8C2211yescharacteristic, complemented
20C2×C8×D167C2211yescharacteristic, complemented
21#19128831C2211yescharacteristic, complemented
22#19128828C2211yescharacteristic
23#19128822—21no—
24#19128812—21no—
25#19128809—21no—
26#19128808C2211yescharacteristic, complemented
27#19128806C2211yescharacteristic
28#19128804—21no—
29#19128803C2211yescharacteristic
30#19128801—21no—
31#19128800—21no—
32Q8×Dic167C2211yescharacteristic, complemented
33D4×Dic167C2211yescharacteristic
34C4×C22⋊D167C2211yescharacteristic
35(C22⋊D167)⋊C4—21no—
36C2×C4⋊Dic167C2211yescharacteristic, complemented
37C4⋊C4×D167C2211yescharacteristic, complemented
38#19128764C2211yescharacteristic, complemented
39#19128764C2211yescharacteristic, complemented
40#19128763C2211yescharacteristic, complemented
41Dic167⋊C8C2211yescharacteristic, complemented
42#19128756C2211yescharacteristic, complemented
43#19128754C2211yescharacteristic, complemented
44#19128754C2211yescharacteristic, complemented
45#19128753C2211yescharacteristic
46#19128753C2211yescharacteristic
47#19128753C2211yescharacteristic, complemented
48#19128753C2211yescharacteristic, complemented
49#19128752C2211yescharacteristic, complemented
50(C2×D167)⋊C8C2211yescharacteristic, complemented
51C8×Dic167C2211yescharacteristic, complemented
52C22×D4×C167C2211yescharacteristic
53C2×SD16×C167C2211yescharacteristic
54C2×SD16×C167C2211yescharacteristic, complemented
55C22⋊Q8×C167C2211yescharacteristic, complemented
56#102×C167C2211yescharacteristic
57#102×C167C2211yescharacteristic, complemented
58C22⋊C8×C167C2211yescharacteristic, complemented
59C23×D167—21no—
60C2×Dic334—21no—
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—
72C22×Dic167—21no—
73C2×C4×D167—21no—
74C2×C4×D167C2311yescharacteristic
75#14581937—41no—
76C2×C167⋊C8—41no—
77C2×C167⋊C8C2311yescharacteristic
78C2×C167⋊C8C2311yescharacteristic
79C8×D167—41no—
80C4⋊Dic167—21no—
81C4⋊Dic167—21no—
82C4⋊Dic167C2311yescharacteristic
83C4⋊Dic167C2311yescharacteristic
84C4⋊Dic167C2311yescharacteristic
85Dic167⋊C4—21no—
86Dic167⋊C4—21no—
87Dic167⋊C4—21no—
88Dic167⋊C4D411yescharacteristic
89C22⋊Dic167—21no—
90C22⋊Dic167—21no—
91C22⋊Dic167D411yescharacteristic
92(C2×D167)⋊C4—21no—
93(C2×D167)⋊C4—21no—
94(C2×D167)⋊C4D411yescharacteristic
95C4×Dic167—21no—
96C4×Dic167C2311yescharacteristic
97C24×C167—21no—
98C2×Q8×C167C2311yescharacteristic
99C2×D4×C167—22no—
100C2×D4×C167—21no—
101C2×D4×C167—41no—
102C2×D4×C167C2311yescharacteristic
103C2×D4×C167C2311yescharacteristic
104C22×C4×C167C2311yescharacteristic
105SD16×C167—22no—
106SD16×C167—41no—
107C2×C8×C167C2311yescharacteristic
108C2×C8×C167C2311yescharacteristic
109C4⋊C4×C167—21no—
110C4⋊C4×C167C2311yescharacteristic
111C22⋊C4×C167—21no—
112C22×D167—21no—
113C22×D167—42no—
114C22×D167—42no—
115C22×D167—81no—
116C22×D167C2×D411yescharacteristic
117Dic334—81no—
118D668—21no—
119D668—21no—
120C22⋊D167—42no—
121C22⋊D167—42no—
122C22⋊D167—42no—
123C22⋊D167—42no—
124C22⋊D167—81no—
125C22⋊D167SD1614yescomplemented
126C2×Dic167—21no—
127C2×Dic167—21no—
128C2×Dic167—21no—
129C2×Dic167—42no—
130C2×Dic167—81no—
131C2×Dic167C2×D411yescharacteristic
132C2×Dic167C2×D411yescharacteristic
133C2×Dic167C2×D411yescharacteristic
134C4×D167—21no—
135C4×D167—21no—
136C4×D167—81no—
137C167⋊C8—21no—
138C167⋊C8—21no—
139C23×C167—21no—
140C23×C167—21no—
141C23×C167—42no—
142C23×C167—42no—
143C23×C167—81no—
144C23×C167C2×D411yescharacteristic
145Q8×C167—21no—
146Q8×C167—21no—
147D4×C167—21no—
148D4×C167—21no—
149D4×C167—21no—
150D4×C167—42no—
151D4×C167#4112yes—
152C2×C4×C167—21no—
153C2×C4×C167—21no—
154C2×C4×C167—21no—
155C2×C4×C167—21no—
156C2×C4×C167C2×D411yescharacteristic
157C2×C4×C167C2411yescharacteristic
158C8×C167—21no—
159C8×C167—21no—
160C22×C167—21no—
161C22×C167—21no—
162C22×C167—22no—
163C22×C167—41no—
164C22×C167—41no—
165C22×C167—41no—
166C22×C167—41no—
167C22×C167—41no—
168C22×C167—84no—
169C22×C167C2×SD1612yes—
170C22×C167C22×D411yescharacteristic
171C2×D167—21no—
172C2×D167—21no—
173C2×D167—41no—
174C2×D167—41no—
175C2×D167—41no—
176C2×D167—84no—
177C2×D167C2×SD1612yes—
178C4×C167—21no—
179C4×C167—41no—
180C4×C167—41no—
181C4×C167C2×#4111yescharacteristic
182C4×C167H(5,𝔽2)11yescharacteristic
183Dic167—21no—
184Dic167—21no—
185Dic167—41no—
186Dic167—41no—
187Dic167—41no—
188Dic167C2×SD1612yes—
189C2×C167—22no—
190C2×C167—42no—
191C2×C167—81no—
192C2×C167#54611yescharacteristic
193C2×C167C22×SD1611yescharacteristic
194C2×C167#57611yescharacteristic
195D167—22no—
196D167—81no—
197C167#329511yescharacteristic, complemented, Sylow
198#3295—1671nomaximal, Sylow
199#554—1671no—
200D42—1671no—
201#486—1671no—
202#481—1671no—
203#481—1671no—
204#476—1671no—
205#465—1671no—
206#465—1671no—
207#460—1671no—
208#450—1671no—
209#450D16711yescharacteristic, complemented
210C4×SD16—1671no—
211C8×D4—1671no—
212C2×#102—1671no—
213C2×#102—1671no—
214C22×D4—3341no—
215C22×D4—1671no—
216C22×D4C2×D16711yescharacteristic
217C2×SD16—1671no—
218C2×SD16C2×D16711yescharacteristic
219C2×SD16C2×D16711yescharacteristic, complemented
220C22×C8—1671no—
221C22×C8—1671no—
222C4⋊Q8—1671no—
223#127—1671no—
224#125—3341no—
225C22⋊Q8—3341no—
226C22⋊Q8—3341no—
227C22⋊Q8—1671no—
228C22⋊Q8C2×D16711yescharacteristic, complemented
229#121—3341no—
230#121—1671no—
231#121—1671no—
232#120—3341no—
233#120—3341no—
234C4×Q8—1671no—
235C4×D4—3341no—
236C4×D4—1671no—
237C4×D4—1671no—
238C2×C4⋊C4—1671no—
239C2×C4⋊C4—1671no—
240#106—1671no—
241#106—1671no—
242#106—1671no—
243C4⋊C8—1671no—
244#103—1671no—
245#102—1671no—
246#102—1671no—
247#102—1671no—
248#102—1671no—
249#102—1671no—
250#102—1671no—
251#102—1671no—
252#102C2×D16711yescharacteristic
253#102C2×D16711yescharacteristic, complemented
254C22⋊C8—1671no—
255C22⋊C8C2×D16711yescharacteristic, complemented
256C4×C8—1671no—
257C24—21no—
258C24—3341no—
259C2×Q8—3341no—
260C2×Q8C22×D16711yescharacteristic
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—3342no—
265C2×D4—3341no—
266C2×D4—3341no—
267C2×D4—3341no—
268C2×D4—3341no—
269C2×D4—3341no—
270C2×D4—6681no—
271C2×D4—6684no—
272C2×D4—1671no—
273C2×D4—1671no—
274C2×D4C22×D16711yescharacteristic
275C2×D4C22×D16711yescharacteristic
276C22×C4—3341no—
277C22×C4—3341no—
278C22×C4—3341no—
279C22×C4—1671no—
280C22×C4C22×D16711yescharacteristic
281SD16—22no—
282SD16—41no—
283SD16—6681no—
284C2×C8—6681no—
285C2×C8—6681no—
286C2×C8—1671no—
287C2×C8—1671no—
288C2×C8C22×D16711yescharacteristic
289C2×C8C22×D16711yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—3341no—
292C4⋊C4—3341no—
293C4⋊C4—3341no—
294C4⋊C4—3341no—
295C4⋊C4—3341no—
296C4⋊C4—1671no—
297C4⋊C4—1671no—
298C4⋊C4—1671no—
299C4⋊C4—1671no—
300C4⋊C4C22×D16711yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—3341no—
303C22⋊C4—3341no—
304C22⋊C4—3341no—
305C22⋊C4—3341no—
306C22⋊C4—1671no—
307C22⋊C4—1671no—
308C42—3341no—
309C42—1671no—
310C23—21no—
311C23—21no—
312C23—42no—
313C23—42no—
314C23—81no—
315C23—13361no—
316C23—3341no—
317C23—6682no—
318C23—6682no—
319C23—1671no—
320C23D4×D16711yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—13361no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—13361no—
329D4—3341no—
330D4—3341no—
331D4—6682no—
332D4—6682no—
333D4—6682no—
334D4—6682no—
335D4—1674no—
336D4#1458194912yes—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—13361no—
342C2×C4—13361no—
343C2×C4—3341no—
344C2×C4—3341no—
345C2×C4—3341no—
346C2×C4—3341no—
347C2×C4—3341no—
348C2×C4—6682no—
349C2×C4—1671no—
350C2×C4—1671no—
351C2×C4—1671no—
352C2×C4D4×D16711yescharacteristic
353C2×C4C23×D16711yescharacteristic
354C8—21no—
355C8—21no—
356C8—3341no—
357C8—3341no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—84no—
367C22—13364no—
368C22—3341no—
369C22—3341no—
370C22—6681no—
371C22—6681no—
372C22—6681no—
373C22—1672no—
374C22SD16×D16712yes—
375C22C2×D4×D16711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—3341no—
380C4—3341no—
381C4—6681no—
382C4—6681no—
383C4—6681no—
384C4—1672no—
385C4C2×#1458194911yescharacteristic
386C4#1912887711yescharacteristic
387C2—22no—
388C2—42no—
389C2—81no—
390C2—13361no—
391C2—3342no—
392C2#2492390611yescharacteristic
393C2C2×SD16×D16711yescharacteristic
394C2#2492407311yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#450×C167C2
socleC22×C167C22×D4
prime radicalD4×C22⋊D167C2
hypercenter#450D167
hypocenterC167#3295
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #24923977C2semidirect
3. #24923614C2semidirect
4. #24923588C2semidirect
5. #24923586C2semidirect
6. #24923557C2semidirect
7. #24923500C2semidirect
8. #24923495C2semidirect
9. #24923467C2semidirect
10. #24923418C2semidirect
11. SD16×Dic167C2semidirect
12. (C22⋊D167)⋊C8C2semidirect
13. C2×#19128753C2semidirect
14. #102×D167C2semidirect
15. #450×C167C2semidirect
18. C2×#14581937C22semidirect
19. C22×C167⋊C8C22semidirect
20. C2×C8×D167C22semidirect
21. #19128831C22semidirect
26. #19128808C22semidirect
32. Q8×Dic167C22semidirect
36. C2×C4⋊Dic167C22semidirect
37. C4⋊C4×D167C22semidirect
38. #19128764C22semidirect
39. #19128764C22semidirect
40. #19128763C22semidirect
41. Dic167⋊C8C22semidirect
42. #19128756C22semidirect
43. #19128754C22semidirect
44. #19128754C22semidirect
47. #19128753C22semidirect
48. #19128753C22semidirect
49. #19128752C22semidirect
50. (C2×D167)⋊C8C22semidirect
51. C8×Dic167C22semidirect
54. C2×SD16×C167C22semidirect
55. C22⋊Q8×C167C22semidirect
57. #102×C167C22semidirect
58. C22⋊C8×C167C22semidirect
125. C22⋊D167SD16semidirect
197. C167#3295semidirect
209. #450D167semidirect
219. C2×SD16C2×D167semidirect
228. C22⋊Q8C2×D167semidirect
253. #102C2×D167semidirect
255. C22⋊C8C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249239772
D4×C22⋊D1672
#249236142
#249235882
#249235862
#249235572
#249235002
#249234952
#249234672
#249234182
SD16×Dic1672
(C22⋊D167)⋊C82
C2×#191287532
#102×D1672
#450×C1672
#3295167