Finite Group Databasebeta

← #36336163

Subgroups — #36336163

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#24923961C211yesmaximal, characteristic, complemented
2D4×C22⋊D167C211yesmaximal, characteristic
3#24923632C211yesmaximal, characteristic, complemented
4#24923591C211yesmaximal, characteristic, complemented
5#24923583C211yesmaximal, characteristic, complemented
6#24923567C211yesmaximal, characteristic, complemented
7#24923512C211yesmaximal, characteristic, complemented
8#24923500C211yesmaximal, characteristic, complemented
9#24923467C211yesmaximal, characteristic, complemented
10#24923418C211yesmaximal, characteristic, complemented
11#24923370C211yesmaximal, characteristic, complemented
12#24923337C211yesmaximal, characteristic, complemented
13#24923272C211yesmaximal, characteristic, complemented
14C2×#19128753C211yesmaximal, characteristic, complemented
15#450×C167C211yesmaximal, characteristic, complemented
16C22×C22⋊D167—21no—
17C2×D4×D167C2211yescharacteristic
18C2×#14581937C2211yescharacteristic, complemented
19C2×#14581933C2211yescharacteristic, complemented
20C22×C167⋊C8C2211yescharacteristic, complemented
21#19128828C2211yescharacteristic
22#19128827—21no—
23#19128823—21no—
24#19128823C2211yescharacteristic, complemented
25#19128816—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#19128790C2211yescharacteristic, complemented
36#19128788—21no—
37C2×C4⋊Dic167C2211yescharacteristic, complemented
38#19128766C2211yescharacteristic, complemented
39#19128766C2211yescharacteristic, 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
51#19128742C2211yescharacteristic, 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×D668C2311yescharacteristic
61C2×C22⋊D167—21no—
62C2×C22⋊D167—21no—
63C2×C22⋊D167—21no—
64C2×C22⋊D167—21no—
65C2×C22⋊D167—21no—
66C2×C22⋊D167—44no—
67C2×C22⋊D167D411yescharacteristic
68D4×D167—22no—
69D4×D167—41no—
70C22×Dic167—21no—
71C22×Dic167—21no—
72C2×C4×D167C2311yescharacteristic
73#14581937—41no—
74#14581933—41no—
75C2×C167⋊C8—41no—
76C2×C167⋊C8C2311yescharacteristic
77C2×C167⋊C8C2311yescharacteristic
78C4⋊Dic167—21no—
79C4⋊Dic167—21no—
80C4⋊Dic167—21no—
81C4⋊Dic167C2311yescharacteristic
82C4⋊Dic167C2311yescharacteristic
83C4⋊Dic167C2311yescharacteristic
84Dic167⋊C4—21no—
85Dic167⋊C4—21no—
86Dic167⋊C4D411yescharacteristic
87C22⋊Dic167—21no—
88C22⋊Dic167—21no—
89C22⋊Dic167D411yescharacteristic
90(C2×D167)⋊C4—21no—
91(C2×D167)⋊C4—21no—
92(C2×D167)⋊C4—21no—
93(C2×D167)⋊C4D411yescharacteristic
94C4×Dic167—21no—
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
117D668—21no—
118D668—21no—
119C22⋊D167—22no—
120C22⋊D167—42no—
121C22⋊D167—42no—
122C22⋊D167—42no—
123C22⋊D167—42no—
124C22⋊D167—81no—
125C2×Dic167—21no—
126C2×Dic167—21no—
127C2×Dic167—21no—
128C2×Dic167—42no—
129C2×Dic167—42no—
130C2×Dic167C2×D411yescharacteristic
131C2×Dic167C2×D411yescharacteristic
132C2×Dic167C2×D411yescharacteristic
133C4×D167—21no—
134C4×D167—21no—
135C167⋊C8—21no—
136C167⋊C8—21no—
137C23×C167—21no—
138C23×C167—21no—
139C23×C167—42no—
140C23×C167—42no—
141C23×C167—81no—
142C23×C167C2×D411yescharacteristic
143Q8×C167—21no—
144Q8×C167—21no—
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—21no—
160C22×C167—22no—
161C22×C167—41no—
162C22×C167—41no—
163C22×C167—41no—
164C22×C167—41no—
165C22×C167—41no—
166C22×C167—84no—
167C22×C167AGL(1,ℤ/8)12yes—
168C22×C167C22×D411yescharacteristic
169C2×D167—21no—
170C2×D167—21no—
171C2×D167—21no—
172C2×D167—41no—
173C2×D167—41no—
174C2×D167—41no—
175C2×D167—84no—
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—22no—
188C2×C167—42no—
189C2×C167—81no—
190C2×C167#54611yescharacteristic
191C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C167#57711yescharacteristic
193D167—41no—
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#450D16711yescharacteristic, complemented
208#440—1671no—
209#435—1671no—
210#418—1671no—
211C2×#102—1671no—
212C22×D4—3341no—
213C22×D4—1671no—
214C22×D4C2×D16711yescharacteristic
215C2×SD16—1671no—
216C2×SD16C2×D16711yescharacteristic
217C2×SD16C2×D16711yescharacteristic, complemented
218C2×M16—1671no—
219C22×C8—1671no—
220#127—1671no—
221#126—3341no—
222#125—3341no—
223#125—1671no—
224#123—3341no—
225C22⋊Q8—1671no—
226C22⋊Q8C2×D16711yescharacteristic, complemented
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#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#102C2×D16711yescharacteristic
251#102C2×D16711yescharacteristic, complemented
252C22⋊C8—1671no—
253C22⋊C8C2×D16711yescharacteristic, complemented
254#97—1671no—
255C24—21no—
256C24—3341no—
257C2×Q8C22×D16711yescharacteristic
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—22no—
278SD16—41no—
279SD16—6681no—
280M16—6681no—
281C2×C8—6681no—
282C2×C8—1671no—
283C2×C8—1671no—
284C2×C8C22×D16711yescharacteristic
285C2×C8C22×D16711yescharacteristic
286C4⋊C4—21no—
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⋊C4—1671no—
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—3341no—
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—21no—
320Q8—21no—
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#1458194912yes—
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—21no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—84no—
363C22—13364no—
364C22—3341no—
365C22—3341no—
366C22—3341no—
367C22—6681no—
368C22—6681no—
369C22—6681no—
370C22#1912885912yes—
371C22C2×D4×D16711yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—3341no—
376C4—3341no—
377C4—3341no—
378C4—6681no—
379C4—6681no—
380C4—6681no—
381C4C2×#1458194911yescharacteristic
382C4#1912887711yescharacteristic
383C2—22no—
384C2—42no—
385C2—81no—
386C2—13361no—
387C2—6681no—
388C2#2492390611yescharacteristic
389C2C2×#1912885911yescharacteristic
390C2#2492407911yescharacteristic
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#3310
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. #24923961C2semidirect
3. #24923632C2semidirect
4. #24923591C2semidirect
5. #24923583C2semidirect
6. #24923567C2semidirect
7. #24923512C2semidirect
8. #24923500C2semidirect
9. #24923467C2semidirect
10. #24923418C2semidirect
11. #24923370C2semidirect
12. #24923337C2semidirect
13. #24923272C2semidirect
14. C2×#19128753C2semidirect
15. #450×C167C2semidirect
18. C2×#14581937C22semidirect
19. C2×#14581933C22semidirect
20. C22×C167⋊C8C22semidirect
24. #19128823C22semidirect
26. #19128808C22semidirect
32. Q8×Dic167C22semidirect
35. #19128790C22semidirect
37. C2×C4⋊Dic167C22semidirect
38. #19128766C22semidirect
39. #19128766C22semidirect
40. #19128763C22semidirect
41. Dic167⋊C8C22semidirect
42. #19128756C22semidirect
43. #19128754C22semidirect
44. #19128754C22semidirect
47. #19128753C22semidirect
48. #19128753C22semidirect
49. #19128752C22semidirect
50. (C2×D167)⋊C8C22semidirect
51. #19128742C22semidirect
54. C2×SD16×C167C22semidirect
55. C22⋊Q8×C167C22semidirect
57. #102×C167C22semidirect
58. C22⋊C8×C167C22semidirect
195. C167#3310semidirect
207. #450D167semidirect
217. C2×SD16C2×D167semidirect
226. C22⋊Q8C2×D167semidirect
251. #102C2×D167semidirect
253. C22⋊C8C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249239612
D4×C22⋊D1672
#249236322
#249235912
#249235832
#249235672
#249235122
#249235002
#249234672
#249234182
#249233702
#249233372
#249232722
C2×#191287532
#450×C1672
#3310167