Finite Group Databasebeta

← #36336165

Subgroups — #36336165

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
2#24923903C211yesmaximal, characteristic
3#24923632C211yesmaximal, characteristic, complemented
4#24923595C211yesmaximal, characteristic, complemented
5#24923580C211yesmaximal, characteristic, complemented
6#24923568C211yesmaximal, characteristic, complemented
7#24923518C211yesmaximal, characteristic, complemented
8#24923467C211yesmaximal, characteristic, complemented
9#24923419C211yesmaximal, characteristic, complemented
10#24923418C211yesmaximal, characteristic, complemented
11#24923372C211yesmaximal, characteristic, complemented
12#24923340C211yesmaximal, characteristic, complemented
13#24923272C211yesmaximal, characteristic, complemented
14C2×#19128754C211yesmaximal, characteristic, complemented
15#465×C167C211yesmaximal, characteristic, complemented
16C22×D668C2211yescharacteristic
17C2×D4×D167—21no—
18C2×D4×D167C2211yescharacteristic
19C2×#14581937C2211yescharacteristic
20C2×#14581937C2211yescharacteristic, complemented
21C2×#14581933C2211yescharacteristic, complemented
22#19128828C2211yescharacteristic
23#19128827—21no—
24#19128823—21no—
25#19128823C2211yescharacteristic, complemented
26#19128816—21no—
27#19128808C2211yescharacteristic, complemented
28#19128807C2211yescharacteristic
29#19128805—21no—
30#19128802—21no—
31#19128800—21no—
32Q8×Dic167C2211yescharacteristic, complemented
33#19128794C2211yescharacteristic
34C4×C22⋊D167C2211yescharacteristic
35#19128790C2211yescharacteristic, complemented
36#19128788—21no—
37C2×C4⋊Dic167C2211yescharacteristic, complemented
38#19128765C2211yescharacteristic, complemented
39#19128765C2211yescharacteristic, complemented
40Dic167⋊C8C2211yescharacteristic, complemented
41#19128754C2211yescharacteristic
42#19128754C2211yescharacteristic
43#19128754C2211yescharacteristic, complemented
44#19128754C2211yescharacteristic, complemented
45#19128753C2211yescharacteristic, complemented
46#19128753C2211yescharacteristic, complemented
47#19128752C2211yescharacteristic
48#19128752C2211yescharacteristic, complemented
49C22⋊(C167⋊C8)C2211yescharacteristic, complemented
50(C2×D167)⋊C8C2211yescharacteristic, complemented
51#19128742C2211yescharacteristic, complemented
52C2×SD16×C167C2211yescharacteristic, complemented
53C22×C8×C167C2211yescharacteristic, complemented
54C22⋊Q8×C167C2211yescharacteristic, complemented
55#121×C167C2211yescharacteristic
56#106×C167C2211yescharacteristic, complemented
57#103×C167C2211yescharacteristic, complemented
58#102×C167C2211yescharacteristic, complemented
59C23×D167—21no—
60C23×D167—21no—
61C2×D668—22no—
62C2×D668—21no—
63C2×D668—21no—
64C2×D668—41no—
65C2×D668C2311yescharacteristic
66C2×D668C2311yescharacteristic
67C2×C22⋊D167—21no—
68C2×C22⋊D167—21no—
69C2×C22⋊D167—21no—
70C2×C22⋊D167D411yescharacteristic
71D4×D167—22no—
72D4×D167—41no—
73D4×D167—44no—
74C22×Dic167—21no—
75C2×C4×D167—21no—
76C2×C4×D167C2311yescharacteristic
77#14581937—22no—
78#14581937—41no—
79#14581933—41no—
80C2×C167⋊C8C2311yescharacteristic
81C2×C167⋊C8C2311yescharacteristic
82C4⋊Dic167—21no—
83C4⋊Dic167—21no—
84C4⋊Dic167—21no—
85C4⋊Dic167C2311yescharacteristic
86C4⋊Dic167C2311yescharacteristic
87Dic167⋊C4—21no—
88Dic167⋊C4—21no—
89Dic167⋊C4D411yescharacteristic
90C22⋊Dic167D411yescharacteristic
91(C2×D167)⋊C4—21no—
92(C2×D167)⋊C4—21no—
93(C2×D167)⋊C4—21no—
94(C2×D167)⋊C4—21no—
95(C2×D167)⋊C4D411yescharacteristic
96C4×Dic167—21no—
97C4×Dic167—21no—
98C4×Dic167C2311yescharacteristic
99C2×Q8×C167C2311yescharacteristic
100C2×D4×C167—21no—
101C2×D4×C167C2311yescharacteristic
102C22×C4×C167C2311yescharacteristic
103SD16×C167—41no—
104C2×C8×C167—41no—
105C2×C8×C167C2311yescharacteristic
106C2×C8×C167C2311yescharacteristic
107C4⋊C4×C167—21no—
108C4⋊C4×C167C2311yescharacteristic
109C4⋊C4×C167C2311yescharacteristic
110C22⋊C4×C167—21no—
111C22⋊C4×C167—21no—
112C22×D167—21no—
113C22×D167—21no—
114C22×D167—42no—
115C22×D167—42no—
116C22×D167—42no—
117C22×D167—42no—
118C22×D167—81no—
119C22×D167—81no—
120C22×D167C2×D411yescharacteristic
121D668—21no—
122D668—21no—
123D668—21no—
124D668—42no—
125D668—42no—
126D668#4112yes—
127C22⋊D167—22no—
128C22⋊D167—42no—
129C22⋊D167—42no—
130C22⋊D167—81no—
131C2×Dic167—21no—
132C2×Dic167—21no—
133C2×Dic167—42no—
134C2×Dic167C2×D411yescharacteristic
135C2×Dic167C2×D411yescharacteristic
136C2×Dic167C2×D411yescharacteristic
137C4×D167—21no—
138C4×D167—21no—
139C4×D167—42no—
140C167⋊C8—21no—
141C167⋊C8—21no—
142C23×C167—21no—
143C23×C167C2×D411yescharacteristic
144Q8×C167—21no—
145Q8×C167—21no—
146D4×C167—21no—
147D4×C167—21no—
148D4×C167—42no—
149C2×C4×C167—21no—
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—41no—
161C22×C167—41no—
162C22×C167—41no—
163C22×C167AGL(1,ℤ/8)12yes—
164C22×C167C22×D411yescharacteristic
165C2×D167—21no—
166C2×D167—22no—
167C2×D167—21no—
168C2×D167—21no—
169C2×D167—41no—
170C2×D167—41no—
171C2×D167—41no—
172C2×D167—41no—
173C2×D167—41no—
174C2×D167—84no—
175C2×D167—84no—
176C4×C167—21no—
177C4×C167—41no—
178C4×C167—41no—
179C4×C167—41no—
180C4×C167C2×#4111yescharacteristic
181C4×C167H(5,𝔽2)11yescharacteristic
182Dic167—21no—
183Dic167—21no—
184Dic167—21no—
185Dic167—41no—
186Dic167—41no—
187C2×C167—22no—
188C2×C167—81no—
189C2×C167#54611yescharacteristic
190C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C167#57711yescharacteristic
192D167—41no—
193D167—42no—
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#465D16711yescharacteristic, complemented
205#460—1671no—
206#450—1671no—
207#450—1671no—
208#440—1671no—
209#435—1671no—
210#418—1671no—
211C2×#102—1671no—
212C22×D4—3341no—
213C22×D4—1671no—
214C22×D4—1671no—
215C2×SD16—1671no—
216C2×SD16—1671no—
217C2×SD16C2×D16711yescharacteristic, complemented
218C2×M16—1671no—
219C22×C8C2×D16711yescharacteristic, complemented
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#121C2×D16711yescharacteristic
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#106C2×D16711yescharacteristic, complemented
241C4⋊C8—1671no—
242#103C2×D16711yescharacteristic, complemented
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#102C2×D16711yescharacteristic, complemented
252C22⋊C8—1671no—
253C22⋊C8—1671no—
254#97—1671no—
255C24—3341no—
256C24—3341no—
257C2×Q8C22×D16711yescharacteristic
258C2×D4—21no—
259C2×D4—3342no—
260C2×D4—3341no—
261C2×D4—3342no—
262C2×D4—3341no—
263C2×D4—3341no—
264C2×D4—3341no—
265C2×D4—3341no—
266C2×D4—6681no—
267C2×D4—6681no—
268C2×D4—6684no—
269C2×D4—1671no—
270C2×D4—1671no—
271C2×D4—1671no—
272C2×D4C22×D16711yescharacteristic
273C22×C4—3341no—
274C22×C4—3341no—
275C22×C4—1671no—
276C22×C4C22×D16711yescharacteristic
277SD16—41no—
278SD16—3342no—
279SD16—6681no—
280M16—6681no—
281C2×C8—41no—
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⋊C4C22×D16711yescharacteristic
296C4⋊C4C22×D16711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
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—13361no—
310C23—13361no—
311C23—3341no—
312C23—3341no—
313C23—6682no—
314C23—6682no—
315C23—6682no—
316C23—6682no—
317C23—1671no—
318C23D4×D16711yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—13361no—
325D4—3342no—
326D4—3341no—
327D4—3341no—
328D4—3341no—
329D4—6682no—
330D4—6682no—
331D4—6682no—
332D4—6682no—
333D4—1672no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
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—41no—
357C22—41no—
358C22—41no—
359C22—13364no—
360C22—13364no—
361C22—3341no—
362C22—3342no—
363C22—3341no—
364C22—3341no—
365C22—6681no—
366C22—6681no—
367C22—6681no—
368C22—6681no—
369C22—6681no—
370C22#1912885912yes—
371C22C2×D4×D16711yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—3341no—
377C4—3341no—
378C4—3341no—
379C4—6681no—
380C4—6681no—
381C4#41×D16711yescharacteristic
382C4#1912887711yescharacteristic
383C2—22no—
384C2—81no—
385C2—13361no—
386C2—6681no—
387C2—6682no—
388C2#2492390811yescharacteristic
389C2C2×#1912885911yescharacteristic
390C2#2492407611yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#465×C167C2
socleC22×C167C22×D4
prime radical#24923903C2
hypercenter#465D167
hypocenterC167#3310
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #24923961C2semidirect
3. #24923632C2semidirect
4. #24923595C2semidirect
5. #24923580C2semidirect
6. #24923568C2semidirect
7. #24923518C2semidirect
8. #24923467C2semidirect
9. #24923419C2semidirect
10. #24923418C2semidirect
11. #24923372C2semidirect
12. #24923340C2semidirect
13. #24923272C2semidirect
14. C2×#19128754C2semidirect
15. #465×C167C2semidirect
20. C2×#14581937C22semidirect
21. C2×#14581933C22semidirect
25. #19128823C22semidirect
27. #19128808C22semidirect
32. Q8×Dic167C22semidirect
35. #19128790C22semidirect
37. C2×C4⋊Dic167C22semidirect
38. #19128765C22semidirect
39. #19128765C22semidirect
40. Dic167⋊C8C22semidirect
43. #19128754C22semidirect
44. #19128754C22semidirect
45. #19128753C22semidirect
46. #19128753C22semidirect
48. #19128752C22semidirect
49. C22⋊(C167⋊C8)C22semidirect
50. (C2×D167)⋊C8C22semidirect
51. #19128742C22semidirect
52. C2×SD16×C167C22semidirect
53. C22×C8×C167C22semidirect
54. C22⋊Q8×C167C22semidirect
56. #106×C167C22semidirect
57. #103×C167C22semidirect
58. #102×C167C22semidirect
195. C167#3310semidirect
204. #465D167semidirect
217. C2×SD16C2×D167semidirect
219. C22×C8C2×D167semidirect
226. C22⋊Q8C2×D167semidirect
240. #106C2×D167semidirect
242. #103C2×D167semidirect
251. #102C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249239612
#249239032
#249236322
#249235952
#249235802
#249235682
#249235182
#249234672
#249234192
#249234182
#249233722
#249233402
#249232722
C2×#191287542
#465×C1672
#3310167