Finite Group Databasebeta

← #36334274

Subgroups — #36334274

Order 21376.

450 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923801C211yesmaximal, characteristic, complemented
2C4×D4×D167C211yesmaximal, characteristic, complemented
3#24923601C211yesmaximal, characteristic, complemented
4#24923581C211yesmaximal, characteristic, complemented
5#24923568C211yesmaximal, characteristic, complemented
6#24923567C211yesmaximal, characteristic, complemented
7#24923483C211yesmaximal, characteristic, complemented
8#24923482C211yesmaximal, characteristic, complemented
9#24923421C211yesmaximal, characteristic, complemented
10#24923401C211yesmaximal, characteristic
11#24923379C211yesmaximal, characteristic, complemented
12#24923371C211yesmaximal, characteristic, complemented
13#24923369C211yesmaximal, characteristic, complemented
14#24923324C211yesmaximal, characteristic, complemented
15#442×C167C211yesmaximal, characteristic, complemented
16C2×D4×D167C2211yescharacteristic
17C22×C4×D167—21no—
18C2×#14581939C2211yescharacteristic
19C2×#14581938C2211yescharacteristic
20C2×#14581935C2211yescharacteristic
21#19128825—21no—
22#19128823C2211yescharacteristic, complemented
23#19128819C2211yescharacteristic, complemented
24#19128814—21no—
25#19128812C2211yescharacteristic
26#19128803C2211yescharacteristic
27C4×Dic334C2211yescharacteristic, complemented
28C4×D668C2211yescharacteristic
29D4×Dic167C2211yescharacteristic
30#19128794C2211yescharacteristic
31C4×C22⋊D167—21no—
32C4×C22⋊D167—21no—
33(C22⋊D167)⋊C4—21no—
34#19128791—21no—
35C4⋊C4×D167C2211yescharacteristic
36C22⋊C4×D167—21no—
37C42×D167C2211yescharacteristic
38#19128766C2211yescharacteristic, complemented
39#19128765C2211yescharacteristic, complemented
40#19128763C2211yescharacteristic, complemented
41Dic167⋊C8C2211yescharacteristic, complemented
42Dic167⋊C8C2211yescharacteristic, complemented
43#19128757C2211yescharacteristic, complemented
44#19128755C2211yescharacteristic, complemented
45#19128754C2211yescharacteristic
46#19128753C2211yescharacteristic
47#19128753C2211yescharacteristic, complemented
48#19128752C2211yescharacteristic
49(C2×D167)⋊C8C2211yescharacteristic
50(C2×D167)⋊C8C2211yescharacteristic
51#19128743C2211yescharacteristic, complemented
52C2×D8×C167C2211yescharacteristic
53C4×D4×C167C2211yescharacteristic
54C4×D4×C167C2211yescharacteristic, complemented
55#106×C167C2211yescharacteristic, complemented
56#102×C167C2211yescharacteristic
57#102×C167C2211yescharacteristic, complemented
58#97×C167C2211yescharacteristic, complemented
59C23×D167—21no—
60C2×Dic334C2311yescharacteristic
61C2×D668C2311yescharacteristic
62C2×C22⋊D167—21no—
63C2×C22⋊D167—21no—
64D4×D167—22no—
65D4×D167—22no—
66C22×Dic167—21no—
67C2×C4×D167—21no—
68C2×C4×D167—21no—
69C2×C4×D167—21no—
70C2×C4×D167—44no—
71C2×C4×D167D411yescharacteristic
72C2×C4×D167D411yescharacteristic
73C2×C4×D167C2311yescharacteristic
74#14581939—22no—
75#14581938—22no—
76#14581935—22no—
77C2×C167⋊C8C2311yescharacteristic
78C2×C167⋊C8C2311yescharacteristic
79C4⋊Dic167—21no—
80C4⋊Dic167—21no—
81C4⋊Dic167C2311yescharacteristic
82C4⋊Dic167C2311yescharacteristic
83Dic167⋊C4—21no—
84Dic167⋊C4—21no—
85Dic167⋊C4—21no—
86C22⋊Dic167—21no—
87C22⋊Dic167—21no—
88C22⋊Dic167—21no—
89C22⋊Dic167—21no—
90(C2×D167)⋊C4—21no—
91(C2×D167)⋊C4—21no—
92(C2×D167)⋊C4—21no—
93C4×Dic167—21no—
94C4×Dic167D411yescharacteristic
95C4×Dic167D411yescharacteristic
96C4×Dic167C2311yescharacteristic
97C2×D4×C167C2311yescharacteristic
98C2×D4×C167C2311yescharacteristic
99C22×C4×C167—21no—
100C22×C4×C167—21no—
101D8×C167—22no—
102C2×C8×C167C2311yescharacteristic
103C2×C8×C167C2311yescharacteristic
104C4⋊C4×C167C2311yescharacteristic
105C4⋊C4×C167C2311yescharacteristic
106C22⋊C4×C167—21no—
107C22⋊C4×C167—21no—
108C42×C167C2311yescharacteristic
109C22×D167—24no—
110C22×D167—21no—
111C22×D167—42no—
112C22×D167—42no—
113C22×D167C2×D411yescharacteristic
114Dic334—21no—
115Dic334—21no—
116D668—21no—
117D668#4112yes—
118C22⋊D167—24no—
119C22⋊D167—42no—
120C2×Dic167—21no—
121C2×Dic167—21no—
122C2×Dic167—21no—
123C2×Dic167—42no—
124C2×Dic167C2×D411yescharacteristic
125C2×Dic167C2×D411yescharacteristic
126C2×Dic167C2×D411yescharacteristic
127C4×D167—22no—
128C4×D167—21no—
129C4×D167—21no—
130C4×D167—41no—
131C4×D167—42no—
132C4×D167—42no—
133C4×D167—42no—
134C167⋊C8—21no—
135C167⋊C8—21no—
136C23×C167—21no—
137C23×C167—21no—
138D4×C167—21no—
139D4×C167—21no—
140D4×C167—21no—
141D4×C167#4112yes—
142C2×C4×C167—21no—
143C2×C4×C167—21no—
144C2×C4×C167—42no—
145C2×C4×C167—42no—
146C2×C4×C167C2×D411yescharacteristic
147C2×C4×C167C2×D411yescharacteristic
148C2×C4×C167C2411yescharacteristic
149C8×C167—21no—
150C8×C167—21no—
151C22×C167—22no—
152C22×C167—41no—
153C22×C167—41no—
154C22×C167—41no—
155C22×C167—41no—
156C22×C167—41no—
157C22×C167C22×D411yescharacteristic
158C2×D167—21no—
159C2×D167—21no—
160C2×D167—22no—
161C2×D167—48no—
162C2×D167—41no—
163C2×D167—41no—
164C2×D167AGL(1,ℤ/8)12yes—
165C4×C167—21no—
166C4×C167—21no—
167C4×C167—41no—
168C4×C167—41no—
169C4×C167C2×#4111yescharacteristic
170C4×C167C2×#4111yescharacteristic
171Dic167—21no—
172Dic167—21no—
173Dic167—21no—
174Dic167—41no—
175Dic167—41no—
176Dic167—41no—
177C2×C167—42no—
178C2×C167—81no—
179C2×C167#52511yescharacteristic
180C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
181C2×C167#57511yescharacteristic
182D167—22no—
183D167—42no—
184C167#311411yescharacteristic, complemented, Sylow
185#3114—1671nomaximal, Sylow
186#529—1671no—
187C2×C4×D4—1671no—
188#483—1671no—
189#480—1671no—
190#478—1671no—
191#478—1671no—
192#463—1671no—
193#463—1671no—
194#450—1671no—
195#447—1671no—
196#442—1671no—
197#442D16711yescharacteristic, complemented
198#440—1671no—
199#440—1671no—
200#432—1671no—
201C22×D4—1671no—
202C23×C4—3341no—
203C2×SD16—1671no—
204C2×SD16—1671no—
205C2×D8—1671no—
206C2×D8C2×D16711yescharacteristic
207#126—3341no—
208#125—1671no—
209#124—1671no—
210#123—3341no—
211C22⋊Q8—1671no—
212#121—1671no—
213C4×Q8—1671no—
214C4×D4—3341no—
215C4×D4—3341no—
216C4×D4—3341no—
217C4×D4—1671no—
218C4×D4—1671no—
219C4×D4—1671no—
220C4×D4C2×D16711yescharacteristic
221C4×D4C2×D16711yescharacteristic, complemented
222#117—3341no—
223C2×C4⋊C4—1671no—
224C2×C22⋊C4—3341no—
225C2×C42—1671no—
226#107—1671no—
227#107—1671no—
228#106—1671no—
229#106C2×D16711yescharacteristic, complemented
230C4⋊C8—1671no—
231C4⋊C8—1671no—
232#103—1671no—
233#103—1671no—
234#102—1671no—
235#102—1671no—
236#102—1671no—
237#102—1671no—
238#102C2×D16711yescharacteristic
239#102C2×D16711yescharacteristic, complemented
240C22⋊C8—1671no—
241C22⋊C8—1671no—
242#97—1671no—
243#97C2×D16711yescharacteristic, complemented
244C24—3341no—
245C2×Q8—1671no—
246C2×D4—3342no—
247C2×D4—3342no—
248C2×D4—3341no—
249C2×D4—3341no—
250C2×D4—1671no—
251C2×D4C22×D16711yescharacteristic
252C2×D4C22×D16711yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—3341no—
256C22×C4—3341no—
257C22×C4—3341no—
258C22×C4—3341no—
259C22×C4—6684no—
260C22×C4—1671no—
261C22×C4—1671no—
262C22×C4—1671no—
263SD16—3342no—
264SD16—3342no—
265D8—22no—
266D8—3342no—
267C2×C8—1671no—
268C2×C8—1671no—
269C2×C8C22×D16711yescharacteristic
270C2×C8C22×D16711yescharacteristic
271C4⋊C4—3341no—
272C4⋊C4—3341no—
273C4⋊C4—3341no—
274C4⋊C4—3341no—
275C4⋊C4—3341no—
276C4⋊C4—1671no—
277C4⋊C4—1671no—
278C4⋊C4C22×D16711yescharacteristic
279C4⋊C4C22×D16711yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—3341no—
283C22⋊C4—3341no—
284C22⋊C4—3341no—
285C22⋊C4—3341no—
286C22⋊C4—3341no—
287C22⋊C4—3341no—
288C22⋊C4—3341no—
289C42—3341no—
290C42—1671no—
291C42—1671no—
292C42—1671no—
293C42C22×D16711yescharacteristic
294C23—21no—
295C23—21no—
296C23—3344no—
297C23—3341no—
298C23—6682no—
299C23—6682no—
300C23—1671no—
301Q8—3341no—
302Q8—3341no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—3344no—
307D4—3341no—
308D4—6682no—
309D4—1672no—
310D4#1458195012yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—3342no—
316C2×C4—3341no—
317C2×C4—3341no—
318C2×C4—3341no—
319C2×C4—3341no—
320C2×C4—3341no—
321C2×C4—6681no—
322C2×C4—6682no—
323C2×C4—6682no—
324C2×C4—6682no—
325C2×C4—6682no—
326C2×C4—1671no—
327C2×C4—1671no—
328C2×C4—1671no—
329C2×C4D4×D16711yescharacteristic
330C2×C4D4×D16711yescharacteristic
331C2×C4C23×D16711yescharacteristic
332C8—21no—
333C8—21no—
334C8—3341no—
335C8—3341no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—3341no—
343C22—3341no—
344C22—3342no—
345C22—6688no—
346C22—6681no—
347C22—6681no—
348C22—1672no—
349C22C2×D4×D16711yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—3341no—
355C4—3341no—
356C4—3341no—
357C4—6681no—
358C4—6681no—
359C4—6681no—
360C4#41×D16711yescharacteristic
361C4C2×#1458195011yescharacteristic
362C2—42no—
363C2—81no—
364C2—3342no—
365C2—6682no—
366C2#2492377811yescharacteristic
367C2AGL(1,ℤ/8)×D16711yescharacteristic
368C2#2492406511yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#442×C167C2
socleC22×C167C22×D4
prime radical#24923401C2
hypercenter#442D167
hypocenterC167#3114
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #24923801C2semidirect
2. C4×D4×D167C2semidirect
3. #24923601C2semidirect
4. #24923581C2semidirect
5. #24923568C2semidirect
6. #24923567C2semidirect
7. #24923483C2semidirect
8. #24923482C2semidirect
9. #24923421C2semidirect
11. #24923379C2semidirect
12. #24923371C2semidirect
13. #24923369C2semidirect
14. #24923324C2semidirect
15. #442×C167C2semidirect
22. #19128823C22semidirect
23. #19128819C22semidirect
27. C4×Dic334C22semidirect
38. #19128766C22semidirect
39. #19128765C22semidirect
40. #19128763C22semidirect
41. Dic167⋊C8C22semidirect
42. Dic167⋊C8C22semidirect
43. #19128757C22semidirect
44. #19128755C22semidirect
47. #19128753C22semidirect
51. #19128743C22semidirect
54. C4×D4×C167C22semidirect
55. #106×C167C22semidirect
57. #102×C167C22semidirect
58. #97×C167C22semidirect
184. C167#3114semidirect
197. #442D167semidirect
221. C4×D4C2×D167semidirect
229. #106C2×D167semidirect
239. #102C2×D167semidirect
243. #97C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249238012
C4×D4×D1672
#249236012
#249235812
#249235682
#249235672
#249234832
#249234822
#249234212
#249234012
#249233792
#249233712
#249233692
#249233242
#442×C1672
#3114167