Finite Group Databasebeta

← #36336129

Subgroups — #36336129

Order 21376.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923923C211yesmaximal, characteristic, complemented
2#24923628C211yesmaximal, characteristic, complemented
3#24923602C211yesmaximal, characteristic, complemented
4#24923580C211yesmaximal, characteristic, complemented
5#24923564C211yesmaximal, characteristic, complemented
6#24923517C211yesmaximal, characteristic, complemented
7#24923503C211yesmaximal, characteristic, complemented
8#24923481C211yesmaximal, characteristic, complemented
9#24923414C211yesmaximal, characteristic, complemented
10#24923408C211yesmaximal, characteristic, complemented
11#24923367C211yesmaximal, characteristic, complemented
12#24923341C211yesmaximal, characteristic, complemented
13#24923270C211yesmaximal, characteristic, complemented
14C2×#19128756C211yesmaximal, characteristic, complemented
15#548×C167C211yesmaximal, characteristic, complemented
16C2×#14581949—21no—
17C2×#14581937C2211yescharacteristic
18C2×#14581937C2211yescharacteristic
19C2×#14581935C2211yescharacteristic
20C2×#14581934C2211yescharacteristic
21C22×C167⋊C8C2211yescharacteristic
22#19128832C2211yescharacteristic
23#19128816—21no—
24#19128816—21no—
25#19128812—21no—
26#19128806C2211yescharacteristic
27#19128806C2211yescharacteristic
28C4×D668C2211yescharacteristic
29C2×C4⋊Dic167C2211yescharacteristic
30C2×C4⋊Dic167C2211yescharacteristic
31#19128765C2211yescharacteristic
32#19128765C2211yescharacteristic
33#19128765C2211yescharacteristic
34C4⋊(C167⋊C8)C2211yescharacteristic
35#19128756C2211yescharacteristic
36#19128756C2211yescharacteristic
37#19128756C2211yescharacteristic
38#19128756C2211yescharacteristic
39#19128756C2211yescharacteristic
40#19128756C2211yescharacteristic
41#19128753C2211yescharacteristic
42#19128753C2211yescharacteristic
43#19128753C2211yescharacteristic
44#19128752C2211yescharacteristic
45C22⋊(C167⋊C8)C2211yescharacteristic
46C22⋊(C167⋊C8)C2211yescharacteristic
47#19128743C2211yescharacteristic
48C2×#41×C167C2211yescharacteristic
49C22×Q8×C167C2211yescharacteristic
50#124×C167—21no—
51#124×C167C2211yescharacteristic
52C22⋊Q8×C167—21no—
53C22⋊Q8×C167C2211yescharacteristic
54#121×C167—21no—
55#121×C167C2211yescharacteristic
56C4×Q8×C167C2211yescharacteristic
57C4×D4×C167—21no—
58C4×D4×C167C2211yescharacteristic
59#14581949—44no—
60C2×Dic334—21no—
61C2×D668C2311yescharacteristic
62C2×C22⋊D167—21no—
63C2×C22⋊D167—21no—
64C22×Dic167—21no—
65C22×Dic167—21no—
66C2×C4×D167—21no—
67#14581937—22no—
68#14581937—41no—
69#14581935—41no—
70#14581934—41no—
71C2×C167⋊C8—41no—
72C2×C167⋊C8C2311yescharacteristic
73C2×C167⋊C8C2311yescharacteristic
74C2×C167⋊C8C2311yescharacteristic
75C2×C167⋊C8C2311yescharacteristic
76C4⋊Dic167—21no—
77C4⋊Dic167—21no—
78C4⋊Dic167—21no—
79C4⋊Dic167C2311yescharacteristic
80C4⋊Dic167C2311yescharacteristic
81C4⋊Dic167C2311yescharacteristic
82(C2×D167)⋊C4—21no—
83(C2×D167)⋊C4—21no—
84(C2×D167)⋊C4—21no—
85#41×C167—22no—
86#41×C167—41no—
87C2×Q8×C167—21no—
88C2×Q8×C167—22no—
89C2×Q8×C167—41no—
90C2×Q8×C167C2311yescharacteristic
91C2×Q8×C167C2311yescharacteristic
92C2×D4×C167—21no—
93C2×D4×C167—21no—
94C2×D4×C167D411yescharacteristic
95C2×D4×C167C2311yescharacteristic
96C22×C4×C167—21no—
97C22×C4×C167—21no—
98C22×C4×C167C2311yescharacteristic
99C22×C4×C167C2311yescharacteristic
100C4⋊C4×C167—21no—
101C4⋊C4×C167—21no—
102C4⋊C4×C167D411yescharacteristic
103C4⋊C4×C167C2311yescharacteristic
104C22⋊C4×C167—21no—
105C22⋊C4×C167—21no—
106C22⋊C4×C167—21no—
107C22⋊C4×C167—21no—
108C22⋊C4×C167D411yescharacteristic
109C22⋊C4×C167D411yescharacteristic
110C42×C167—21no—
111C42×C167C2311yescharacteristic
112C22×D167—21no—
113Dic334—42no—
114D668—21no—
115D668—21no—
116C22⋊D167—42no—
117C22⋊D167—42no—
118C2×Dic167—21no—
119C2×Dic167—21no—
120C2×Dic167—21no—
121C2×Dic167—42no—
122C2×Dic167—42no—
123C4×D167—42no—
124C167⋊C8—21no—
125C167⋊C8—21no—
126C167⋊C8—21no—
127C167⋊C8—21no—
128C23×C167—21no—
129C23×C167C2×D411yescharacteristic
130C23×C167C2×D411yescharacteristic
131Q8×C167—21no—
132Q8×C167—21no—
133Q8×C167—21no—
134Q8×C167—42no—
135Q8×C167#4112yescomplemented
136D4×C167—22no—
137D4×C167—21no—
138D4×C167—21no—
139D4×C167—42no—
140D4×C167—42no—
141C2×C4×C167—21no—
142C2×C4×C167—21no—
143C2×C4×C167—21no—
144C2×C4×C167—21no—
145C2×C4×C167—21no—
146C2×C4×C167—21no—
147C2×C4×C167—21no—
148C2×C4×C167—42no—
149C2×C4×C167—42no—
150C2×C4×C167C2×D411yescharacteristic
151C2×C4×C167C2×D411yescharacteristic
152C2×C4×C167C2×D411yescharacteristic
153C2×C4×C167C2×D411yescharacteristic
154C2×C4×C167C2411yescharacteristic
155C22×C167—21no—
156C22×C167—21no—
157C22×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—41no—
166C2×D167—41no—
167C2×D167—41no—
168C4×C167—21no—
169C4×C167—21no—
170C4×C167—22no—
171C4×C167—21no—
172C4×C167—21no—
173C4×C167—41no—
174C4×C167—41no—
175C4×C167C2×#4111yescharacteristic
176C4×C167H(5,𝔽2)11yescharacteristic
177Dic167—41no—
178Dic167—41no—
179Dic167—41no—
180C2×C167—22no—
181C2×C167—41no—
182C2×C167—81no—
183C2×C167#54611yescharacteristic
184C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C167#57811yescharacteristic
186D167—81no—
187C167#330811yescharacteristic, complemented, Sylow
188#3308—1671nomaximal, Sylow
189#548D16711yescharacteristic, complemented
190#547—1671no—
191#489—1671no—
192#484—1671no—
193#480—1671no—
194#477—1671no—
195#468—1671no—
196#466—1671no—
197#463—1671no—
198#449—1671no—
199#448—1671no—
200#440—1671no—
201#435—1671no—
202#417—1671no—
203C2×#103—1671no—
204C2×#41—3341no—
205C2×#41C2×D16711yescharacteristic
206C22×Q8C2×D16711yescharacteristic
207C2×SD16—1671no—
208C2×SD16—1671no—
209C2×D8—1671no—
210C2×M16—1671no—
211C22×C8—1671no—
212C4⋊Q8—1671no—
213#124—21no—
214#124C2×D16711yescharacteristic
215#123—3341no—
216#123—3341no—
217C22⋊Q8—21no—
218C22⋊Q8—3341no—
219C22⋊Q8C2×D16711yescharacteristic
220#121—21no—
221#121—1671no—
222#121—1671no—
223#121C2×D16711yescharacteristic
224C4×Q8C2×D16711yescharacteristic
225C4×D4—21no—
226C4×D4—1671no—
227C4×D4C2×D16711yescharacteristic
228C2×C4⋊C4—1671no—
229C2×C4⋊C4—1671no—
230#107—1671no—
231#107—1671no—
232#107—1671no—
233C4⋊C8—1671no—
234#103—1671no—
235#103—1671no—
236#103—1671no—
237#103—1671no—
238#103—1671no—
239#103—1671no—
240#102—1671no—
241#102—1671no—
242#102—1671no—
243#102—1671no—
244C22⋊C8—1671no—
245C22⋊C8—1671no—
246#97—1671no—
247#41—22no—
248#41—41no—
249#41—6684no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—3341no—
254C2×Q8C22×D16711yescharacteristic
255C2×Q8C22×D16711yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—3341no—
259C2×D4—3341no—
260C2×D4—1671no—
261C2×D4C22⋊D16711yescharacteristic
262C2×D4C22×D16711yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—3341no—
266C22×C4—3341no—
267C22×C4—3341no—
268C22×C4C22×D16711yescharacteristic
269C22×C4C22×D16711yescharacteristic
270SD16—3342no—
271SD16—6681no—
272D8—6681no—
273M16—6681no—
274C2×C8—6681no—
275C2×C8—1671no—
276C2×C8—1671no—
277C2×C8—1671no—
278C2×C8—1671no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—3341no—
282C4⋊C4—3341no—
283C4⋊C4—3341no—
284C4⋊C4—1671no—
285C4⋊C4—1671no—
286C4⋊C4—1671no—
287C4⋊C4C22⋊D16711yescharacteristic
288C4⋊C4C22×D16711yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—3341no—
294C22⋊C4—3341no—
295C22⋊C4—3341no—
296C22⋊C4C22⋊D16711yescharacteristic
297C22⋊C4C22⋊D16711yescharacteristic
298C42—21no—
299C42C22×D16711yescharacteristic
300C23—21no—
301C23—3341no—
302C23C2×C22⋊D16711yescharacteristic
303C23C2×C22⋊D16711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—6682no—
309Q8#1458194912yescomplemented
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—3341no—
316D4—3341no—
317D4—6682no—
318D4—6682no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—3341no—
329C2×C4—3341no—
330C2×C4—3341no—
331C2×C4—6682no—
332C2×C4—6682no—
333C2×C4—6682no—
334C2×C4C2×C22⋊D16711yescharacteristic
335C2×C4C2×C22⋊D16711yescharacteristic
336C2×C4C2×C22⋊D16711yescharacteristic
337C2×C4C2×C22⋊D16711yescharacteristic
338C2×C4C23×D16711yescharacteristic
339C8—3341no—
340C8—3341no—
341C8—3341no—
342C8—3341no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—6681no—
352C22—6681no—
353C22—6681no—
354C22#1912885712yes—
355C22C22×C22⋊D16711yescharacteristic
356C4—21no—
357C4—21no—
358C4—22no—
359C4—21no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—6681no—
364C4—6681no—
365C4—6681no—
366C4C2×#1458194911yescharacteristic
367C4#1912887711yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—13361no—
372C2#2492391311yescharacteristic
373C2C2×#1912885711yescharacteristic
374C2#2492408411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#548×C167C2
socleC22×C167C22×D4
hypercenter#548D167
hypocenterC167#3308
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #24923923C2semidirect
2. #24923628C2semidirect
3. #24923602C2semidirect
4. #24923580C2semidirect
5. #24923564C2semidirect
6. #24923517C2semidirect
7. #24923503C2semidirect
8. #24923481C2semidirect
9. #24923414C2semidirect
10. #24923408C2semidirect
11. #24923367C2semidirect
12. #24923341C2semidirect
13. #24923270C2semidirect
14. C2×#19128756C2semidirect
15. #548×C167C2semidirect
135. Q8×C167#41semidirect
187. C167#3308semidirect
189. #548D167semidirect
309. Q8#14581949semidirect

Maximal subgroups

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

subgroupindex
#249239232
#249236282
#249236022
#249235802
#249235642
#249235172
#249235032
#249234812
#249234142
#249234082
#249233672
#249233412
#249232702
C2×#191287562
#548×C1672
#3308167