Finite Group Databasebeta

← #36335739

Subgroups — #36335739

Order 21376.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128856C212yesmaximal, complemented
2#24923923C211yesmaximal, characteristic, complemented
3D4×D668C211yesmaximal, characteristic, complemented
4#24923642C211yesmaximal, characteristic, complemented
5#24923516C212yesmaximal, complemented
6#24923474C211yesmaximal, characteristic, complemented
7#24923464C211yesmaximal, characteristic, complemented
8#24923421C212yesmaximal, complemented
9#24923415C212yesmaximal, complemented
10#24923348C211yesmaximal, characteristic, complemented
11C4×D8×C167C211yesmaximal, characteristic, complemented
12C2×#14581949C2212yescomplemented
13C22×D668—21no—
14C2×D4×D167C2212yes—
15#19128856—28no—
16C2×#14581939C2212yescomplemented
17C2×#14581939C2212yescomplemented
18C2×#14581938C2212yes—
19C2×#14581935C2212yescomplemented
20C2×#14581933C2212yescomplemented
21#19128832C2211yescharacteristic, complemented
22#19128829C2211yescharacteristic
23#19128816—22no—
24#19128812C2212yescomplemented
25#19128807C2212yes—
26#19128806—21no—
27#19128806—21no—
28#19128802—22no—
29C4×D668C2211yescharacteristic
30C2×C4⋊Dic167—21no—
31C4⋊(C167⋊C8)C2211yescharacteristic, complemented
32#19128755C2212yescomplemented
33#19128752C2212yescomplemented
34(C2×D167)⋊C8C2212yescomplemented
35C2×D8×C167C2211yescharacteristic
36C4×D4×C167C2211yescharacteristic
37C4×D4×C167C2211yescharacteristic, complemented
38#107×C167C2211yescharacteristic, complemented
39#102×C167C2211yescharacteristic
40#102×C167C2211yescharacteristic, complemented
41C4×C8×C167C2211yescharacteristic, complemented
42C23×D167—22no—
43#14581949—24no—
44#14581949—42no—
45C2×Dic334C2312yes—
46C2×D668—22no—
47C2×D668—21no—
48C2×D668—21no—
49C2×D668—44no—
50C2×D668D411yescharacteristic
51C2×D668C2312yes—
52C2×C22⋊D167—22no—
53C2×C22⋊D167—22no—
54D4×D167—24no—
55D4×D167—42no—
56C22×Dic167—22no—
57C2×C4×D167C2312yes—
58#14581939—24no—
59#14581939—42no—
60#14581938—24no—
61#14581935—24no—
62#14581933—24no—
63C2×C167⋊C8C2312yes—
64C4⋊Dic167—22no—
65C4⋊Dic167—21no—
66C4⋊Dic167—21no—
67C4⋊Dic167D411yescharacteristic
68(C2×D167)⋊C4—22no—
69(C2×D167)⋊C4—22no—
70(C2×D167)⋊C4D412yes—
71C2×D4×C167C2311yescharacteristic
72C2×D4×C167C2311yescharacteristic
73C22×C4×C167—21no—
74C22×C4×C167—21no—
75D8×C167D414yescomplemented
76C2×C8×C167C2311yescharacteristic
77C2×C8×C167C2311yescharacteristic
78C4⋊C4×C167C2311yescharacteristic
79C4⋊C4×C167C2311yescharacteristic
80C22⋊C4×C167—21no—
81C22⋊C4×C167—21no—
82C42×C167C2311yescharacteristic
83C22×D167—22no—
84C22×D167—44no—
85C22×D167—44no—
86C22×D167—82no—
87C22×D167C2×D412yes—
88Dic334—22no—
89Dic334—22no—
90D668—22no—
91D668—22no—
92D668—22no—
93D668—44no—
94D668—42no—
95D668—81no—
96C22⋊D167—44no—
97C22⋊D167—44no—
98C2×Dic167—22no—
99C2×Dic167—44no—
100C2×Dic167C2×D412yes—
101C4×D167—22no—
102C4×D167—22no—
103C167⋊C8—22no—
104C23×C167—21no—
105C23×C167—21no—
106D4×C167—21no—
107D4×C167—21no—
108D4×C167C2×D412yes—
109D4×C167C2×D412yescomplemented
110C2×C4×C167—21no—
111C2×C4×C167—21no—
112C2×C4×C167—42no—
113C2×C4×C167—42no—
114C2×C4×C167C2×D411yescharacteristic
115C2×C4×C167C2×D411yescharacteristic
116C2×C4×C167C2411yescharacteristic
117C8×C167—21no—
118C8×C167C2×D412yescomplemented
119C22×C167—22no—
120C22×C167—22no—
121C22×C167—41no—
122C22×C167—41no—
123C22×C167—41no—
124C22×C167—41no—
125C22×C167C22×D411yescharacteristic
126C2×D167—22no—
127C2×D167—22no—
128C2×D167—22no—
129C2×D167—42no—
130C2×D167—42no—
131C2×D167—42no—
132C2×D167—88no—
133C4×C167—21no—
134C4×C167—41no—
135C4×C167—41no—
136C4×C167AGL(1,ℤ/8)12yescomplemented
137C4×C167C22×D411yescharacteristic
138C4×C167H(5,𝔽2)11yescharacteristic
139Dic167—22no—
140Dic167—42no—
141C2×C167—42no—
142C2×C167—42no—
143C2×C167D4211yescharacteristic
144C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C167#57711yescharacteristic
146D167—42no—
147D167—82no—
148C167#327211yescharacteristic, complemented, Sylow
149#3272—1671nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1672no—
151#547—1671no—
152D42—1671no—
153#492—1671no—
154#468—1672no—
155#461—1671no—
156#459—1671no—
157#450—1672no—
158#449—1672no—
159C4×D8D16711yescharacteristic, complemented
160#436—1671no—
161C2×#41—1672no—
162C22×D4—3341no—
163C22×D4—1672no—
164AGL(1,ℤ/8)—3348no—
165C2×SD16—1672no—
166C2×SD16—1672no—
167C2×SD16—1672no—
168C2×D8—1672no—
169C2×D8C2×D16711yescharacteristic
170C2×M16—1672no—
171C4⋊Q8—1671no—
172#127—1671no—
173#123—3342no—
174C22⋊Q8—1672no—
175#121—3341no—
176#121—3341no—
177#121—1672no—
178#120—3342no—
179C4×D4—1671no—
180C4×D4C2×D16711yescharacteristic
181C4×D4C2×D16711yescharacteristic, complemented
182C2×C4⋊C4—3341no—
183#107C2×D16711yescharacteristic, complemented
184C4⋊C8—1671no—
185#103—1672no—
186#102—1672no—
187#102C2×D16711yescharacteristic
188#102C2×D16711yescharacteristic, complemented
189C22⋊C8—1672no—
190C4×C8C2×D16711yescharacteristic, complemented
191C24—3342no—
192#41—3344no—
193#41—6682no—
194C2×Q8—1672no—
195C2×D4—3342no—
196C2×D4—3344no—
197C2×D4—3342no—
198C2×D4—3342no—
199C2×D4—3341no—
200C2×D4—3341no—
201C2×D4—6684no—
202C2×D4—6682no—
203C2×D4—1671no—
204C2×D4—1672no—
205C2×D4C22×D16711yescharacteristic
206C2×D4C22×D16711yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—3342no—
210C22×C4—1672no—
211SD16—3344no—
212SD16—3344no—
213SD16—6682no—
214D8—3344no—
215D8D66814yescomplemented
216M16—3344no—
217C2×C8—1672no—
218C2×C8C22×D16711yescharacteristic
219C2×C8C22×D16711yescharacteristic
220C4⋊C4—3342no—
221C4⋊C4—3341no—
222C4⋊C4—3341no—
223C4⋊C4—1671no—
224C4⋊C4C22×D16711yescharacteristic
225C4⋊C4C22×D16711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—3342no—
229C22⋊C4—3342no—
230C22⋊C4—1672no—
231C42C22×D16711yescharacteristic
232C23—21no—
233C23—21no—
234C23—13362no—
235C23—3342no—
236C23—6684no—
237C23—6684no—
238C23—1672no—
239Q8—3342no—
240Q8—3342no—
241D4—21no—
242D4—21no—
243D4—13361no—
244D4—3342no—
245D4—3342no—
246D4—3342no—
247D4—6684no—
248D4—6684no—
249D4—6684no—
250D4—6682no—
251D4C2×D66812yes—
252D4C2×D66812yescomplemented
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—3342no—
258C2×C4—3342no—
259C2×C4—3342no—
260C2×C4—6684no—
261C2×C4—1672no—
262C2×C4D4×D16711yescharacteristic
263C2×C4D4×D16711yescharacteristic
264C2×C4C23×D16711yescharacteristic
265C8—21no—
266C8—3342no—
267C8C2×D66812yescomplemented
268C22—22no—
269C22—22no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—13368no—
275C22—3342no—
276C22—3342no—
277C22—3342no—
278C22—6682no—
279C22—6682no—
280C22—6682no—
281C22C2×D4×D16711yescharacteristic
282C4—21no—
283C4—41no—
284C4—41no—
285C4—3342no—
286C4—6682no—
287C4#1912885612yescomplemented
288C4C22×D66811yescharacteristic
289C4#1912887811yescharacteristic
290C2—42no—
291C2—42no—
292C2—13362no—
293C2—6682no—
294C2D4×D66811yescharacteristic
295C2C2×#1912885611yescharacteristic
296C2#2492407611yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC4×D8×C167C2
socleC22×C167C22×D4
hypercenterC4×D8D167
hypocenterC167#3272
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#19128856C2semidirect
2. #24923923C2semidirect
3. D4×D668C2semidirect
4. #24923642C2semidirect
5. #24923516C2semidirect
6. #24923474C2semidirect
7. #24923464C2semidirect
8. #24923421C2semidirect
9. #24923415C2semidirect
10. #24923348C2semidirect
11. C4×D8×C167C2semidirect
12. C2×#14581949C22semidirect
16. C2×#14581939C22semidirect
17. C2×#14581939C22semidirect
19. C2×#14581935C22semidirect
20. C2×#14581933C22semidirect
21. #19128832C22semidirect
24. #19128812C22semidirect
31. C4⋊(C167⋊C8)C22semidirect
32. #19128755C22semidirect
33. #19128752C22semidirect
34. (C2×D167)⋊C8C22semidirect
37. C4×D4×C167C22semidirect
38. #107×C167C22semidirect
40. #102×C167C22semidirect
41. C4×C8×C167C22semidirect
75. D8×C167D4semidirect
109. D4×C167C2×D4semidirect
118. C8×C167C2×D4semidirect
136. C4×C167AGL(1,ℤ/8)semidirect
148. C167#3272semidirect
159. C4×D8D167semidirect
181. C4×D4C2×D167semidirect
183. #107C2×D167semidirect
188. #102C2×D167semidirect
190. C4×C8C2×D167semidirect
215. D8D668semidirect
252. D4C2×D668semidirect
267. C8C2×D668semidirect
287. C4#19128856semidirect

Maximal subgroups

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

subgroupindex
C2×#191288562
#249239232
D4×D6682
#249236422
#249235162
#249234742
#249234642
#249234212
#249234152
#249233482
C4×D8×C1672
#3272167