Finite Group Databasebeta

← #36335740

Subgroups — #36335740

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×#19128860C212yesmaximal, complemented
2#24923924C211yesmaximal, characteristic, complemented
3D4×D668C211yesmaximal, characteristic, complemented
4#24923642C211yesmaximal, characteristic, complemented
5#24923515C212yesmaximal, complemented
6#24923475C211yesmaximal, characteristic, complemented
7#24923465C211yesmaximal, characteristic, complemented
8#24923422C212yesmaximal, complemented
9#24923416C212yesmaximal, complemented
10C4×D1336C211yesmaximal, characteristic, complemented
11#436×C167C211yesmaximal, characteristic, complemented
12C2×#14581950C2212yescomplemented
13C22×D668C2212yes—
14C2×D4×D167—21no—
15#19128860—28no—
16C2×#14581939C2212yes—
17C2×#14581939C2212yescomplemented
18C2×#14581939C2212yescomplemented
19C2×D1336C2211yescharacteristic
20C2×D1336C2212yescomplemented
21#19128832C2211yescharacteristic, complemented
22#19128829C2211yescharacteristic
23#19128814—22no—
24#19128811C2212yescomplemented
25#19128807—21no—
26#19128806C2212yes—
27#19128803—21no—
28#19128802—22no—
29C4×D668C2211yescharacteristic
30C4×D668C2211yescharacteristic, complemented
31C4⋊C4×D167—21no—
32#19128766C2211yescharacteristic, complemented
33#19128757C2212yescomplemented
34#19128754C2211yescharacteristic
35#19128754C2211yescharacteristic, complemented
36#19128754C2212yescomplemented
37C2×M16×C167C2212yescomplemented
38C4×D4×C167C2211yescharacteristic
39C4⋊C8×C167C2211yescharacteristic, complemented
40C22⋊C8×C167C2212yescomplemented
41C4×C8×C167C2211yescharacteristic, complemented
42C23×D167—22no—
43#14581950—24no—
44#14581950—42no—
45C2×Dic334C2312yes—
46C2×D668—24no—
47C2×D668—21no—
48C2×D668—22no—
49C2×D668—21no—
50C2×D668—42no—
51C2×D668C2312yes—
52C2×D668C2311yescharacteristic
53C2×D668C2311yescharacteristic
54C2×C22⋊D167—22no—
55C2×C22⋊D167—22no—
56D4×D167—44no—
57C2×C4×D167—21no—
58C2×C4×D167—22no—
59C2×C4×D167—21no—
60#14581939—24no—
61#14581939—24no—
62#14581939—42no—
63D1336—24no—
64D1336D414yescomplemented
65C4⋊Dic167—21no—
66C4⋊Dic167—21no—
67C4⋊Dic167C2311yescharacteristic
68C4⋊Dic167C2311yescharacteristic
69Dic167⋊C4—22no—
70C22⋊Dic167—22no—
71(C2×D167)⋊C4—21no—
72(C2×D167)⋊C4—21no—
73(C2×D167)⋊C4—22no—
74C2×D4×C167D411yescharacteristic
75C22×C4×C167C2312yes—
76M16×C167—24no—
77C2×C8×C167C2312yes—
78C2×C8×C167C2311yescharacteristic
79C2×C8×C167C2311yescharacteristic
80C4⋊C4×C167D411yescharacteristic
81C22⋊C4×C167D412yes—
82C42×C167C2311yescharacteristic
83C22×D167—21no—
84C22×D167—21no—
85C22×D167—22no—
86C22×D167—44no—
87C22×D167—44no—
88C22×D167—82no—
89Dic334—22no—
90Dic334—22no—
91D668—22no—
92D668—21no—
93D668—21no—
94D668—22no—
95D668—44no—
96D668—42no—
97D668—81no—
98D668C2×D412yes—
99D668C2×D412yescomplemented
100C22⋊D167—44no—
101C22⋊D167—44no—
102C2×Dic167—21no—
103C2×Dic167—21no—
104C2×Dic167—22no—
105C4×D167—44no—
106C4×D167—42no—
107C4×D167—42no—
108C23×C167C2×D412yes—
109D4×C167—22no—
110C2×C4×C167—22no—
111C2×C4×C167—22no—
112C2×C4×C167C2×D411yescharacteristic
113C2×C4×C167C2×D412yes—
114C2×C4×C167C2×D411yescharacteristic
115C2×C4×C167C2411yescharacteristic
116C8×C167—22no—
117C8×C167—21no—
118C8×C167C2×D412yescomplemented
119C22×C167—22no—
120C22×C167—22no—
121C22×C167—22no—
122C22×C167C22×D411yescharacteristic
123C2×D167—22no—
124C2×D167—22no—
125C2×D167—41no—
126C2×D167—41no—
127C2×D167—41no—
128C2×D167—41no—
129C2×D167—42no—
130C2×D167—42no—
131C2×D167—42no—
132C2×D167—88no—
133C4×C167—22no—
134C4×C167—21no—
135C4×C167AGL(1,ℤ/8)12yescomplemented
136C4×C167C22×D411yescharacteristic
137C4×C167H(5,𝔽2)11yescharacteristic
138Dic167—42no—
139Dic167—41no—
140Dic167—41no—
141C2×C167—42no—
142C2×C167D4211yescharacteristic
143C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C167#57711yescharacteristic
145D167—42no—
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×D8—1671no—
160#436D16711yescharacteristic, complemented
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×D8—1671no—
170C2×M16C2×D16712yescomplemented
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×D4—1671no—
181C4×D4C2×D16711yescharacteristic
182C2×C4⋊C4—3341no—
183#107—1671no—
184C4⋊C8C2×D16711yescharacteristic, complemented
185#103—1672no—
186#102—1671no—
187#102—1671no—
188#102—1672no—
189C22⋊C8C2×D16712yescomplemented
190C4×C8C2×D16711yescharacteristic, complemented
191C24—3342no—
192#41—3344no—
193#41—6682no—
194C2×Q8—1672no—
195C2×D4—3344no—
196C2×D4—3342no—
197C2×D4—3342no—
198C2×D4—3341no—
199C2×D4—3342no—
200C2×D4—3341no—
201C2×D4—6682no—
202C2×D4—6684no—
203C2×D4—1672no—
204C2×D4—1671no—
205C2×D4—1671no—
206C2×D4D66811yescharacteristic
207C22×C4—3341no—
208C22×C4—3342no—
209C22×C4—3341no—
210C22×C4C22×D16712yes—
211SD16—3344no—
212SD16—3344no—
213SD16—6682no—
214D8—3344no—
215D8—1674no—
216M16—24no—
217C2×C8C22×D16711yescharacteristic
218C2×C8C22×D16712yes—
219C2×C8C22×D16711yescharacteristic
220C4⋊C4—3341no—
221C4⋊C4—3342no—
222C4⋊C4—3341no—
223C4⋊C4—1671no—
224C4⋊C4—1671no—
225C4⋊C4D66811yescharacteristic
226C22⋊C4—3342no—
227C22⋊C4—3341no—
228C22⋊C4—3341no—
229C22⋊C4—3342no—
230C22⋊C4D66812yes—
231C42C22×D16711yescharacteristic
232C23—13362no—
233C23—3341no—
234C23—3341no—
235C23—3342no—
236C23—6684no—
237C23—6684no—
238C23C2×D66812yes—
239Q8—3342no—
240Q8—3342no—
241D4—22no—
242D4—13361no—
243D4—3342no—
244D4—3341no—
245D4—3341no—
246D4—3342no—
247D4—6684no—
248D4—6684no—
249D4—6684no—
250D4—6682no—
251D4—1672no—
252D4—1672no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—3341no—
256C2×C4—3341no—
257C2×C4—3342no—
258C2×C4—6684no—
259C2×C4—6682no—
260C2×C4—6682no—
261C2×C4C2×D66811yescharacteristic
262C2×C4C2×D66812yes—
263C2×C4C2×D66811yescharacteristic
264C2×C4C23×D16711yescharacteristic
265C8—22no—
266C8—21no—
267C8D4×D16712yescomplemented
268C22—22no—
269C22—22no—
270C22—22no—
271C22—13368no—
272C22—3342no—
273C22—3342no—
274C22—6682no—
275C22—6682no—
276C22—6682no—
277C22—6681no—
278C22—6681no—
279C22—6681no—
280C22—6681no—
281C22C22×D66811yescharacteristic
282C4—22no—
283C4—21no—
284C4—6682no—
285C4—6681no—
286C4—6681no—
287C4#1912886012yescomplemented
288C4C2×D4×D16711yescharacteristic
289C4#1912887811yescharacteristic
290C2—42no—
291C2—13362no—
292C2—6682no—
293C2—6682no—
294C2D4×D66811yescharacteristic
295C2C2×#1912886011yescharacteristic
296C2#2492408111yescharacteristic
subgroupG/H
centerC22C22×D668
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#436×C167C2
socleC22×C167C22×D4
hypercenter#436D167
hypocenterC167#3272
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#19128860C2semidirect
2. #24923924C2semidirect
3. D4×D668C2semidirect
4. #24923642C2semidirect
5. #24923515C2semidirect
6. #24923475C2semidirect
7. #24923465C2semidirect
8. #24923422C2semidirect
9. #24923416C2semidirect
10. C4×D1336C2semidirect
11. #436×C167C2semidirect
12. C2×#14581950C22semidirect
17. C2×#14581939C22semidirect
18. C2×#14581939C22semidirect
20. C2×D1336C22semidirect
21. #19128832C22semidirect
24. #19128811C22semidirect
30. C4×D668C22semidirect
32. #19128766C22semidirect
33. #19128757C22semidirect
35. #19128754C22semidirect
36. #19128754C22semidirect
37. C2×M16×C167C22semidirect
39. C4⋊C8×C167C22semidirect
40. C22⋊C8×C167C22semidirect
41. C4×C8×C167C22semidirect
64. D1336D4semidirect
99. D668C2×D4semidirect
118. C8×C167C2×D4semidirect
135. C4×C167AGL(1,ℤ/8)semidirect
148. C167#3272semidirect
160. #436D167semidirect
170. C2×M16C2×D167semidirect
184. C4⋊C8C2×D167semidirect
189. C22⋊C8C2×D167semidirect
190. C4×C8C2×D167semidirect
267. C8D4×D167semidirect
287. C4#19128860semidirect

Maximal subgroups

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

subgroupindex
C2×#191288602
#249239242
D4×D6682
#249236422
#249235152
#249234752
#249234652
#249234222
#249234162
C4×D13362
#436×C1672
#3272167