Finite Group Databasebeta

← #36335738

Subgroups — #36335738

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×#19128857C212yesmaximal, complemented
2D4×D668C211yesmaximal, characteristic, complemented
3#24923643C211yesmaximal, characteristic, complemented
4#24923517C212yesmaximal, complemented
5#24923471C211yesmaximal, characteristic, complemented
6#24923464C211yesmaximal, characteristic, complemented
7#24923419C212yesmaximal, complemented
8#24923414C212yesmaximal, complemented
9C4×#14581935C211yesmaximal, characteristic, complemented
10#24923347C211yesmaximal, characteristic, complemented
11#547×C167C211yesmaximal, characteristic, complemented
12C22×D668C2212yes—
13C2×D4×D167—21no—
14#19128857—28no—
15C2×#14581937C2212yes—
16C2×#14581937C2212yescomplemented
17C2×#14581937C2212yescomplemented
18C2×#14581935C2211yescharacteristic
19C2×#14581935C2212yescomplemented
20C2×#14581934C2212yescomplemented
21#19128829C2211yescharacteristic
22#19128807—21no—
23#19128806C2212yes—
24#19128802—22no—
25C4×D668C2211yescharacteristic
26#19128765C2211yescharacteristic, complemented
27C4⋊(C167⋊C8)C2211yescharacteristic, complemented
28#19128756C2212yescomplemented
29#19128753C2211yescharacteristic, complemented
30#19128752C2211yescharacteristic
31#19128752C2212yescomplemented
32C22⋊(C167⋊C8)C2212yescomplemented
33C4×C167⋊C8C2211yescharacteristic, complemented
34C2×#41×C167C2212yescomplemented
35C4⋊Q8×C167C2211yescharacteristic, complemented
36#123×C167—22no—
37C22⋊Q8×C167C2212yescomplemented
38#121×C167—21no—
39C4×D4×C167C2211yescharacteristic
40C4×D4×C167C2211yescharacteristic, complemented
41C2×C4⋊C4×C167—21no—
42C23×D167—22no—
43C2×D668—24no—
44C2×D668—21no—
45C2×D668—22no—
46C2×D668—21no—
47C2×D668—42no—
48C2×D668C2312yes—
49C2×D668C2311yescharacteristic
50C2×C22⋊D167—22no—
51D4×D167—44no—
52C2×C4×D167—21no—
53#14581937—24no—
54#14581937—24no—
55#14581937—42no—
56#14581935—24no—
57#14581935D414yescomplemented
58#14581934—24no—
59C2×C167⋊C8C2312yes—
60C2×C167⋊C8C2311yescharacteristic
61C2×C167⋊C8C2311yescharacteristic
62C4⋊Dic167C2311yescharacteristic
63(C2×D167)⋊C4—21no—
64(C2×D167)⋊C4—22no—
65#41×C167—24no—
66#41×C167—42no—
67C2×Q8×C167C2312yes—
68C2×D4×C167—22no—
69C2×D4×C167D411yescharacteristic
70C2×D4×C167C2311yescharacteristic
71C22×C4×C167—21no—
72C22×C4×C167—22no—
73C22×C4×C167C2312yes—
74C4⋊C4×C167—21no—
75C4⋊C4×C167—22no—
76C4⋊C4×C167—21no—
77C4⋊C4×C167D411yescharacteristic
78C4⋊C4×C167C2311yescharacteristic
79C22⋊C4×C167—22no—
80C22⋊C4×C167—21no—
81C22⋊C4×C167D412yes—
82C42×C167C2311yescharacteristic
83C22×D167—21no—
84C22×D167—22no—
85C22×D167—44no—
86C22×D167—44no—
87C22×D167—82no—
88D668—22no—
89D668—21no—
90D668—22no—
91D668—44no—
92D668—42no—
93D668—81no—
94D668C2×D412yes—
95C22⋊D167—44no—
96C2×Dic167—21no—
97C4×D167—42no—
98C167⋊C8—22no—
99C167⋊C8—21no—
100C167⋊C8C2×D412yescomplemented
101C23×C167—21no—
102C23×C167C2×D412yes—
103Q8×C167—22no—
104Q8×C167—22no—
105D4×C167—22no—
106D4×C167—21no—
107D4×C167—44no—
108D4×C167C2×D412yescomplemented
109C2×C4×C167—22no—
110C2×C4×C167—22no—
111C2×C4×C167—21no—
112C2×C4×C167—22no—
113C2×C4×C167—44no—
114C2×C4×C167—42no—
115C2×C4×C167C2×D411yescharacteristic
116C2×C4×C167C2×D412yes—
117C2×C4×C167C2×D411yescharacteristic
118C2×C4×C167C2411yescharacteristic
119C22×C167—22no—
120C22×C167—22no—
121C22×C167—22no—
122C22×C167—22no—
123C22×C167—41no—
124C22×C167—41no—
125C22×C167C22×D411yescharacteristic
126C2×D167—22no—
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×C167—42no—
136C4×C167—41no—
137C4×C167AGL(1,ℤ/8)12yescomplemented
138C4×C167C22×D411yescharacteristic
139C4×C167H(5,𝔽2)11yescharacteristic
140Dic167—41no—
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#547D16711yescharacteristic, complemented
152D42—1671no—
153#492—1671no—
154#468—1672no—
155#461—1671no—
156#459—1671no—
157#450—1672no—
158#449—1672no—
159C4×D8—1671no—
160#436—1671no—
161C2×#41C2×D16712yescomplemented
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×M16—1672no—
171C4⋊Q8C2×D16711yescharacteristic, complemented
172#127—1671no—
173#123—22no—
174C22⋊Q8C2×D16712yescomplemented
175#121—21no—
176#121—3341no—
177#121—1672no—
178#120—3342no—
179C4×D4—1671no—
180C4×D4C2×D16711yescharacteristic
181C4×D4C2×D16711yescharacteristic, complemented
182C2×C4⋊C4—21no—
183#107—1671no—
184C4⋊C8—1671no—
185#103—1672no—
186#102—1671no—
187#102—1671no—
188#102—1672no—
189C22⋊C8—1672no—
190C4×C8—1671no—
191C24—3342no—
192#41—24no—
193#41—42no—
194C2×Q8C22×D16712yes—
195C2×D4—22no—
196C2×D4—3344no—
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×D4C22⋊D16711yescharacteristic
206C2×D4C22×D16711yescharacteristic
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—3341no—
210C22×C4C22×D16712yes—
211SD16—3344no—
212SD16—3344no—
213SD16—6682no—
214D8—3344no—
215D8—1674no—
216M16—3344no—
217C2×C8—1671no—
218C2×C8—1672no—
219C2×C8—1671no—
220C4⋊C4—21no—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C4⋊C4—1671no—
224C4⋊C4C22⋊D16711yescharacteristic
225C4⋊C4C22×D16711yescharacteristic
226C22⋊C4—22no—
227C22⋊C4—21no—
228C22⋊C4—3341no—
229C22⋊C4—3342no—
230C22⋊C4C22⋊D16712yes—
231C42C22×D16711yescharacteristic
232C23—21no—
233C23—13362no—
234C23—3341no—
235C23—3342no—
236C23—6684no—
237C23—6684no—
238C23C2×C22⋊D16712yes—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—21no—
243D4—44no—
244D4—13361no—
245D4—3342no—
246D4—3341no—
247D4—3342no—
248D4—6684no—
249D4—6684no—
250D4—6682no—
251D4—1672no—
252D4D4×D16712yescomplemented
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—3341no—
260C2×C4—6682no—
261C2×C4C2×C22⋊D16711yescharacteristic
262C2×C4C2×C22⋊D16712yes—
263C2×C4C2×C22⋊D16711yescharacteristic
264C2×C4C23×D16711yescharacteristic
265C8—3342no—
266C8—3341no—
267C8—1672no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—13368no—
275C22—3342no—
276C22—6682no—
277C22—6682no—
278C22—6682no—
279C22—6681no—
280C22—6681no—
281C22C22×C22⋊D16711yescharacteristic
282C4—21no—
283C4—22no—
284C4—41no—
285C4—42no—
286C4—6681no—
287C4#1912885712yescomplemented
288C4C2×D4×D16711yescharacteristic
289C4#1912887711yescharacteristic
290C2—42no—
291C2—42no—
292C2—13362no—
293C2—6682no—
294C2D4×C22⋊D16711yescharacteristic
295C2C2×#1912885711yescharacteristic
296C2#2492407711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#547×C167C2
socleC22×C167C22×D4
hypercenter#547D167
hypocenterC167#3272
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#19128857C2semidirect
2. D4×D668C2semidirect
3. #24923643C2semidirect
4. #24923517C2semidirect
5. #24923471C2semidirect
6. #24923464C2semidirect
7. #24923419C2semidirect
8. #24923414C2semidirect
9. C4×#14581935C2semidirect
10. #24923347C2semidirect
11. #547×C167C2semidirect
16. C2×#14581937C22semidirect
17. C2×#14581937C22semidirect
19. C2×#14581935C22semidirect
20. C2×#14581934C22semidirect
26. #19128765C22semidirect
27. C4⋊(C167⋊C8)C22semidirect
28. #19128756C22semidirect
29. #19128753C22semidirect
31. #19128752C22semidirect
32. C22⋊(C167⋊C8)C22semidirect
33. C4×C167⋊C8C22semidirect
34. C2×#41×C167C22semidirect
35. C4⋊Q8×C167C22semidirect
37. C22⋊Q8×C167C22semidirect
40. C4×D4×C167C22semidirect
57. #14581935D4semidirect
100. C167⋊C8C2×D4semidirect
108. D4×C167C2×D4semidirect
137. C4×C167AGL(1,ℤ/8)semidirect
148. C167#3272semidirect
151. #547D167semidirect
161. C2×#41C2×D167semidirect
171. C4⋊Q8C2×D167semidirect
174. C22⋊Q8C2×D167semidirect
181. C4×D4C2×D167semidirect
252. D4D4×D167semidirect
287. C4#19128857semidirect

Maximal subgroups

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

subgroupindex
C2×#191288572
D4×D6682
#249236432
#249235172
#249234712
#249234642
#249234192
#249234142
C4×#145819352
#249233472
#547×C1672
#3272167