Finite Group Databasebeta

← #36335824

Subgroups — #36335824

Order 21376.

526 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 131 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×#19128851C211yesmaximal, characteristic, complemented
2C2×D8×D167C211yesmaximal, characteristic, complemented
3#24923914C212yesmaximal, complemented
4#24923653C211yesmaximal, characteristic, complemented
5#24923511C211yesmaximal, characteristic, complemented
6#24923506C211yesmaximal, characteristic, complemented
7#24923484C212yesmaximal, complemented
8#24923421C212yesmaximal, complemented
9#24923415C212yesmaximal, complemented
10C8×D668C211yesmaximal, characteristic, complemented
11C4×D8×C167C211yesmaximal, characteristic, complemented
12C2×#14581949C2212yescomplemented
13C2×D4×D167C2212yes—
14#19128851—24no—
15C2×Dic668C2211yescharacteristic, complemented
16C2×#14581939C2212yescomplemented
17C2×#14581938C2212yescomplemented
18C2×D1336C2211yescharacteristic
19C2×#14581935C2212yes—
20D8×D167—24no—
21C2×C8×D167C2211yescharacteristic
22C2×C8×D167C2211yescharacteristic, complemented
23#19128820C2212yescomplemented
24#19128816—22no—
25#19128812C2212yes—
26#19128807C2212yescomplemented
27#19128806—22no—
28#19128802—22no—
29C4×D668C2211yescharacteristic
30C2×(C2×D167)⋊C4—22no—
31C4⋊(C167⋊C8)C2211yescharacteristic, complemented
32#19128755C2212yescomplemented
33#19128752C2212yescomplemented
34(C2×D167)⋊C8C2211yescharacteristic
35(C2×D167)⋊C8C2211yescharacteristic, complemented
36C2×D8×C167C2211yescharacteristic
37C4×D4×C167C2212yescomplemented
38#107×C167C2211yescharacteristic, complemented
39#102×C167C2212yescomplemented
40C4×C8×C167C2211yescharacteristic, complemented
41C23×D167—22no—
42#14581949—24no—
43#14581949—42no—
44C2×Dic334C2312yes—
45C2×D668—22no—
46C2×D668D411yescharacteristic
47C2×D668C2312yes—
48C2×C22⋊D167—22no—
49C2×C22⋊D167—22no—
50D4×D167—24no—
51D4×D167—42no—
52C22×Dic167—22no—
53C2×C4×D167C2311yescharacteristic
54C2×C4×D167C2311yescharacteristic
55Dic668—22no—
56#14581939—42no—
57#14581938—24no—
58D1336—22no—
59#14581935—24no—
60C2×C167⋊C8C2311yescharacteristic
61C2×C167⋊C8C2311yescharacteristic
62C8×D167—22no—
63C8×D167—22no—
64C4⋊Dic167—22no—
65C4⋊Dic167D411yescharacteristic
66(C2×D167)⋊C4—22no—
67(C2×D167)⋊C4—22no—
68(C2×D167)⋊C4—22no—
69(C2×D167)⋊C4—22no—
70(C2×D167)⋊C4D411yescharacteristic
71(C2×D167)⋊C4D411yescharacteristic
72C2×D4×C167C2312yes—
73C22×C4×C167—22no—
74D8×C167D414yescomplemented
75C2×C8×C167C2311yescharacteristic
76C2×C8×C167C2311yescharacteristic
77C4⋊C4×C167C2312yes—
78C22⋊C4×C167—22no—
79C42×C167C2311yescharacteristic
80C22×D167—22no—
81C22×D167—44no—
82C22×D167—44no—
83C22×D167—82no—
84C22×D167C2×D411yescharacteristic
85C22×D167C2×D411yescharacteristic
86Dic334—22no—
87Dic334—22no—
88D668—22no—
89D668—22no—
90D668—22no—
91D668—82no—
92C22⋊D167—44no—
93C22⋊D167—44no—
94C2×Dic167—22no—
95C2×Dic167—44no—
96C2×Dic167C2×D411yescharacteristic
97C2×Dic167C2×D411yescharacteristic
98C4×D167—21no—
99C4×D167—21no—
100C4×D167—21no—
101C4×D167—21no—
102C167⋊C8—21no—
103C167⋊C8—21no—
104C23×C167—22no—
105D4×C167—22no—
106D4×C167C2×D414yes—
107C2×C4×C167—22no—
108C2×C4×C167—82no—
109C2×C4×C167C2×D411yescharacteristic
110C2×C4×C167C2×D411yescharacteristic
111C2×C4×C167C2411yescharacteristic
112C8×C167—21no—
113C8×C167C2×D412yes—
114C22×C167—24no—
115C22×C167—42no—
116C22×C167—42no—
117C22×C167C22×D411yescharacteristic
118C2×D167—21no—
119C2×D167—21no—
120C2×D167—21no—
121C2×D167—21no—
122C2×D167—21no—
123C2×D167—42no—
124C2×D167—42no—
125C2×D167—42no—
126C2×D167—88no—
127C2×D167#13512yescomplemented
128C4×C167—21no—
129C4×C167—21no—
130C4×C167—42no—
131C4×C167C22×D411yescharacteristic
132C4×C167H(5,𝔽2)11yescharacteristic
133Dic167—21no—
134Dic167—21no—
135Dic167—42no—
136C2×C167—44no—
137C2×C167D4211yescharacteristic
138C2×C167C2×#13511yescharacteristic
139C2×C167#57611yescharacteristic
140D167—22no—
141D167—41no—
142D167—82no—
143C167#328011yescharacteristic, complemented, Sylow
144#3280—1671nomaximal, Sylow
145C2×#135—1671no—
146C22×D8—1671no—
147#546—1672no—
148#495—1671no—
149#467—1671no—
150#466—1671no—
151#463—1672no—
152#450—1672no—
153#449—1672no—
154C4×D8D16711yescharacteristic, complemented
155C8×D4—1671no—
156C2×#41—1672no—
157C22×D4—1672no—
158#135—3344no—
159C2×Q16—1671no—
160C2×SD16—1672no—
161C2×SD16—1672no—
162C2×D8—3344no—
163C2×D8—1672no—
164C2×D8—1671no—
165C2×D8C2×D16711yescharacteristic
166C22×C8—1671no—
167C22×C8—1671no—
168#124—1672no—
169#123—3342no—
170C22⋊Q8—1672no—
171#121—3342no—
172#121—1672no—
173#120—3342no—
174C4×D4—1671no—
175C4×D4C2×D16712yescomplemented
176C2×C22⋊C4—3342no—
177#107C2×D16711yescharacteristic, complemented
178C4⋊C8—1671no—
179#103—1672no—
180#102—1672no—
181#102C2×D16712yescomplemented
182C22⋊C8—1671no—
183C22⋊C8—1671no—
184C4×C8C2×D16711yescharacteristic, complemented
185C24—3342no—
186#41—3344no—
187#41—6682no—
188C2×Q8—1672no—
189C2×D4—3342no—
190C2×D4—3344no—
191C2×D4—3342no—
192C2×D4—3342no—
193C2×D4—6682no—
194C2×D4—1671no—
195C2×D4—1672no—
196C2×D4C22×D16712yes—
197C22×C4—22no—
198C22×C4—3342no—
199C22×C4—1671no—
200C22×C4—1671no—
201Q16—3342no—
202SD16—3344no—
203SD16—6682no—
204D8—3344no—
205D8—3342no—
206D8D66814yescomplemented
207C2×C8—3342no—
208C2×C8—3342no—
209C2×C8—1671no—
210C2×C8—1671no—
211C2×C8C22×D16711yescharacteristic
212C2×C8C22×D16711yescharacteristic
213C4⋊C4—3342no—
214C4⋊C4—1671no—
215C4⋊C4C22×D16712yes—
216C22⋊C4—22no—
217C22⋊C4—3342no—
218C22⋊C4—3342no—
219C22⋊C4—3342no—
220C22⋊C4—3342no—
221C22⋊C4—1671no—
222C22⋊C4—1671no—
223C42C22×D16711yescharacteristic
224C23—22no—
225C23—13362no—
226C23—3342no—
227C23—6684no—
228C23—6684no—
229C23—1671no—
230C23—1671no—
231Q8—3342no—
232Q8—3342no—
233D4—22no—
234D4—13362no—
235D4—3342no—
236D4—3342no—
237D4—3342no—
238D4—6684no—
239D4—6684no—
240D4C2×D66814yes—
241C2×C4—22no—
242C2×C4—82no—
243C2×C4—3341no—
244C2×C4—3341no—
245C2×C4—3341no—
246C2×C4—3341no—
247C2×C4—3342no—
248C2×C4—6684no—
249C2×C4—1671no—
250C2×C4—1671no—
251C2×C4D4×D16711yescharacteristic
252C2×C4D4×D16711yescharacteristic
253C2×C4C23×D16711yescharacteristic
254C8—21no—
255C8—3341no—
256C8—3341no—
257C8C2×D66812yes—
258C22—24no—
259C22—42no—
260C22—42no—
261C22—13368no—
262C22—3341no—
263C22—3341no—
264C22—3341no—
265C22—3341no—
266C22—3341no—
267C22—6682no—
268C22—6682no—
269C22—6682no—
270C22—1672no—
271C22C2×D4×D16711yescharacteristic
272C4—21no—
273C4—21no—
274C4—42no—
275C4—3341no—
276C4—3341no—
277C4—6682no—
278C4C22×D66811yescharacteristic
279C4#1912887811yescharacteristic
280C2—44no—
281C2—13362no—
282C2—3342no—
283C2—6681no—
284C2D4×D66811yescharacteristic
285C2#135×D16711yescharacteristic
286C2#2492407011yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC4×D8×C167C2
socleC22×C167C22×D4
hypercenterC4×D8D167
hypocenterC167#3280
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×#19128851C2semidirect
2. C2×D8×D167C2semidirect
3. #24923914C2semidirect
4. #24923653C2semidirect
5. #24923511C2semidirect
6. #24923506C2semidirect
7. #24923484C2semidirect
8. #24923421C2semidirect
9. #24923415C2semidirect
10. C8×D668C2semidirect
11. C4×D8×C167C2semidirect
12. C2×#14581949C22semidirect
15. C2×Dic668C22semidirect
16. C2×#14581939C22semidirect
17. C2×#14581938C22semidirect
22. C2×C8×D167C22semidirect
23. #19128820C22semidirect
26. #19128807C22semidirect
31. C4⋊(C167⋊C8)C22semidirect
32. #19128755C22semidirect
33. #19128752C22semidirect
35. (C2×D167)⋊C8C22semidirect
37. C4×D4×C167C22semidirect
38. #107×C167C22semidirect
39. #102×C167C22semidirect
40. C4×C8×C167C22semidirect
74. D8×C167D4semidirect
127. C2×D167#135semidirect
143. C167#3280semidirect
154. C4×D8D167semidirect
175. C4×D4C2×D167semidirect
177. #107C2×D167semidirect
181. #102C2×D167semidirect
184. C4×C8C2×D167semidirect
206. D8D668semidirect

Maximal subgroups

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

subgroupindex
C2×#191288512
C2×D8×D1672
#249239142
#249236532
#249235112
#249235062
#249234842
#249234212
#249234152
C8×D6682
C4×D8×C1672
#3280167