Finite Group Databasebeta

← #36335952

Subgroups — #36335952

Order 21376.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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
1#24923923C211yesmaximal, characteristic, complemented
2D4×D668C211yesmaximal, characteristic, complemented
3#24923616C211yesmaximal, characteristic, complemented
4#24923583C212yesmaximal, complemented
5#24923550C211yesmaximal, characteristic, complemented
6#24923505C212yesmaximal, complemented
7#24923465C211yesmaximal, characteristic, complemented
8#24923405C212yesmaximal, complemented
9C4×D1336C211yesmaximal, characteristic, complemented
10C2×#19128754C212yesmaximal, complemented
11C8×D4×C167C211yesmaximal, characteristic, complemented
12C2×#14581949—21no—
13C22×D668C2212yes—
14C2×D4×D167—21no—
15C2×D1336C2211yescharacteristic
16C2×D1336C2212yescomplemented
17#19128832C2211yescharacteristic, complemented
18#19128829C2211yescharacteristic
19#19128816—22no—
20#19128812—21no—
21#19128807—21no—
22#19128806C2212yes—
23#19128806C2212yescomplemented
24#19128802—22no—
25C4×D668C2211yescharacteristic
26C4×D668C2211yescharacteristic, complemented
27C2×C4⋊Dic167C2212yescomplemented
28#19128766C2212yescomplemented
29#19128766C2211yescharacteristic, complemented
30#19128754C2212yes—
31#19128754C2211yescharacteristic
32#19128754C2212yescomplemented
33#19128754C2212yescomplemented
34#19128754C2211yescharacteristic, complemented
35#19128754C2212yescomplemented
36C22×C8×C167C2212yescomplemented
37C4×D4×C167C2211yescharacteristic
38C4⋊C8×C167C2211yescharacteristic, complemented
39C22⋊C8×C167C2212yescomplemented
40C4×C8×C167C2211yescharacteristic, complemented
41C23×D167—22no—
42#14581949—44no—
43C2×Dic334—21no—
44C2×D668—24no—
45C2×D668—21no—
46C2×D668—22no—
47C2×D668—21no—
48C2×D668—42no—
49C2×D668C2311yescharacteristic
50C2×D668C2312yes—
51C2×D668C2311yescharacteristic
52C2×C22⋊D167—22no—
53C2×C22⋊D167—22no—
54D4×D167—44no—
55C22×Dic167—22no—
56C2×C4×D167—21no—
57C2×C4×D167—21no—
58D1336—22no—
59D1336—42no—
60C4⋊Dic167—21no—
61C4⋊Dic167—22no—
62C4⋊Dic167C2312yes—
63C4⋊Dic167C2311yescharacteristic
64C4⋊Dic167C2311yescharacteristic
65(C2×D167)⋊C4—21no—
66(C2×D167)⋊C4—21no—
67(C2×D167)⋊C4—22no—
68(C2×D167)⋊C4—22no—
69C2×D4×C167D411yescharacteristic
70C22×C4×C167C2312yes—
71C2×C8×C167—42no—
72C2×C8×C167C2311yescharacteristic
73C2×C8×C167C2312yes—
74C2×C8×C167C2311yescharacteristic
75C4⋊C4×C167D411yescharacteristic
76C22⋊C4×C167D412yes—
77C42×C167C2311yescharacteristic
78C22×D167—21no—
79C22×D167—21no—
80C22×D167—22no—
81C22×D167—44no—
82C22×D167—44no—
83C22×D167—82no—
84Dic334—42no—
85D668—21no—
86D668—22no—
87D668—21no—
88D668—21no—
89D668—22no—
90D668—44no—
91D668—42no—
92D668—81no—
93D668#4112yescomplemented
94C22⋊D167—44no—
95C22⋊D167—44no—
96C2×Dic167—21no—
97C2×Dic167—21no—
98C2×Dic167—22no—
99C2×Dic167—44no—
100C4×D167—42no—
101C4×D167—42no—
102C23×C167C2×D412yes—
103D4×C167D814yescomplemented
104C2×C4×C167—22no—
105C2×C4×C167—22no—
106C2×C4×C167C2×D411yescharacteristic
107C2×C4×C167C2×D412yes—
108C2×C4×C167C2×D411yescharacteristic
109C2×C4×C167C2411yescharacteristic
110C8×C167—21no—
111C8×C167—21no—
112C8×C167—22no—
113C22×C167—22no—
114C22×C167—22no—
115C22×C167C2×D814yes—
116C22×C167C22×D411yescharacteristic
117C2×D167—22no—
118C2×D167—41no—
119C2×D167—41no—
120C2×D167—42no—
121C2×D167—41no—
122C2×D167—41no—
123C2×D167—42no—
124C2×D167—42no—
125C2×D167—41no—
126C2×D167—88no—
127C4×C167—22no—
128C4×C167—21no—
129C4×C167C2×D812yes—
130C4×C167C2×#4111yescharacteristic
131C4×C167H(5,𝔽2)11yescharacteristic
132Dic167—41no—
133Dic167—41no—
134Dic167—42no—
135C2×C167—24no—
136C2×C167#54611yescharacteristic
137C2×C167C22×D811yescharacteristic
138C2×C167#57711yescharacteristic
139D167—42no—
140D167—81no—
141D167—82no—
142C167#329411yescharacteristic, complemented, Sylow
143#3294—1671nomaximal, Sylow
144#547—1671no—
145D42—1671no—
146#486—1671no—
147#480—1672no—
148#474—1671no—
149#466—1672no—
150#459—1671no—
151#447—1672no—
152C4×D8—1671no—
153C8×D4D16711yescharacteristic, complemented
154C2×#102—1672no—
155C2×#41—3341no—
156C22×D4—3341no—
157C22×D4—1672no—
158C2×D8—1671no—
159C2×D8—1672no—
160C22×C8C2×D16712yescomplemented
161C4⋊Q8—1671no—
162#127—1671no—
163#123—3342no—
164C22⋊Q8—3341no—
165#121—3341no—
166#121—1672no—
167#121—1672no—
168#120—3342no—
169C4×D4—1671no—
170C4×D4—1671no—
171C4×D4C2×D16711yescharacteristic
172C2×C4⋊C4—1672no—
173#107—1671no—
174#107—1672no—
175C4⋊C8C2×D16711yescharacteristic, complemented
176#102—1671no—
177#102—1672no—
178#102—1672no—
179#102—1672no—
180#102—1671no—
181#102—1672no—
182C22⋊C8C2×D16712yescomplemented
183C4×C8C2×D16711yescharacteristic, complemented
184C24—3342no—
185#41—6684no—
186C2×Q8—3341no—
187C2×D4—3344no—
188C2×D4—3342no—
189C2×D4—3342no—
190C2×D4—3341no—
191C2×D4—3342no—
192C2×D4—3341no—
193C2×D4—6682no—
194C2×D4—6684no—
195C2×D4—1672no—
196C2×D4—1671no—
197C2×D4—1671no—
198C2×D4D66811yescharacteristic
199C22×C4—3342no—
200C22×C4—3341no—
201C22×C4—3341no—
202C22×C4C22×D16712yes—
203D8—3342no—
204D8—6682no—
205C2×C8—42no—
206C2×C8C22×D16711yescharacteristic
207C2×C8C22×D16711yescharacteristic
208C2×C8C22×D16712yes—
209C4⋊C4—3342no—
210C4⋊C4—3341no—
211C4⋊C4—1671no—
212C4⋊C4—1672no—
213C4⋊C4—1671no—
214C4⋊C4D66811yescharacteristic
215C22⋊C4—3341no—
216C22⋊C4—3341no—
217C22⋊C4—3342no—
218C22⋊C4—3342no—
219C22⋊C4D66812yes—
220C42C22×D16711yescharacteristic
221C23—13362no—
222C23—3341no—
223C23—3341no—
224C23—3342no—
225C23—6684no—
226C23—6684no—
227C23C2×D66812yes—
228Q8—6682no—
229D4—13361no—
230D4—3341no—
231D4—3342no—
232D4—3341no—
233D4—3341no—
234D4—3342no—
235D4—6684no—
236D4—6684no—
237D4—6684no—
238D4—6682no—
239D4—1672no—
240D4D133614yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—3341no—
244C2×C4—3341no—
245C2×C4—3342no—
246C2×C4—6684no—
247C2×C4—6682no—
248C2×C4—6682no—
249C2×C4C2×D66811yescharacteristic
250C2×C4C2×D66812yes—
251C2×C4C2×D66811yescharacteristic
252C2×C4C23×D16711yescharacteristic
253C8—21no—
254C8—21no—
255C8—22no—
256C22—22no—
257C22—22no—
258C22—13368no—
259C22—3342no—
260C22—6682no—
261C22—6681no—
262C22—6681no—
263C22—6682no—
264C22—6682no—
265C22—6681no—
266C22—6681no—
267C22—6681no—
268C22C2×D133614yes—
269C22C22×D66811yescharacteristic
270C4—22no—
271C4—21no—
272C4—6681no—
273C4—6681no—
274C4—6682no—
275C4C2×D133612yes—
276C4#41×D16711yescharacteristic
277C4#1912887711yescharacteristic
278C2—24no—
279C2—13362no—
280C2—13361no—
281C2—6682no—
282C2#2492391411yescharacteristic
283C2C22×D133611yescharacteristic
284C2#2492408111yescharacteristic
subgroupG/H
centerC22C22×D668
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC8×D4×C167C2
socleC22×C167C22×D4
hypercenterC8×D4D167
hypocenterC167#3294
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #24923923C2semidirect
2. D4×D668C2semidirect
3. #24923616C2semidirect
4. #24923583C2semidirect
5. #24923550C2semidirect
6. #24923505C2semidirect
7. #24923465C2semidirect
8. #24923405C2semidirect
9. C4×D1336C2semidirect
10. C2×#19128754C2semidirect
11. C8×D4×C167C2semidirect
16. C2×D1336C22semidirect
17. #19128832C22semidirect
23. #19128806C22semidirect
26. C4×D668C22semidirect
27. C2×C4⋊Dic167C22semidirect
28. #19128766C22semidirect
29. #19128766C22semidirect
32. #19128754C22semidirect
33. #19128754C22semidirect
34. #19128754C22semidirect
35. #19128754C22semidirect
36. C22×C8×C167C22semidirect
38. C4⋊C8×C167C22semidirect
39. C22⋊C8×C167C22semidirect
40. C4×C8×C167C22semidirect
93. D668#41semidirect
103. D4×C167D8semidirect
142. C167#3294semidirect
153. C8×D4D167semidirect
160. C22×C8C2×D167semidirect
175. C4⋊C8C2×D167semidirect
182. C22⋊C8C2×D167semidirect
183. C4×C8C2×D167semidirect
240. D4D1336semidirect

Maximal subgroups

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

subgroupindex
#249239232
D4×D6682
#249236162
#249235832
#249235502
#249235052
#249234652
#249234052
C4×D13362
C2×#191287542
C8×D4×C1672
#3294167