Finite Group Databasebeta

← #35226350

Subgroups — #35226350

Order 20864.

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#23782319C211yesmaximal, characteristic, complemented
2D4×D652C211yesmaximal, characteristic, complemented
3#23782012C211yesmaximal, characteristic, complemented
4#23781979C212yesmaximal, complemented
5#23781946C211yesmaximal, characteristic, complemented
6#23781901C212yesmaximal, complemented
7#23781861C211yesmaximal, characteristic, complemented
8#23781801C212yesmaximal, complemented
9C4×D1304C211yesmaximal, characteristic, complemented
10C2×#19104657C212yesmaximal, complemented
11C8×D4×C163C211yesmaximal, characteristic, complemented
12C2×#14579698—21no—
13C22×D652C2212yes—
14C2×D4×D163—21no—
15C2×D1304C2211yescharacteristic
16C2×D1304C2212yescomplemented
17#19104735C2211yescharacteristic, complemented
18#19104732C2211yescharacteristic
19#19104719—22no—
20#19104715—21no—
21#19104710—21no—
22#19104709C2212yes—
23#19104709C2212yescomplemented
24#19104705—22no—
25C4×D652C2211yescharacteristic
26C4×D652C2211yescharacteristic, complemented
27C2×C4⋊Dic163C2212yescomplemented
28#19104669C2212yescomplemented
29#19104669C2211yescharacteristic, complemented
30#19104657C2211yescharacteristic
31#19104657C2212yes—
32#19104657C2211yescharacteristic, complemented
33#19104657C2212yescomplemented
34#19104657C2212yescomplemented
35#19104657C2212yescomplemented
36C22×C8×C163C2212yescomplemented
37C4×D4×C163C2211yescharacteristic
38C4⋊C8×C163C2211yescharacteristic, complemented
39C22⋊C8×C163C2212yescomplemented
40C4×C8×C163C2211yescharacteristic, complemented
41C23×D163—22no—
42#14579698—44no—
43C2×Dic326—21no—
44C2×D652—24no—
45C2×D652—21no—
46C2×D652—22no—
47C2×D652—21no—
48C2×D652—42no—
49C2×D652C2312yes—
50C2×D652C2311yescharacteristic
51C2×D652C2311yescharacteristic
52C2×C22⋊D163—22no—
53C2×C22⋊D163—22no—
54D4×D163—44no—
55C22×Dic163—22no—
56C2×C4×D163—21no—
57C2×C4×D163—21no—
58D1304—22no—
59D1304—42no—
60C4⋊Dic163—22no—
61C4⋊Dic163—21no—
62C4⋊Dic163C2311yescharacteristic
63C4⋊Dic163C2312yes—
64C4⋊Dic163C2311yescharacteristic
65(C2×D163)⋊C4—21no—
66(C2×D163)⋊C4—21no—
67(C2×D163)⋊C4—22no—
68(C2×D163)⋊C4—22no—
69C2×D4×C163D411yescharacteristic
70C22×C4×C163C2312yes—
71C2×C8×C163—42no—
72C2×C8×C163C2311yescharacteristic
73C2×C8×C163C2311yescharacteristic
74C2×C8×C163C2312yes—
75C4⋊C4×C163D411yescharacteristic
76C22⋊C4×C163D412yes—
77C42×C163C2311yescharacteristic
78C22×D163—21no—
79C22×D163—21no—
80C22×D163—22no—
81C22×D163—44no—
82C22×D163—44no—
83C22×D163—82no—
84Dic326—42no—
85D652—21no—
86D652—22no—
87D652—21no—
88D652—21no—
89D652—22no—
90D652—44no—
91D652—42no—
92D652—81no—
93D652#4112yescomplemented
94C22⋊D163—44no—
95C22⋊D163—44no—
96C2×Dic163—21no—
97C2×Dic163—21no—
98C2×Dic163—22no—
99C2×Dic163—44no—
100C4×D163—42no—
101C4×D163—42no—
102C23×C163C2×D412yes—
103D4×C163D814yescomplemented
104C2×C4×C163—22no—
105C2×C4×C163—22no—
106C2×C4×C163C2×D411yescharacteristic
107C2×C4×C163C2×D412yes—
108C2×C4×C163C2×D411yescharacteristic
109C2×C4×C163C2411yescharacteristic
110C8×C163—21no—
111C8×C163—21no—
112C8×C163—22no—
113C22×C163—22no—
114C22×C163—22no—
115C22×C163C2×D814yes—
116C22×C163C22×D411yescharacteristic
117C2×D163—22no—
118C2×D163—41no—
119C2×D163—41no—
120C2×D163—42no—
121C2×D163—41no—
122C2×D163—41no—
123C2×D163—42no—
124C2×D163—42no—
125C2×D163—41no—
126C2×D163—88no—
127C4×C163—22no—
128C4×C163—21no—
129C4×C163C2×D812yes—
130C4×C163C2×#4111yescharacteristic
131C4×C163H(5,𝔽2)11yescharacteristic
132Dic163—41no—
133Dic163—41no—
134Dic163—42no—
135C2×C163—24no—
136C2×C163#54611yescharacteristic
137C2×C163C22×D811yescharacteristic
138C2×C163#57711yescharacteristic
139D163—42no—
140D163—81no—
141D163—82no—
142C163#329411yescharacteristic, complemented, Sylow
143#3294—1631nomaximal, Sylow
144#547—1631no—
145D42—1631no—
146#486—1631no—
147#480—1632no—
148#474—1631no—
149#466—1632no—
150#459—1631no—
151#447—1632no—
152C4×D8—1631no—
153C8×D4D16311yescharacteristic, complemented
154C2×#102—1632no—
155C2×#41—3261no—
156C22×D4—3261no—
157C22×D4—1632no—
158C2×D8—1631no—
159C2×D8—1632no—
160C22×C8C2×D16312yescomplemented
161C4⋊Q8—1631no—
162#127—1631no—
163#123—3262no—
164C22⋊Q8—3261no—
165#121—3261no—
166#121—1632no—
167#121—1632no—
168#120—3262no—
169C4×D4—1631no—
170C4×D4—1631no—
171C4×D4C2×D16311yescharacteristic
172C2×C4⋊C4—1632no—
173#107—1631no—
174#107—1632no—
175C4⋊C8C2×D16311yescharacteristic, complemented
176#102—1631no—
177#102—1632no—
178#102—1632no—
179#102—1632no—
180#102—1631no—
181#102—1632no—
182C22⋊C8C2×D16312yescomplemented
183C4×C8C2×D16311yescharacteristic, complemented
184C24—3262no—
185#41—6524no—
186C2×Q8—3261no—
187C2×D4—3264no—
188C2×D4—3262no—
189C2×D4—3262no—
190C2×D4—3261no—
191C2×D4—3262no—
192C2×D4—3261no—
193C2×D4—6522no—
194C2×D4—6524no—
195C2×D4—1632no—
196C2×D4—1631no—
197C2×D4—1631no—
198C2×D4D65211yescharacteristic
199C22×C4—3262no—
200C22×C4—3261no—
201C22×C4—3261no—
202C22×C4C22×D16312yes—
203D8—3262no—
204D8—6522no—
205C2×C8—42no—
206C2×C8C22×D16311yescharacteristic
207C2×C8C22×D16311yescharacteristic
208C2×C8C22×D16312yes—
209C4⋊C4—3262no—
210C4⋊C4—3261no—
211C4⋊C4—1631no—
212C4⋊C4—1632no—
213C4⋊C4—1631no—
214C4⋊C4D65211yescharacteristic
215C22⋊C4—3261no—
216C22⋊C4—3261no—
217C22⋊C4—3262no—
218C22⋊C4—3262no—
219C22⋊C4D65212yes—
220C42C22×D16311yescharacteristic
221C23—13042no—
222C23—3261no—
223C23—3261no—
224C23—3262no—
225C23—6524no—
226C23—6524no—
227C23C2×D65212yes—
228Q8—6522no—
229D4—13041no—
230D4—3261no—
231D4—3262no—
232D4—3261no—
233D4—3261no—
234D4—3262no—
235D4—6524no—
236D4—6524no—
237D4—6524no—
238D4—6522no—
239D4—1632no—
240D4D130414yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—3261no—
244C2×C4—3261no—
245C2×C4—3262no—
246C2×C4—6524no—
247C2×C4—6522no—
248C2×C4—6522no—
249C2×C4C2×D65211yescharacteristic
250C2×C4C2×D65212yes—
251C2×C4C2×D65211yescharacteristic
252C2×C4C23×D16311yescharacteristic
253C8—21no—
254C8—21no—
255C8—22no—
256C22—22no—
257C22—22no—
258C22—13048no—
259C22—3262no—
260C22—6521no—
261C22—6522no—
262C22—6522no—
263C22—6521no—
264C22—6522no—
265C22—6521no—
266C22—6521no—
267C22—6521no—
268C22C2×D130414yes—
269C22C22×D65211yescharacteristic
270C4—21no—
271C4—22no—
272C4—6521no—
273C4—6521no—
274C4—6522no—
275C4C2×D130412yes—
276C4#41×D16311yescharacteristic
277C4#1910478011yescharacteristic
278C2—24no—
279C2—13042no—
280C2—13041no—
281C2—6522no—
282C2#2378231011yescharacteristic
283C2C22×D130411yescharacteristic
284C2#2378247711yescharacteristic
subgroupG/H
centerC22C22×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC8×D4×C163C2
socleC22×C163C22×D4
hypercenterC8×D4D163
hypocenterC163#3294
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #23782319C2semidirect
2. D4×D652C2semidirect
3. #23782012C2semidirect
4. #23781979C2semidirect
5. #23781946C2semidirect
6. #23781901C2semidirect
7. #23781861C2semidirect
8. #23781801C2semidirect
9. C4×D1304C2semidirect
10. C2×#19104657C2semidirect
11. C8×D4×C163C2semidirect
16. C2×D1304C22semidirect
17. #19104735C22semidirect
23. #19104709C22semidirect
26. C4×D652C22semidirect
27. C2×C4⋊Dic163C22semidirect
28. #19104669C22semidirect
29. #19104669C22semidirect
32. #19104657C22semidirect
33. #19104657C22semidirect
34. #19104657C22semidirect
35. #19104657C22semidirect
36. C22×C8×C163C22semidirect
38. C4⋊C8×C163C22semidirect
39. C22⋊C8×C163C22semidirect
40. C4×C8×C163C22semidirect
93. D652#41semidirect
103. D4×C163D8semidirect
142. C163#3294semidirect
153. C8×D4D163semidirect
160. C22×C8C2×D163semidirect
175. C4⋊C8C2×D163semidirect
182. C22⋊C8C2×D163semidirect
183. C4×C8C2×D163semidirect
240. D4D1304semidirect

Maximal subgroups

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

subgroupindex
#237823192
D4×D6522
#237820122
#237819792
#237819462
#237819012
#237818612
#237818012
C4×D13042
C2×#191046572
C8×D4×C1632
#3294163