Finite Group Databasebeta

← #35224463

Subgroups — #35224463

Order 20864.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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#23782178C211yesmaximal, characteristic
2C2×C4⋊C4×D163C211yesmaximal, characteristic, complemented
3#23781998C211yesmaximal, characteristic, complemented
4#23781994C212yesmaximal, complemented
5#23781984C211yesmaximal, characteristic, complemented
6#23781983C211yesmaximal, characteristic, complemented
7#23781980C212yesmaximal, complemented
8#23781706C212yesmaximal, complemented
9#23781665C212yesmaximal, complemented
10#23781627C211yesmaximal, characteristic, complemented
11#484×C163C211yesmaximal, characteristic, complemented
12C2×#14579700C2211yescharacteristic
13C22×C4×D163—21no—
14C22×C4×D163C2211yescharacteristic
15C2×#14579682C2212yes—
16#19104720—21no—
17#19104711C2211yescharacteristic
18#19104710C2211yescharacteristic
19#19104709C2211yescharacteristic
20#19104705—21no—
21#19104697—21no—
22(C22⋊D163)⋊C4—21no—
23#19104693C2211yescharacteristic
24C2×C4⋊Dic163C2211yescharacteristic
25C2×Dic163⋊C4—21no—
26C4⋊C4×D163—21no—
27C4⋊C4×D163—21no—
28C4⋊C4×D163C2212yes—
29C4⋊C4×D163C2212yes—
30#19104668C2212yescomplemented
31#19104667C2212yescomplemented
32#19104666C2212yescomplemented
33#19104659C2212yescomplemented
34#19104657C2212yescomplemented
35#19104655C2212yescomplemented
36C22⋊(C163⋊C8)C2211yescharacteristic
37(C2×D163)⋊C8C2212yes—
38C22⋊Q8×C163C2211yescharacteristic
39C2×C4⋊C4×C163C2211yescharacteristic
40#107×C163C2212yescomplemented
41#103×C163C2212yescomplemented
42C22⋊C8×C163C2211yescharacteristic
43C23×D163D411yescharacteristic
44#14579700—24no—
45C2×D652—21no—
46C2×D652—21no—
47C2×D652C2311yescharacteristic
48C2×C22⋊D163—21no—
49C22×Dic163—21no—
50C22×Dic163D411yescharacteristic
51C2×C4×D163—24no—
52C2×C4×D163—22no—
53C2×C4×D163—22no—
54C2×C4×D163—21no—
55C2×C4×D163—44no—
56C2×C4×D163D411yescharacteristic
57C2×C4×D163D411yescharacteristic
58C2×C4×D163C2311yescharacteristic
59C2×C4×D163C2311yescharacteristic
60#14579682—24no—
61C2×C163⋊C8C2312yes—
62C4⋊Dic163—21no—
63C4⋊Dic163C2312yes—
64C4⋊Dic163C2311yescharacteristic
65Dic163⋊C4—22no—
66Dic163⋊C4—22no—
67Dic163⋊C4—21no—
68(C2×D163)⋊C4—21no—
69(C2×D163)⋊C4—21no—
70(C2×D163)⋊C4—21no—
71(C2×D163)⋊C4—21no—
72C4×Dic163—21no—
73C2×Q8×C163C2311yescharacteristic
74C22×C4×C163—21no—
75C22×C4×C163C2311yescharacteristic
76C2×C8×C163C2312yes—
77C4⋊C4×C163—21no—
78C4⋊C4×C163—21no—
79C4⋊C4×C163C2312yes—
80C4⋊C4×C163C2311yescharacteristic
81C22⋊C4×C163—21no—
82C22×D163—22no—
83C22×D163—22no—
84C22×D163—22no—
85C22×D163—21no—
86C22×D163C2×D411yescharacteristic
87C22×D163C2×D411yescharacteristic
88D652—22no—
89D652—42no—
90D652—42no—
91C22⋊D163—42no—
92C2×Dic163—22no—
93C2×Dic163—21no—
94C2×Dic163—22no—
95C2×Dic163—42no—
96C2×Dic163C2×D411yescharacteristic
97C2×Dic163C2×D411yescharacteristic
98C4×D163—22no—
99C4×D163—22no—
100C4×D163—22no—
101C4×D163—44no—
102C4×D163—44no—
103C4×D163—42no—
104C4×D163#4114yes—
105C163⋊C8—22no—
106C23×C163C2×D411yescharacteristic
107Q8×C163—22no—
108C2×C4×C163—22no—
109C2×C4×C163—21no—
110C2×C4×C163—21no—
111C2×C4×C163—22no—
112C2×C4×C163—42no—
113C2×C4×C163C2×D411yescharacteristic
114C2×C4×C163C2411yescharacteristic
115C8×C163—22no—
116C22×C163—21no—
117C22×C163—21no—
118C22×C163#13712yes—
119C22×C163C22×D411yescharacteristic
120C2×D163—28no—
121C2×D163—21no—
122C2×D163—21no—
123C2×D163—21no—
124C2×D163—21no—
125C2×D163—41no—
126C2×D163—41no—
127C2×D163—41no—
128C2×D163#12312yes—
129C2×D163AGL(1,ℤ/8)12yes—
130C4×C163—21no—
131C4×C163—41no—
132C4×C163—42no—
133C4×C163—41no—
134C4×C163C2×#4112yes—
135Dic163—21no—
136Dic163—42no—
137Dic163—41no—
138Dic163#12312yes—
139C2×C163—22no—
140C2×C163C2×#12311yescharacteristic
141C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
142C2×C163C2×#13711yescharacteristic
143D163—22no—
144D163—22no—
145D163—81no—
146C163#309211yescharacteristic, complemented, Sylow
147#3092—1631nomaximal, Sylow
148#525—1631no—
149C22×C4⋊C4—1631no—
150#484—1631no—
151#484D16311yescharacteristic, complemented
152#483—1632no—
153#481—1631no—
154#481—1631no—
155#480—1632no—
156#428—1632no—
157#417—1632no—
158#407—1631no—
159C2×#41—1631no—
160C23×C4—3261no—
161C23×C4—1631no—
162C2×M16—1632no—
163#123—3261no—
164C22⋊Q8—1631no—
165C22⋊Q8C2×D16311yescharacteristic
166#121—1631no—
167#121—1631no—
168#120—3261no—
169C4×D4—3261no—
170C4×D4—3261no—
171#117—1631no—
172C2×C4⋊C4—3261no—
173C2×C4⋊C4—3261no—
174C2×C4⋊C4—3261no—
175C2×C4⋊C4—1631no—
176C2×C4⋊C4—1632no—
177C2×C4⋊C4—1632no—
178C2×C4⋊C4C2×D16311yescharacteristic
179#107—1632no—
180#107C2×D16312yescomplemented
181#106—1632no—
182#106—1632no—
183#103—1632no—
184#103C2×D16312yescomplemented
185#102—1632no—
186#102—1632no—
187C22⋊C8—1631no—
188C22⋊C8—1632no—
189C22⋊C8C2×D16311yescharacteristic
190C24—1631no—
191#41—3264no—
192C2×Q8C22×D16311yescharacteristic
193C2×D4—3261no—
194C2×D4—3261no—
195C2×D4—3261no—
196C2×D4—1631no—
197C22×C4—21no—
198C22×C4—3264no—
199C22×C4—3262no—
200C22×C4—3262no—
201C22×C4—3261no—
202C22×C4—3261no—
203C22×C4—6524no—
204C22×C4—1631no—
205C22×C4—1631no—
206C22×C4—1631no—
207C22×C4—1631no—
208C22×C4—1631no—
209C22×C4C22×D16311yescharacteristic
210M16—3264no—
211C2×C8—1632no—
212C2×C8C22×D16312yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—3262no—
216C4⋊C4—3262no—
217C4⋊C4—3261no—
218C4⋊C4—3261no—
219C4⋊C4—1632no—
220C4⋊C4—1631no—
221C4⋊C4C22×D16312yes—
222C4⋊C4C22×D16311yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—3261no—
225C22⋊C4—3261no—
226C22⋊C4—3261no—
227C22⋊C4—3261no—
228C42—3261no—
229C23—3262no—
230C23—3262no—
231C23—3262no—
232C23—3261no—
233C23—1631no—
234C23—1631no—
235C23D4×D16311yescharacteristic
236Q8—22no—
237D4—3262no—
238D4—6522no—
239D4—6522no—
240D4—6522no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—3262no—
247C2×C4—3262no—
248C2×C4—3262no—
249C2×C4—3262no—
250C2×C4—3261no—
251C2×C4—3262no—
252C2×C4—6524no—
253C2×C4—6524no—
254C2×C4—6522no—
255C2×C4—6522no—
256C2×C4—1631no—
257C2×C4—1631no—
258C2×C4—1634no—
259C2×C4D4×D16311yescharacteristic
260C2×C4C23×D16311yescharacteristic
261C8—22no—
262C8—3262no—
263C22—21no—
264C22—21no—
265C22—3268no—
266C22—3261no—
267C22—3261no—
268C22—3261no—
269C22—3261no—
270C22—6521no—
271C22—6521no—
272C22—6521no—
273C22—1632no—
274C22—1632no—
275C22#1910476712yes—
276C22C2×D4×D16311yescharacteristic
277C4—21no—
278C4—41no—
279C4—42no—
280C4—41no—
281C4—3261no—
282C4—6522no—
283C4—6521no—
284C4—1632no—
285C4#41×D16312yes—
286C2—22no—
287C2—13041no—
288C2—3262no—
289C2—3262no—
290C2#123×D16311yescharacteristic
291C2AGL(1,ℤ/8)×D16311yescharacteristic
292C2C2×#1910476711yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#484×C163C2
socleC22×C163C22×D4
prime radical#23782178C2
hypercenter#484D163
hypocenterC163#3092
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D163C2semidirect
3. #23781998C2semidirect
4. #23781994C2semidirect
5. #23781984C2semidirect
6. #23781983C2semidirect
7. #23781980C2semidirect
8. #23781706C2semidirect
9. #23781665C2semidirect
10. #23781627C2semidirect
11. #484×C163C2semidirect
30. #19104668C22semidirect
31. #19104667C22semidirect
32. #19104666C22semidirect
33. #19104659C22semidirect
34. #19104657C22semidirect
35. #19104655C22semidirect
40. #107×C163C22semidirect
41. #103×C163C22semidirect
146. C163#3092semidirect
151. #484D163semidirect
180. #107C2×D163semidirect
184. #103C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237821782
C2×C4⋊C4×D1632
#237819982
#237819942
#237819842
#237819832
#237819802
#237817062
#237816652
#237816272
#484×C1632
#3092163