Finite Group Databasebeta

← #35223765

Subgroups — #35223765

Order 20864.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×D4×D163C211yesmaximal, characteristic, complemented
2C2×#19104763C211yesmaximal, characteristic, complemented
3C2×#19104761C211yesmaximal, characteristic, complemented
4#23782177C211yesmaximal, characteristic, complemented
5#23781817C214yesmaximal, complemented
6#23781800C214yesmaximal, complemented
7#23781669C211yesmaximal, characteristic, complemented
8#23781628C211yesmaximal, characteristic, complemented
9#418×C163C211yesmaximal, characteristic, complemented
10C24×D163—21no—
11C2×#14579699C2211yescharacteristic, complemented
12C22×D652C2211yescharacteristic
13C22×C22⋊D163—21no—
14C2×D4×D163—24no—
15C2×D4×D163—22no—
16C2×D4×D163—24no—
17C2×D4×D163C2214yes—
18C22×C4×D163C2211yescharacteristic
19#19104763—24no—
20#19104761—24no—
21C2×#14579688C2212yescomplemented
22C2×#14579687C2212yescomplemented
23C2×D1304C2212yescomplemented
24C2×#14579684C2212yescomplemented
25C2×#14579683C2211yescharacteristic, complemented
26#19104719—21no—
27#19104715C2212yescomplemented
28#19104710C2212yescomplemented
29#19104705—21no—
30C4×D652—22no—
31#19104655C2214yescomplemented
32(C2×D163)⋊C8C2214yescomplemented
33C22×D4×C163C2211yescharacteristic
34C2×M16×C163C2211yescharacteristic, complemented
35#117×C163C2211yescharacteristic, complemented
36#102×C163C2214yescomplemented
37C23×D163—28no—
38C23×D163—24no—
39C23×D163—21no—
40C23×D163—48no—
41C23×D163D411yescharacteristic
42#14579699—22no—
43#14579699—22no—
44C2×Dic326C2311yescharacteristic, complemented
45C2×D652—22no—
46C2×D652—21no—
47C2×D652—22no—
48C2×D652D414yescomplemented
49C2×D652C2312yes—
50C2×D652C2311yescharacteristic, complemented
51C2×C22⋊D163—28no—
52C2×C22⋊D163—24no—
53C2×C22⋊D163—21no—
54D4×D163—216no—
55D4×D163—28no—
56D4×D163—28no—
57C22×Dic163D411yescharacteristic
58C2×C4×D163—22no—
59C2×C4×D163—22no—
60C2×C4×D163—21no—
61C2×C4×D163D412yes—
62C2×C4×D163C2312yes—
63#14579688—24no—
64#14579687—24no—
65D1304—24no—
66#14579684—24no—
67#14579683—22no—
68C2×C163⋊C8C2312yescomplemented
69C4⋊Dic163—22no—
70(C2×D163)⋊C4—22no—
71(C2×D163)⋊C4—22no—
72C24×C163—21no—
73C2×D4×C163—21no—
74C2×D4×C163—22no—
75C2×D4×C163D414yescomplemented
76C2×D4×C163C2312yes—
77C22×C4×C163C2311yescharacteristic
78M16×C163—22no—
79C2×C8×C163C2312yescomplemented
80C4⋊C4×C163C2312yescomplemented
81C22⋊C4×C163—21no—
82C42×C163—21no—
83C22×D163—22no—
84C22×D163—24no—
85C22×D163—216no—
86C22×D163—24no—
87C22×D163—22no—
88C22×D163—21no—
89C22×D163—432no—
90C22×D163—48no—
91C22×D163—44no—
92C22×D163—48no—
93C22×D163C2×D412yes—
94Dic326—21no—
95Dic326—21no—
96D652—24no—
97D652—22no—
98D652—21no—
99D652—21no—
100D652—44no—
101D652C2×D414yes—
102C22⋊D163—216no—
103C22⋊D163—42no—
104C2×Dic163—22no—
105C2×Dic163—21no—
106C2×Dic163C2×D412yes—
107C4×D163—24no—
108C4×D163—22no—
109C4×D163—22no—
110C4×D163—42no—
111C163⋊C8—22no—
112C23×C163—24no—
113C23×C163—22no—
114C23×C163—44no—
115C23×C163C2×D411yescharacteristic
116D4×C163—24no—
117D4×C163—22no—
118D4×C163C2×D414yes—
119C2×C4×C163—22no—
120C2×C4×C163C2×D411yescharacteristic
121C2×C4×C163C2×D412yes—
122C2×C4×C163C2×D412yes—
123C2×C4×C163C2411yescharacteristic
124C8×C163—22no—
125C22×C163—22no—
126C22×C163—24no—
127C22×C163—48no—
128C22×C163—44no—
129C22×C163#12012yes—
130C22×C163C22×D411yescharacteristic
131C2×D163—28no—
132C2×D163—22no—
133C2×D163—22no—
134C2×D163—24no—
135C2×D163—432no—
136C2×D163—48no—
137C2×D163—44no—
138C2×D163—41no—
139C2×D163—42no—
140C2×D163AGL(1,ℤ/8)14yescomplemented
141C4×C163—42no—
142C4×C163#12012yes—
143C4×C163C22×D411yescharacteristic
144C4×C163C22×D411yescharacteristic
145Dic163—22no—
146Dic163—41no—
147C2×C163—22no—
148C2×C163—44no—
149C2×C163C2×#12011yescharacteristic
150C2×C163C2×AGL(1,ℤ/8)12yes—
151D163—24no—
152D163—44no—
153D163—81no—
154C163#300311yescharacteristic, complemented, Sylow
155#3003—1631nomaximal, Sylow
156C23×D4—1631no—
157C2×AGL(1,ℤ/8)—1631no—
158C2×AGL(1,ℤ/8)—1631no—
159#525—1631no—
160#450—1634no—
161#447—1634no—
162#418—1631no—
163#418D16311yescharacteristic, complemented
164#407—1631no—
165C25—3261no—
166C2×#41—1631no—
167C22×D4—3264no—
168C22×D4—3264no—
169C22×D4—3262no—
170C22×D4—3261no—
171C22×D4—1634no—
172C22×D4—1631no—
173C22×D4C2×D16311yescharacteristic
174C23×C4—1631no—
175AGL(1,ℤ/8)—3264no—
176AGL(1,ℤ/8)—3264no—
177C2×SD16—1632no—
178C2×SD16—1632no—
179C2×D8—1632no—
180C2×D8—1632no—
181C2×M16—1631no—
182C2×M16C2×D16311yescharacteristic, complemented
183#123—3261no—
184C22⋊Q8—1632no—
185#121—1632no—
186#120—3261no—
187C4×D4—3262no—
188#117C2×D16311yescharacteristic, complemented
189#102—1634no—
190#102C2×D16314yescomplemented
191C22⋊C8—1634no—
192C24—21no—
193C24—3268no—
194C24—3264no—
195C24—3261no—
196C24—6528no—
197C24—1631no—
198#41—3262no—
199#41—3262no—
200C2×Q8—1631no—
201C2×D4—22no—
202C2×D4—21no—
203C2×D4—3262no—
204C2×D4—32616no—
205C2×D4—3268no—
206C2×D4—3268no—
207C2×D4—3268no—
208C2×D4—3261no—
209C2×D4—3264no—
210C2×D4—3261no—
211C2×D4—3262no—
212C2×D4—1634no—
213C2×D4—1631no—
214C2×D4—1632no—
215C2×D4D65214yescomplemented
216C2×D4C22×D16312yes—
217C22×C4—3262no—
218C22×C4—3262no—
219C22×C4—3261no—
220C22×C4—1631no—
221C22×C4—1632no—
222C22×C4—1632no—
223C22×C4C22×D16311yescharacteristic
224SD16—3264no—
225SD16—3264no—
226D8—3264no—
227D8—3264no—
228M16—22no—
229M16—3262no—
230C2×C8—1632no—
231C2×C8C22×D16312yescomplemented
232C4⋊C4—3262no—
233C4⋊C4C22×D16312yescomplemented
234C22⋊C4—21no—
235C22⋊C4—3262no—
236C22⋊C4—3262no—
237C42—21no—
238C23—22no—
239C23—24no—
240C23—44no—
241C23—3264no—
242C23—32616no—
243C23—3262no—
244C23—3262no—
245C23—3264no—
246C23—3261no—
247C23—6524no—
248C23—65232no—
249C23—6528no—
250C23—6528no—
251C23—1632no—
252C23D4×D16311yescharacteristic
253Q8—3261no—
254Q8—3261no—
255D4—22no—
256D4—24no—
257D4—32616no—
258D4—3261no—
259D4—3262no—
260D4—3261no—
261D4—3264no—
262D4—6524no—
263D4—6522no—
264D4—1634no—
265D4C2×D65214yes—
266C2×C4—22no—
267C2×C4—3262no—
268C2×C4—3262no—
269C2×C4—3262no—
270C2×C4—3264no—
271C2×C4—3261no—
272C2×C4—6522no—
273C2×C4—1632no—
274C2×C4D4×D16312yes—
275C2×C4D4×D16311yescharacteristic
276C2×C4C2×D65212yes—
277C2×C4C23×D16311yescharacteristic
278C8—22no—
279C8—3262no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—3268no—
285C22—3262no—
286C22—3262no—
287C22—3264no—
288C22—6528no—
289C22—65232no—
290C22—6521no—
291C22—6522no—
292C22—6524no—
293C22—1634no—
294C22#1910470512yes—
295C22C2×D4×D16311yescharacteristic
296C4—42no—
297C4—3262no—
298C4—6521no—
299C4#1910470512yes—
300C4C2×D4×D16311yescharacteristic
301C4C22×D65211yescharacteristic
302C2—22no—
303C2—44no—
304C2—13041no—
305C2—3264no—
306C2—6524no—
307C2C2×#1910470511yescharacteristic
308C2AGL(1,ℤ/8)×D16312yes—
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#418×C163C2
socleC22×C163C22×D4
hypercenter#418D163
hypocenterC163#3003
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×D4×D163C2semidirect
2. C2×#19104763C2semidirect
3. C2×#19104761C2semidirect
4. #23782177C2semidirect
5. #23781817C2semidirect
6. #23781800C2semidirect
7. #23781669C2semidirect
8. #23781628C2semidirect
9. #418×C163C2semidirect
11. C2×#14579699C22semidirect
21. C2×#14579688C22semidirect
22. C2×#14579687C22semidirect
23. C2×D1304C22semidirect
24. C2×#14579684C22semidirect
25. C2×#14579683C22semidirect
27. #19104715C22semidirect
28. #19104710C22semidirect
31. #19104655C22semidirect
32. (C2×D163)⋊C8C22semidirect
34. C2×M16×C163C22semidirect
35. #117×C163C22semidirect
36. #102×C163C22semidirect
44. C2×Dic326C23semidirect
48. C2×D652D4semidirect
50. C2×D652C23semidirect
68. C2×C163⋊C8C23semidirect
75. C2×D4×C163D4semidirect
79. C2×C8×C163C23semidirect
80. C4⋊C4×C163C23semidirect
140. C2×D163AGL(1,ℤ/8)semidirect
154. C163#3003semidirect
163. #418D163semidirect
182. C2×M16C2×D163semidirect
188. #117C2×D163semidirect
190. #102C2×D163semidirect
215. C2×D4D652semidirect
231. C2×C8C22×D163semidirect
233. C4⋊C4C22×D163semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1632
C2×#191047632
C2×#191047612
#237821772
#237818172
#237818002
#237816692
#237816282
#418×C1632
#3003163