Finite Group Databasebeta

← #35224011

Subgroups — #35224011

Order 20864.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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×#19104759C212yesmaximal, complemented
2#127×D163C211yesmaximal, characteristic, complemented
3C4×#14579698C211yesmaximal, characteristic, complemented
4#23781866C211yesmaximal, characteristic, complemented
5#23781864C212yesmaximal, complemented
6#23781862C211yesmaximal, characteristic, complemented
7#23781860C211yesmaximal, characteristic, complemented
8#23781858C212yesmaximal, complemented
9#23781687C211yesmaximal, characteristic, complemented
10#23781668C212yesmaximal, complemented
11#459×C163C211yesmaximal, characteristic, complemented
12C2×#14579698C2211yescharacteristic
13C2×D4×D163—21no—
14C2×D4×D163—21no—
15C2×D4×D163C2212yes—
16#19104759—28no—
17C2×#14579688C2212yescomplemented
18C2×#14579687C2212yescomplemented
19C2×#14579684C2212yescomplemented
20C2×#14579682C2212yes—
21#19104732C2211yescharacteristic
22#19104731—21no—
23#19104731—21no—
24#19104731C2212yes—
25C4×Dic326C2211yescharacteristic, complemented
26#19104700C2211yescharacteristic, complemented
27D4×Dic163C2211yescharacteristic, complemented
28C4×C22⋊D163—21no—
29(C22⋊D163)⋊C4—21no—
30#19104693C2211yescharacteristic
31#19104691—21no—
32C2×C4×Dic163—21no—
33C42×D163C2211yescharacteristic
34C4⋊(C163⋊C8)C2211yescharacteristic, complemented
35Dic163⋊C8C2212yescomplemented
36#19104657C2212yescomplemented
37#19104656C2212yescomplemented
38#19104655C2212yescomplemented
39C2×D8×C163C2212yescomplemented
40#127×C163C2211yescharacteristic
41C4×D4×C163C2211yescharacteristic, complemented
42C4⋊C8×C163C2211yescharacteristic, complemented
43#102×C163C2212yescomplemented
44C23×D163—22no—
45#14579698—22no—
46#14579698D414yescomplemented
47C2×Dic326C2311yescharacteristic
48C2×D652—21no—
49C2×D652—21no—
50C2×D652C2312yes—
51C2×C22⋊D163—21no—
52C2×C22⋊D163—22no—
53C2×C22⋊D163—22no—
54C2×C22⋊D163—22no—
55D4×D163—24no—
56D4×D163—24no—
57D4×D163—44no—
58D4×D163—44no—
59C22×Dic163—21no—
60C22×Dic163—21no—
61C2×C4×D163D411yescharacteristic
62C2×C4×D163D411yescharacteristic
63C2×C4×D163C2311yescharacteristic
64#14579688—24no—
65#14579687—24no—
66#14579684—24no—
67#14579682—24no—
68C2×C163⋊C8C2312yes—
69C4⋊Dic163C2311yescharacteristic
70Dic163⋊C4—21no—
71Dic163⋊C4—21no—
72C22⋊Dic163—21no—
73(C2×D163)⋊C4—21no—
74C4×Dic163—21no—
75C4×Dic163—21no—
76C4×Dic163—21no—
77C4×Dic163D411yescharacteristic
78C4×Dic163D411yescharacteristic
79C4×Dic163C2311yescharacteristic
80C2×D4×C163—21no—
81C2×D4×C163—21no—
82C2×D4×C163C2312yes—
83C2×D4×C163C2311yescharacteristic
84C22×C4×C163—21no—
85D8×C163—24no—
86C2×C8×C163C2312yes—
87C4⋊C4×C163C2311yescharacteristic
88C22⋊C4×C163—21no—
89C42×C163C2311yescharacteristic
90C22×D163—22no—
91C22×D163—44no—
92C22×D163—44no—
93C22×D163—44no—
94C22×D163C2×D411yescharacteristic
95Dic326—21no—
96Dic326C2×D412yescomplemented
97D652—22no—
98D652—22no—
99D652—42no—
100D652—42no—
101C22⋊D163—24no—
102C22⋊D163—44no—
103C22⋊D163—44no—
104C22⋊D163—44no—
105C2×Dic163—21no—
106C2×Dic163—24no—
107C2×Dic163—21no—
108C2×Dic163—42no—
109C2×Dic163C2×D411yescharacteristic
110C2×Dic163C2×D411yescharacteristic
111C2×Dic163C2×D411yescharacteristic
112C4×D163—22no—
113C4×D163—22no—
114C4×D163C2×D412yes—
115C4×D163#4112yes—
116C163⋊C8—22no—
117C23×C163—21no—
118C23×C163—22no—
119D4×C163—22no—
120D4×C163—21no—
121D4×C163—22no—
122D4×C163—42no—
123D4×C163—42no—
124D4×C163C2×D412yescomplemented
125C2×C4×C163—21no—
126C2×C4×C163—42no—
127C2×C4×C163C2×D411yescharacteristic
128C2×C4×C163C2×D411yescharacteristic
129C2×C4×C163C2411yescharacteristic
130C8×C163—22no—
131C22×C163—22no—
132C22×C163—41no—
133C22×C163—41no—
134C22×C163—42no—
135C22×C163—42no—
136C22×C163—42no—
137C22×C163C22×D411yescharacteristic
138C2×D163—21no—
139C2×D163—21no—
140C2×D163—42no—
141C2×D163—42no—
142C2×D163—42no—
143C2×D163—88no—
144C2×D163#12112yes—
145C4×C163—21no—
146C4×C163—41no—
147C4×C163AGL(1,ℤ/8)12yescomplemented
148C4×C163C22×D411yescharacteristic
149C4×C163C2×#4111yescharacteristic
150Dic163—22no—
151Dic163—21no—
152Dic163—41no—
153Dic163#12112yes—
154Dic163AGL(1,ℤ/8)12yescomplemented
155C2×C163—42no—
156C2×C163—82no—
157C2×C163C2×#12111yescharacteristic
158C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
160D163—22no—
161D163—82no—
162C163#303811yescharacteristic, complemented, Sylow
163#3038—1631nomaximal, Sylow
164C2×AGL(1,ℤ/8)—1632no—
165C2×#127—1631no—
166C4×#41—1631no—
167#460—1631no—
168#460—1631no—
169#460—1632no—
170#459—1632no—
171#459—1631no—
172#459D16311yescharacteristic, complemented
173#423—1631no—
174#418—1632no—
175C2×#41—1631no—
176C22×D4—3261no—
177C22×D4—3261no—
178C22×D4—1632no—
179AGL(1,ℤ/8)—3268no—
180C2×SD16—1632no—
181C2×SD16—1632no—
182C2×D8—1632no—
183C2×D8C2×D16312yescomplemented
184C2×M16—1632no—
185#127—3261no—
186#127—3261no—
187#127—1632no—
188#127—1631no—
189#127C2×D16311yescharacteristic
190C4×Q8—1631no—
191C4×Q8—1631no—
192C4×D4—3261no—
193C4×D4—3261no—
194C4×D4—1631no—
195C4×D4C2×D16311yescharacteristic, complemented
196#117—3261no—
197#117—1631no—
198C2×C42—3261no—
199C2×C42—1631no—
200C4⋊C8—1632no—
201C4⋊C8—1631no—
202C4⋊C8C2×D16311yescharacteristic, complemented
203#102—1632no—
204#102—1632no—
205#102—1632no—
206#102C2×D16312yescomplemented
207C24—3262no—
208#41—3262no—
209#41—1634no—
210C2×Q8—1631no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—3261no—
214C2×D4—3264no—
215C2×D4—3264no—
216C2×D4—3262no—
217C2×D4—3262no—
218C2×D4—3262no—
219C2×D4—3261no—
220C2×D4—3261no—
221C2×D4—6524no—
222C2×D4—6524no—
223C2×D4—1632no—
224C2×D4C22×D16312yes—
225C2×D4C22×D16311yescharacteristic
226C22×C4—21no—
227C22×C4—3261no—
228C22×C4—3261no—
229C22×C4—1631no—
230C22×C4—1631no—
231C22×C4—1631no—
232SD16—3264no—
233SD16—3264no—
234D8—24no—
235D8—3264no—
236M16—3264no—
237C2×C8—1632no—
238C2×C8C22×D16312yes—
239C4⋊C4—3261no—
240C4⋊C4—3261no—
241C4⋊C4—1631no—
242C4⋊C4C22×D16311yescharacteristic
243C22⋊C4—21no—
244C22⋊C4—3261no—
245C22⋊C4—3261no—
246C42—3261no—
247C42—3261no—
248C42—3261no—
249C42—1631no—
250C42—1631no—
251C42—1631no—
252C42C22×D16311yescharacteristic
253C23—21no—
254C23—22no—
255C23—3262no—
256C23—6524no—
257C23—6524no—
258C23—6524no—
259C23—1631no—
260Q8—3261no—
261Q8—1632no—
262D4—22no—
263D4—21no—
264D4—22no—
265D4—42no—
266D4—42no—
267D4—3264no—
268D4—3262no—
269D4—3262no—
270D4—6522no—
271D4—6524no—
272D4—6524no—
273D4—6524no—
274D4—6522no—
275D4D4×D16312yescomplemented
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—3262no—
279C2×C4—3261no—
280C2×C4—3264no—
281C2×C4—3262no—
282C2×C4—3261no—
283C2×C4—6522no—
284C2×C4—1631no—
285C2×C4—1631no—
286C2×C4—1631no—
287C2×C4—1632no—
288C2×C4—1632no—
289C2×C4D4×D16311yescharacteristic
290C2×C4D4×D16311yescharacteristic
291C2×C4C23×D16311yescharacteristic
292C8—22no—
293C8—3262no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—13048no—
301C22—3261no—
302C22—3261no—
303C22—6522no—
304C22—6522no—
305C22—6522no—
306C22—1632no—
307C22C2×D4×D16311yescharacteristic
308C4—21no—
309C4—41no—
310C4—3262no—
311C4—3261no—
312C4—6521no—
313C4—1632no—
314C4—1632no—
315C4#1910475912yescomplemented
316C4C2×D4×D16311yescharacteristic
317C4#41×D16311yescharacteristic
318C2—42no—
319C2—82no—
320C2—13042no—
321C2—3262no—
322C2#121×D16311yescharacteristic
323C2AGL(1,ℤ/8)×D16311yescharacteristic
324C2C2×#1910475911yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#459×C163C2
socleC22×C163C22×D4
hypercenter#459D163
hypocenterC163#3038
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#19104759C2semidirect
2. #127×D163C2semidirect
3. C4×#14579698C2semidirect
4. #23781866C2semidirect
5. #23781864C2semidirect
6. #23781862C2semidirect
7. #23781860C2semidirect
8. #23781858C2semidirect
9. #23781687C2semidirect
10. #23781668C2semidirect
11. #459×C163C2semidirect
17. C2×#14579688C22semidirect
18. C2×#14579687C22semidirect
19. C2×#14579684C22semidirect
25. C4×Dic326C22semidirect
26. #19104700C22semidirect
27. D4×Dic163C22semidirect
34. C4⋊(C163⋊C8)C22semidirect
35. Dic163⋊C8C22semidirect
36. #19104657C22semidirect
37. #19104656C22semidirect
38. #19104655C22semidirect
39. C2×D8×C163C22semidirect
41. C4×D4×C163C22semidirect
42. C4⋊C8×C163C22semidirect
43. #102×C163C22semidirect
46. #14579698D4semidirect
96. Dic326C2×D4semidirect
124. D4×C163C2×D4semidirect
147. C4×C163AGL(1,ℤ/8)semidirect
154. Dic163AGL(1,ℤ/8)semidirect
162. C163#3038semidirect
172. #459D163semidirect
183. C2×D8C2×D163semidirect
195. C4×D4C2×D163semidirect
202. C4⋊C8C2×D163semidirect
206. #102C2×D163semidirect
275. D4D4×D163semidirect
315. C4#19104759semidirect

Maximal subgroups

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

subgroupindex
C2×#191047592
#127×D1632
C4×#145796982
#237818662
#237818642
#237818622
#237818602
#237818582
#237816872
#237816682
#459×C1632
#3038163