Finite Group Databasebeta

← #35223780

Subgroups — #35223780

Order 20864.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14579698C211yesmaximal, characteristic, complemented
2C2×#19104768C211yesmaximal, characteristic, complemented
3C2×#19104761C211yesmaximal, characteristic, complemented
4#23782177C211yesmaximal, characteristic, complemented
5#23781828C214yesmaximal, complemented
6#23781811C214yesmaximal, complemented
7#23781673C211yesmaximal, characteristic, complemented
8#23781628C211yesmaximal, characteristic, complemented
9#418×C163C211yesmaximal, characteristic, complemented
10C2×#14579699C2211yescharacteristic
11C2×#14579698—24no—
12C2×#14579698—22no—
13C2×#14579698—24no—
14C2×#14579698C2214yes—
15C22×Dic326C2211yescharacteristic
16C22×C22⋊D163—21no—
17C23×Dic163—21no—
18C22×C4×D163C2211yescharacteristic
19#19104768—24no—
20#19104761—24no—
21C2×Dic652C2212yescomplemented
22C2×#14579688C2212yescomplemented
23C2×#14579687C2212yescomplemented
24C2×#14579684C2212yescomplemented
25C2×#14579683C2211yescharacteristic
26#19104719—21no—
27#19104715C2212yescomplemented
28#19104710C2212yescomplemented
29#19104705—21no—
30C4×D652—22no—
31#19104658C2214yescomplemented
32(C2×D163)⋊C8C2214yescomplemented
33C22×D4×C163C2211yescharacteristic
34C2×M16×C163C2211yescharacteristic
35#117×C163C2211yescharacteristic
36#102×C163C2214yescomplemented
37C23×D163D411yescharacteristic
38#14579699—22no—
39#14579699—22no—
40#14579698—216no—
41#14579698—28no—
42#14579698—28no—
43C2×Dic326—21no—
44C2×Dic326—22no—
45C2×Dic326D414yescomplemented
46C2×Dic326C2311yescharacteristic
47C2×Dic326C2312yes—
48C2×D652—22no—
49C2×D652C2311yescharacteristic
50C2×C22⋊D163—28no—
51C2×C22⋊D163—24no—
52C2×C22⋊D163—21no—
53C22×Dic163—28no—
54C22×Dic163—24no—
55C22×Dic163—21no—
56C22×Dic163D411yescharacteristic
57C2×C4×D163—22no—
58C2×C4×D163—22no—
59C2×C4×D163—21no—
60C2×C4×D163D412yes—
61C2×C4×D163C2312yes—
62Dic652—24no—
63#14579688—24no—
64#14579687—24no—
65#14579684—24no—
66#14579683—22no—
67C2×C163⋊C8C2312yes—
68C4⋊Dic163—22no—
69(C2×D163)⋊C4—22no—
70(C2×D163)⋊C4—22no—
71C24×C163—21no—
72C2×D4×C163—21no—
73C2×D4×C163—22no—
74C2×D4×C163D414yescomplemented
75C2×D4×C163C2312yes—
76C22×C4×C163C2311yescharacteristic
77M16×C163—22no—
78C2×C8×C163C2312yes—
79C4⋊C4×C163C2312yes—
80C22⋊C4×C163—21no—
81C42×C163—21no—
82C22×D163—22no—
83C22×D163—21no—
84C22×D163—42no—
85C22×D163C2×D412yes—
86Dic326—24no—
87Dic326—22no—
88Dic326—21no—
89Dic326—21no—
90Dic326C2×D414yes—
91D652—21no—
92D652—21no—
93D652—44no—
94C22⋊D163—216no—
95C22⋊D163—42no—
96C2×Dic163—216no—
97C2×Dic163—22no—
98C2×Dic163—24no—
99C2×Dic163—22no—
100C2×Dic163—21no—
101C2×Dic163C2×D412yes—
102C4×D163—24no—
103C4×D163—22no—
104C4×D163—22no—
105C4×D163—42no—
106C163⋊C8—22no—
107C23×C163—24no—
108C23×C163—22no—
109C23×C163—44no—
110C23×C163C2×D411yescharacteristic
111D4×C163—24no—
112D4×C163—22no—
113D4×C163C2×D414yes—
114C2×C4×C163—22no—
115C2×C4×C163C2×D411yescharacteristic
116C2×C4×C163C2×D412yes—
117C2×C4×C163C2×D412yes—
118C2×C4×C163C2411yescharacteristic
119C8×C163—22no—
120C22×C163—22no—
121C22×C163—24no—
122C22×C163—48no—
123C22×C163—44no—
124C22×C163#12012yes—
125C22×C163C22×D411yescharacteristic
126C2×D163—22no—
127C2×D163—22no—
128C2×D163—22no—
129C2×D163—44no—
130C2×D163—41no—
131C2×D163—42no—
132C4×C163—42no—
133C4×C163#12012yes—
134C4×C163C22×D411yescharacteristic
135C4×C163C22×D411yescharacteristic
136Dic163—22no—
137Dic163—24no—
138Dic163—41no—
139C2×C163—22no—
140C2×C163—44no—
141C2×C163C2×#12011yescharacteristic
142C2×C163#57512yes—
143D163—42no—
144D163—81no—
145C163#300511yescharacteristic, complemented, Sylow
146#3005—1631nomaximal, Sylow
147C22×#41—1631no—
148C2×#137—1631no—
149C2×AGL(1,ℤ/8)—1631no—
150#525—1631no—
151#452—1634no—
152#449—1634no—
153#419—1631no—
154#418D16311yescharacteristic, complemented
155#407—1631no—
156C2×#41—3264no—
157C2×#41—3264no—
158C2×#41—3262no—
159C2×#41—1634no—
160C2×#41—1631no—
161C22×Q8—1631no—
162C22×D4—3261no—
163C22×D4C2×D16311yescharacteristic
164C23×C4—3261no—
165C23×C4—1631no—
166#137—3264no—
167AGL(1,ℤ/8)—3264no—
168C2×Q16—1632no—
169C2×SD16—1632no—
170C2×SD16—1632no—
171C2×D8—1632no—
172C2×M16—1631no—
173C2×M16C2×D16311yescharacteristic
174#123—3261no—
175C22⋊Q8—1632no—
176#121—1632no—
177#120—3261no—
178C4×D4—3262no—
179#117C2×D16311yescharacteristic
180#103—1634no—
181#102C2×D16314yescomplemented
182C22⋊C8—1634no—
183C24—21no—
184C24—1631no—
185#41—32616no—
186#41—3268no—
187#41—3268no—
188#41—3262no—
189#41—3262no—
190C2×Q8—3261no—
191C2×Q8—3262no—
192C2×Q8—1634no—
193C2×Q8—1631no—
194C2×Q8—1632no—
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—3262no—
198C2×D4—3268no—
199C2×D4—3264no—
200C2×D4—3261no—
201C2×D4—1631no—
202C2×D4D65214yescomplemented
203C2×D4C22×D16312yes—
204C22×C4—3268no—
205C22×C4—3261no—
206C22×C4—3264no—
207C22×C4—3262no—
208C22×C4—3262no—
209C22×C4—3261no—
210C22×C4—1631no—
211C22×C4—1632no—
212C22×C4—1632no—
213C22×C4C22×D16311yescharacteristic
214Q16—3264no—
215SD16—3264no—
216SD16—3264no—
217D8—3264no—
218M16—22no—
219M16—3262no—
220C2×C8—1632no—
221C2×C8C22×D16312yes—
222C4⋊C4—3262no—
223C4⋊C4C22×D16312yes—
224C22⋊C4—21no—
225C22⋊C4—3262no—
226C22⋊C4—3262no—
227C42—21no—
228C23—22no—
229C23—24no—
230C23—44no—
231C23—3262no—
232C23—3261no—
233C23—6522no—
234C23—1632no—
235C23D4×D16311yescharacteristic
236Q8—3262no—
237Q8—3264no—
238Q8—3261no—
239Q8—3261no—
240Q8—1634no—
241D4—22no—
242D4—24no—
243D4—32616no—
244D4—3261no—
245D4—3261no—
246D4—6524no—
247D4—6522no—
248D4C2×D65214yes—
249C2×C4—22no—
250C2×C4—32616no—
251C2×C4—3262no—
252C2×C4—3262no—
253C2×C4—3264no—
254C2×C4—3262no—
255C2×C4—3262no—
256C2×C4—3264no—
257C2×C4—3261no—
258C2×C4—6522no—
259C2×C4—1632no—
260C2×C4D4×D16312yes—
261C2×C4D4×D16311yescharacteristic
262C2×C4C2×D65212yes—
263C2×C4C23×D16311yescharacteristic
264C8—22no—
265C8—3262no—
266C22—22no—
267C22—24no—
268C22—48no—
269C22—44no—
270C22—3262no—
271C22—3262no—
272C22—3262no—
273C22—6524no—
274C22—6521no—
275C22—6522no—
276C22#1910470512yes—
277C22C2×D4×D16311yescharacteristic
278C4—42no—
279C4—3262no—
280C4—3264no—
281C4—6521no—
282C4#1910470512yes—
283C4C2×D4×D16311yescharacteristic
284C4C22×D65211yescharacteristic
285C2—22no—
286C2—44no—
287C2—13041no—
288C2—6522no—
289C2C2×#1910470511yescharacteristic
290C2#2378246112yes—
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#418×C163C2
socleC22×C163C22×D4
hypercenter#418D163
hypocenterC163#3005
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×#14579698C2semidirect
2. C2×#19104768C2semidirect
3. C2×#19104761C2semidirect
4. #23782177C2semidirect
5. #23781828C2semidirect
6. #23781811C2semidirect
7. #23781673C2semidirect
8. #23781628C2semidirect
9. #418×C163C2semidirect
21. C2×Dic652C22semidirect
22. C2×#14579688C22semidirect
23. C2×#14579687C22semidirect
24. C2×#14579684C22semidirect
27. #19104715C22semidirect
28. #19104710C22semidirect
31. #19104658C22semidirect
32. (C2×D163)⋊C8C22semidirect
36. #102×C163C22semidirect
45. C2×Dic326D4semidirect
74. C2×D4×C163D4semidirect
145. C163#3005semidirect
154. #418D163semidirect
181. #102C2×D163semidirect
202. C2×D4D652semidirect

Maximal subgroups

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

subgroupindex
C22×#145796982
C2×#191047682
C2×#191047612
#237821772
#237818282
#237818112
#237816732
#237816282
#418×C1632
#3005163