Finite Group Databasebeta

← #35224156

Subgroups — #35224156

Order 20864.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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×#19104763C211yesmaximal, characteristic, complemented
2C2×M16×D163C211yesmaximal, characteristic, complemented
3#23782177C211yesmaximal, characteristic, complemented
4C2×#19104710C211yesmaximal, characteristic, complemented
5#23781919C212yesmaximal, complemented
6#23781912C212yesmaximal, complemented
7#23781902C212yesmaximal, complemented
8#23781895C212yesmaximal, complemented
9#23781669C211yesmaximal, characteristic, complemented
10#23781664C211yesmaximal, characteristic, complemented
11#428×C163C211yesmaximal, characteristic, complemented
12C2×#14579699C2211yescharacteristic
13C22×D652—21no—
14C22×D652C2211yescharacteristic
15C22×C4×D163C2211yescharacteristic
16#19104763—24no—
17C2×#14579688C2212yescomplemented
18C2×D1304C2212yescomplemented
19C2×#14579683C2211yescharacteristic
20C2×#14579682C2212yescomplemented
21M16×D163—24no—
22C2×C8×D163C2212yescomplemented
23#19104719—21no—
24#19104715C2211yescharacteristic, complemented
25#19104715C2211yescharacteristic, complemented
26#19104710—21no—
27#19104710—21no—
28#19104710C2212yes—
29#19104710C2212yes—
30#19104710C2211yescharacteristic, complemented
31#19104710C2211yescharacteristic, complemented
32#19104705—21no—
33C4×D652—21no—
34C4×D652—21no—
35C2×(C2×D163)⋊C4—21no—
36#19104658C2212yescomplemented
37#19104655C2212yescomplemented
38#19104655C2212yescomplemented
39#19104655C2212yescomplemented
40C2×M16×C163C2211yescharacteristic
41#117×C163C2211yescharacteristic
42C2×C4⋊C4×C163C2211yescharacteristic
43#107×C163C2212yescomplemented
44#106×C163C2212yescomplemented
45C23×D163—21no—
46C23×D163D411yescharacteristic
47#14579699—22no—
48#14579699—22no—
49C2×Dic326C2311yescharacteristic
50C2×D652—22no—
51C2×D652—22no—
52C2×D652—21no—
53C2×D652—21no—
54C2×D652—22no—
55C2×D652—21no—
56C2×D652—22no—
57C2×D652—44no—
58C2×D652C2312yes—
59C2×D652C2311yescharacteristic
60C2×C22⋊D163—21no—
61C22×Dic163D411yescharacteristic
62C2×C4×D163—22no—
63C2×C4×D163—21no—
64C2×C4×D163—22no—
65C2×C4×D163D411yescharacteristic
66C2×C4×D163D411yescharacteristic
67C2×C4×D163C2311yescharacteristic
68C2×C4×D163C2311yescharacteristic
69#14579688—24no—
70D1304—24no—
71#14579683—22no—
72#14579682—24no—
73C2×C163⋊C8C2312yes—
74C8×D163—24no—
75C4⋊Dic163—21no—
76C4⋊Dic163—21no—
77(C2×D163)⋊C4—21no—
78(C2×D163)⋊C4—21no—
79(C2×D163)⋊C4—22no—
80(C2×D163)⋊C4—22no—
81(C2×D163)⋊C4—21no—
82(C2×D163)⋊C4—21no—
83C22×C4×C163—21no—
84C22×C4×C163C2311yescharacteristic
85M16×C163D414yescomplemented
86C2×C8×C163C2312yes—
87C4⋊C4×C163—21no—
88C4⋊C4×C163C2311yescharacteristic
89C4⋊C4×C163C2312yes—
90C4⋊C4×C163C2311yescharacteristic
91C22⋊C4×C163—21no—
92C42×C163—21no—
93C22×D163—22no—
94C22×D163—22no—
95C22×D163—21no—
96C22×D163—22no—
97C22×D163—41no—
98C22×D163—42no—
99C22×D163—42no—
100C22×D163—42no—
101C22×D163C2×D411yescharacteristic
102C22×D163C2×D411yescharacteristic
103Dic326—21no—
104Dic326—21no—
105D652—21no—
106D652—22no—
107D652—22no—
108D652—21no—
109D652—42no—
110D652—44no—
111D652—42no—
112D652—44no—
113D652—42no—
114C22⋊D163—42no—
115C2×Dic163—22no—
116C2×Dic163—21no—
117C2×Dic163C2×D411yescharacteristic
118C2×Dic163C2×D411yescharacteristic
119C4×D163—22no—
120C4×D163—22no—
121C4×D163—21no—
122C4×D163—21no—
123C4×D163—21no—
124C4×D163—21no—
125C4×D163—42no—
126C163⋊C8—22no—
127C23×C163C2×D411yescharacteristic
128C2×C4×C163—21no—
129C2×C4×C163—21no—
130C2×C4×C163—22no—
131C2×C4×C163—42no—
132C2×C4×C163C2×D411yescharacteristic
133C2×C4×C163C2×D412yes—
134C2×C4×C163#4112yes—
135C2×C4×C163C2411yescharacteristic
136C8×C163C2×D414yescomplemented
137C22×C163—21no—
138C22×C163—21no—
139C22×C163#12112yes—
140C22×C163C22×D411yescharacteristic
141C2×D163—21no—
142C2×D163—21no—
143C2×D163—21no—
144C2×D163—21no—
145C2×D163—21no—
146C2×D163—44no—
147C2×D163—41no—
148C2×D163—42no—
149C2×D163—42no—
150C2×D163—41no—
151C2×D163—41no—
152C2×D163—42no—
153C2×D163—84no—
154C2×D163AGL(1,ℤ/8)12yescomplemented
155C4×C163—42no—
156C4×C163—41no—
157C4×C163—41no—
158C4×C163#12112yes—
159C4×C163C22×D411yescharacteristic
160C4×C163C2×#4111yescharacteristic
161Dic163—21no—
162Dic163—21no—
163Dic163—41no—
164C2×C163—22no—
165C2×C163C2×#12111yescharacteristic
166C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
167C2×C163#57511yescharacteristic
168D163—22no—
169D163—41no—
170D163—81no—
171D163—82no—
172C163#305511yescharacteristic, complemented, Sylow
173#3055—1631nomaximal, Sylow
174C2×AGL(1,ℤ/8)—1631no—
175C22×M16—1631no—
176#525—1631no—
177C2×#121—1631no—
178#469—1632no—
179#468—1632no—
180#466—1632no—
181#465—1632no—
182#428D16311yescharacteristic, complemented
183#418—1631no—
184#417—1631no—
185C2×#41—1631no—
186C22×D4—3261no—
187C22×D4—1631no—
188C23×C4—1631no—
189AGL(1,ℤ/8)—3264no—
190C2×SD16—1632no—
191C2×D8—1632no—
192C2×M16—3264no—
193C2×M16—1631no—
194C2×M16—1632no—
195C2×M16C2×D16311yescharacteristic
196C22×C8—1632no—
197#123—3261no—
198C22⋊Q8—1631no—
199C22⋊Q8—1631no—
200#121—3261no—
201#121—3261no—
202#121—1632no—
203#121—1632no—
204#121—1631no—
205#121—1631no—
206#120—3261no—
207C4×D4—3261no—
208C4×D4—3261no—
209#117C2×D16311yescharacteristic
210C2×C4⋊C4C2×D16311yescharacteristic
211C2×C22⋊C4—3261no—
212#107C2×D16312yescomplemented
213#106C2×D16312yescomplemented
214#103—1632no—
215#102—1632no—
216#102—1632no—
217#102—1632no—
218C24—3261no—
219C24—1631no—
220#41—3262no—
221#41—3262no—
222C2×Q8—1631no—
223C2×D4—3261no—
224C2×D4—3261no—
225C2×D4—3262no—
226C2×D4—3262no—
227C2×D4—3261no—
228C2×D4—3261no—
229C2×D4—3262no—
230C2×D4—3262no—
231C2×D4—6524no—
232C2×D4—1631no—
233C2×D4—1632no—
234C22×C4—21no—
235C22×C4—3262no—
236C22×C4—3262no—
237C22×C4—3261no—
238C22×C4—1631no—
239C22×C4—1631no—
240C22×C4—1631no—
241C22×C4—1631no—
242C22×C4—1631no—
243C22×C4C22×D16311yescharacteristic
244SD16—3264no—
245D8—3264no—
246M16—3262no—
247M16—3264no—
248M16D65214yescomplemented
249C2×C8—3264no—
250C2×C8—1632no—
251C2×C8C22×D16312yes—
252C4⋊C4—21no—
253C4⋊C4—3261no—
254C4⋊C4—3261no—
255C4⋊C4C22×D16311yescharacteristic
256C4⋊C4C22×D16311yescharacteristic
257C4⋊C4C22×D16312yes—
258C22⋊C4—21no—
259C22⋊C4—3261no—
260C22⋊C4—3261no—
261C22⋊C4—3262no—
262C22⋊C4—3262no—
263C22⋊C4—3261no—
264C22⋊C4—3261no—
265C42—21no—
266C23—3262no—
267C23—3262no—
268C23—3261no—
269C23—3262no—
270C23—6522no—
271C23—6522no—
272C23—6521no—
273C23—6522no—
274C23—1631no—
275C23—1631no—
276C23D4×D16311yescharacteristic
277Q8—3261no—
278Q8—3261no—
279D4—3261no—
280D4—3262no—
281D4—3262no—
282D4—3261no—
283D4—6524no—
284D4—6522no—
285D4—6524no—
286D4—6522no—
287D4—6522no—
288D4—6522no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—3262no—
294C2×C4—3261no—
295C2×C4—3261no—
296C2×C4—3261no—
297C2×C4—3261no—
298C2×C4—3262no—
299C2×C4—3262no—
300C2×C4—3261no—
301C2×C4—6522no—
302C2×C4—1631no—
303C2×C4—1631no—
304C2×C4D4×D16311yescharacteristic
305C2×C4C2×D65212yes—
306C2×C4#1457970012yes—
307C2×C4C23×D16311yescharacteristic
308C8—3262no—
309C8C2×D65214yescomplemented
310C22—21no—
311C22—21no—
312C22—13044no—
313C22—3261no—
314C22—3261no—
315C22—3261no—
316C22—3261no—
317C22—3261no—
318C22—6524no—
319C22—6521no—
320C22—6522no—
321C22—6522no—
322C22—6521no—
323C22—6521no—
324C22—6522no—
325C22—1632no—
326C22#1910471012yes—
327C22C2×D4×D16311yescharacteristic
328C4—42no—
329C4—41no—
330C4—41no—
331C4—3261no—
332C4—3261no—
333C4—6521no—
334C4#1910471012yes—
335C4C22×D65211yescharacteristic
336C4C2×#1457970011yescharacteristic
337C2—22no—
338C2—13042no—
339C2—13041no—
340C2—3262no—
341C2—6521no—
342C2C2×#1910471011yescharacteristic
343C2AGL(1,ℤ/8)×D16311yescharacteristic
344C2#2378246411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#428×C163C2
socleC22×C163C22×D4
hypercenter#428D163
hypocenterC163#3055
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#19104763C2semidirect
2. C2×M16×D163C2semidirect
3. #23782177C2semidirect
4. C2×#19104710C2semidirect
5. #23781919C2semidirect
6. #23781912C2semidirect
7. #23781902C2semidirect
8. #23781895C2semidirect
9. #23781669C2semidirect
10. #23781664C2semidirect
11. #428×C163C2semidirect
17. C2×#14579688C22semidirect
18. C2×D1304C22semidirect
20. C2×#14579682C22semidirect
22. C2×C8×D163C22semidirect
24. #19104715C22semidirect
25. #19104715C22semidirect
30. #19104710C22semidirect
31. #19104710C22semidirect
36. #19104658C22semidirect
37. #19104655C22semidirect
38. #19104655C22semidirect
39. #19104655C22semidirect
43. #107×C163C22semidirect
44. #106×C163C22semidirect
85. M16×C163D4semidirect
136. C8×C163C2×D4semidirect
154. C2×D163AGL(1,ℤ/8)semidirect
172. C163#3055semidirect
182. #428D163semidirect
212. #107C2×D163semidirect
213. #106C2×D163semidirect
248. M16D652semidirect
309. C8C2×D652semidirect

Maximal subgroups

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

subgroupindex
C2×#191047632
C2×M16×D1632
#237821772
C2×#191047102
#237819192
#237819122
#237819022
#237818952
#237816692
#237816642
#428×C1632
#3055163