Finite Group Databasebeta

← #35224149

Subgroups — #35224149

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×M16×D163C211yesmaximal, characteristic, complemented
2#23782172C211yesmaximal, characteristic, complemented
3C2×#19104706C211yesmaximal, characteristic, complemented
4#23781915C212yesmaximal, complemented
5#23781908C212yesmaximal, complemented
6#23781898C212yesmaximal, complemented
7#23781891C212yesmaximal, complemented
8#23781707C211yesmaximal, characteristic, complemented
9#23781670C211yesmaximal, characteristic, complemented
10#23781666C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C163C211yesmaximal, characteristic, complemented
12C22×C22⋊D163—21no—
13C22×C4×D163C2211yescharacteristic
14C2×#14579683C2211yescharacteristic
15C2×#14579682C2212yescomplemented
16M16×D163—24no—
17C2×C8×D163C2212yescomplemented
18#19104716—21no—
19#19104711C2211yescharacteristic, complemented
20#19104711C2211yescharacteristic, complemented
21#19104706—21no—
22#19104706—21no—
23#19104706C2212yes—
24#19104706C2212yes—
25#19104706C2211yescharacteristic, complemented
26#19104706C2211yescharacteristic, complemented
27#19104703—21no—
28C4×C22⋊D163—21no—
29C4×C22⋊D163—21no—
30#19104694C2211yescharacteristic
31C2×C4⋊Dic163C2211yescharacteristic
32C2×C22⋊Dic163—21no—
33#19104669C2212yescomplemented
34#19104667C2212yescomplemented
35#19104659C2212yescomplemented
36#19104656C2212yescomplemented
37#19104656C2212yescomplemented
38#19104656C2212yescomplemented
39C2×#41×C163C2211yescharacteristic
40C22×D4×C163C2211yescharacteristic
41AGL(1,ℤ/8)×C163—24no—
42C2×SD16×C163C2212yescomplemented
43C2×D8×C163C2212yescomplemented
44C2×M16×C163C2211yescharacteristic
45C23×D163D411yescharacteristic
46C2×C22⋊D163—22no—
47C2×C22⋊D163—22no—
48C2×C22⋊D163—21no—
49C2×C22⋊D163—21no—
50C2×C22⋊D163—44no—
51C22×Dic163—21no—
52C22×Dic163D411yescharacteristic
53C2×C4×D163—22no—
54C2×C4×D163—22no—
55C2×C4×D163D411yescharacteristic
56C2×C4×D163D411yescharacteristic
57C2×C4×D163C2311yescharacteristic
58C2×C4×D163C2311yescharacteristic
59#14579683—22no—
60#14579682—24no—
61C2×C163⋊C8C2312yes—
62C8×D163—24no—
63C4⋊Dic163—21no—
64C4⋊Dic163C2311yescharacteristic
65C4⋊Dic163C2311yescharacteristic
66C4⋊Dic163C2312yes—
67Dic163⋊C4—21no—
68Dic163⋊C4—21no—
69C22⋊Dic163—22no—
70C22⋊Dic163—22no—
71C22⋊Dic163—21no—
72C22⋊Dic163—21no—
73C22⋊Dic163—21no—
74(C2×D163)⋊C4—21no—
75(C2×D163)⋊C4—21no—
76C4×Dic163—21no—
77C24×C163—21no—
78#41×C163—22no—
79#41×C163—22no—
80C2×Q8×C163C2311yescharacteristic
81C2×D4×C163—21no—
82C2×D4×C163—22no—
83C2×D4×C163—21no—
84C2×D4×C163—22no—
85C2×D4×C163C2311yescharacteristic
86C2×D4×C163C2312yes—
87C22×C4×C163—21no—
88C22×C4×C163C2311yescharacteristic
89SD16×C163—24no—
90D8×C163—24no—
91M16×C163D414yescomplemented
92C2×C8×C163C2312yes—
93C22×D163—22no—
94C22×D163—22no—
95C22×D163—41no—
96C22×D163C2×D411yescharacteristic
97C22×D163C2×D411yescharacteristic
98C22⋊D163—42no—
99C22⋊D163—44no—
100C22⋊D163—42no—
101C22⋊D163—44no—
102C2×Dic163—22no—
103C2×Dic163—21no—
104C2×Dic163—21no—
105C2×Dic163—22no—
106C2×Dic163—42no—
107C2×Dic163C2×D411yescharacteristic
108C2×Dic163C2×D411yescharacteristic
109C4×D163—22no—
110C4×D163—22no—
111C4×D163—21no—
112C4×D163—21no—
113C4×D163—21no—
114C4×D163—21no—
115C163⋊C8—22no—
116C23×C163—21no—
117C23×C163—22no—
118C23×C163—42no—
119C23×C163—42no—
120C23×C163—42no—
121C23×C163C2×D411yescharacteristic
122Q8×C163—21no—
123Q8×C163—21no—
124D4×C163—21no—
125D4×C163—22no—
126D4×C163—22no—
127D4×C163—21no—
128D4×C163—42no—
129D4×C163—42no—
130C2×C4×C163—21no—
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—41no—
140C22×C163—42no—
141C22×C163—42no—
142C22×C163—41no—
143C22×C163—42no—
144C22×C163—41no—
145C22×C163—84no—
146C22×C163#12112yes—
147C22×C163C22×D411yescharacteristic
148C2×D163—21no—
149C2×D163—21no—
150C2×D163—21no—
151C2×D163—21no—
152C2×D163—21no—
153C2×D163—44no—
154C2×D163AGL(1,ℤ/8)12yescomplemented
155C4×C163—41no—
156C4×C163#12112yes—
157C4×C163C22×D411yescharacteristic
158C4×C163C2×#4111yescharacteristic
159Dic163—21no—
160Dic163—21no—
161Dic163—42no—
162Dic163—41no—
163Dic163—41no—
164C2×C163—22no—
165C2×C163—81no—
166C2×C163—82no—
167C2×C163C2×#12111yescharacteristic
168C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
169C2×C163#57511yescharacteristic
170D163—22no—
171D163—41no—
172C163#305511yescharacteristic, complemented, Sylow
173#3055—1631nomaximal, Sylow
174C2×AGL(1,ℤ/8)D16311yescharacteristic, complemented
175C22×M16—1631no—
176#525—1631no—
177C2×#121—1631no—
178#469—1632no—
179#468—1632no—
180#466—1632no—
181#465—1632no—
182#428—1631no—
183#418—1631no—
184#417—1631no—
185C2×#41C2×D16311yescharacteristic
186C22×D4—3261no—
187C22×D4C2×D16311yescharacteristic
188C23×C4—1631no—
189AGL(1,ℤ/8)—24no—
190C2×SD16C2×D16312yescomplemented
191C2×D8C2×D16312yescomplemented
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#117—1631no—
210C2×C4⋊C4—1631no—
211C2×C22⋊C4—3261no—
212#107—1632no—
213#106—1632no—
214#103—1632no—
215#102—1632no—
216#102—1632no—
217#102—1632no—
218C24—21no—
219C24—1631no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D16311yescharacteristic
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—3261no—
228C2×D4—3261no—
229C2×D4—3262no—
230C2×D4—3262no—
231C2×D4—6524no—
232C2×D4C22×D16311yescharacteristic
233C2×D4C22×D16312yes—
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—24no—
245D8—24no—
246M16—3262no—
247M16—3264no—
248M16C22⋊D16314yescomplemented
249C2×C8—3264no—
250C2×C8—1632no—
251C2×C8C22×D16312yes—
252C4⋊C4—3261no—
253C4⋊C4—3261no—
254C4⋊C4—3261no—
255C4⋊C4—1631no—
256C4⋊C4—1631no—
257C4⋊C4—1632no—
258C22⋊C4—3261no—
259C22⋊C4—3261no—
260C22⋊C4—3261no—
261C22⋊C4—3262no—
262C22⋊C4—3262no—
263C22⋊C4—3261no—
264C22⋊C4—3261no—
265C42—3261no—
266C23—21no—
267C23—22no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—3262no—
272C23—3262no—
273C23—6521no—
274C23—1631no—
275C23—1631no—
276C23D4×D16311yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—22no—
281D4—22no—
282D4—21no—
283D4—42no—
284D4—42no—
285D4—6524no—
286D4—6522no—
287D4—6524no—
288D4—6522no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—3262no—
292C2×C4—3261no—
293C2×C4—3261no—
294C2×C4—3261no—
295C2×C4—3261no—
296C2×C4—3262no—
297C2×C4—3262no—
298C2×C4—3261no—
299C2×C4—3261no—
300C2×C4—3262no—
301C2×C4—6522no—
302C2×C4—1631no—
303C2×C4—1631no—
304C2×C4D4×D16311yescharacteristic
305C2×C4C2×C22⋊D16312yes—
306C2×C4#1457969812yes—
307C2×C4C23×D16311yescharacteristic
308C8—3262no—
309C8C2×C22⋊D16314yescomplemented
310C22—21no—
311C22—21no—
312C22—41no—
313C22—42no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—84no—
319C22—3261no—
320C22—3261no—
321C22—3261no—
322C22—3261no—
323C22—3261no—
324C22—6524no—
325C22—1632no—
326C22#1910470612yes—
327C22C2×D4×D16311yescharacteristic
328C4—41no—
329C4—3261no—
330C4—3261no—
331C4—6522no—
332C4—6521no—
333C4—6521no—
334C4#1910470612yes—
335C4C22×C22⋊D16311yescharacteristic
336C4C2×#1457969811yescharacteristic
337C2—22no—
338C2—82no—
339C2—81no—
340C2—3262no—
341C2—6521no—
342C2C2×#1910470611yescharacteristic
343C2AGL(1,ℤ/8)×D16311yescharacteristic
344C2#2378246111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×AGL(1,ℤ/8)×C163C2
socleC22×C163C22×D4
hypercenterC2×AGL(1,ℤ/8)D163
hypocenterC163#3055
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×M16×D163C2semidirect
2. #23782172C2semidirect
3. C2×#19104706C2semidirect
4. #23781915C2semidirect
5. #23781908C2semidirect
6. #23781898C2semidirect
7. #23781891C2semidirect
8. #23781707C2semidirect
9. #23781670C2semidirect
10. #23781666C2semidirect
11. C2×AGL(1,ℤ/8)×C163C2semidirect
15. C2×#14579682C22semidirect
17. C2×C8×D163C22semidirect
19. #19104711C22semidirect
20. #19104711C22semidirect
25. #19104706C22semidirect
26. #19104706C22semidirect
33. #19104669C22semidirect
34. #19104667C22semidirect
35. #19104659C22semidirect
36. #19104656C22semidirect
37. #19104656C22semidirect
38. #19104656C22semidirect
42. C2×SD16×C163C22semidirect
43. C2×D8×C163C22semidirect
91. M16×C163D4semidirect
136. C8×C163C2×D4semidirect
154. C2×D163AGL(1,ℤ/8)semidirect
172. C163#3055semidirect
174. C2×AGL(1,ℤ/8)D163semidirect
190. C2×SD16C2×D163semidirect
191. C2×D8C2×D163semidirect
248. M16C22⋊D163semidirect
309. C8C2×C22⋊D163semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D1632
#237821722
C2×#191047062
#237819152
#237819082
#237818982
#237818912
#237817072
#237816702
#237816662
C2×AGL(1,ℤ/8)×C1632
#3055163