Finite Group Databasebeta

← #35225012

Subgroups — #35225012

Order 20864.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#19104768C211yesmaximal, characteristic, complemented
2C2×#19104763C211yesmaximal, characteristic, complemented
3C2×#19104757C211yesmaximal, characteristic, complemented
4C22×#14579688C211yesmaximal, characteristic, complemented
5#23782241C211yesmaximal, characteristic, complemented
6#23782234C211yesmaximal, characteristic, complemented
7#23782059C212yesmaximal, complemented
8#23782054C212yesmaximal, complemented
9#23782049C212yesmaximal, complemented
10#23782038C212yesmaximal, complemented
11#405×C163C211yesmaximal, characteristic, complemented
12C2×#14579699C2211yescharacteristic
13C2×#14579699C2211yescharacteristic
14C22×Dic326C2211yescharacteristic
15C22×D652C2211yescharacteristic
16#19104768—24no—
17#19104763—24no—
18#19104757—24no—
19C2×Dic652C2211yescharacteristic, complemented
20C2×Dic652C2212yescomplemented
21C2×#14579688—24no—
22C2×#14579688C2212yes—
23C2×#14579688C2212yes—
24C2×#14579688C2212yes—
25C2×#14579688C2211yescharacteristic, complemented
26C2×#14579688C2211yescharacteristic, complemented
27C2×#14579688C2212yescomplemented
28C2×D1304C2212yes—
29C2×D1304C2211yescharacteristic, complemented
30#19104735C2212yescomplemented
31#19104732C2212yes—
32#19104723C2212yes—
33#19104723C2212yescomplemented
34#19104719—22no—
35#19104715—22no—
36#19104710—22no—
37#19104705—22no—
38C2×M16×C163C2211yescharacteristic
39C22×C8×C163C2211yescharacteristic
40#117×C163C2211yescharacteristic
41#97×C163C2212yescomplemented
42C4×C8×C163C2212yescomplemented
43C23×D163—21no—
44#14579699—22no—
45#14579699—22no—
46#14579699—22no—
47#14579699—22no—
48C2×Dic326—21no—
49C2×Dic326—22no—
50C2×Dic326—22no—
51C2×Dic326C2312yes—
52C2×Dic326C2311yescharacteristic
53C2×Dic326C2311yescharacteristic
54C2×D652—22no—
55C2×D652—22no—
56C2×D652—21no—
57C2×D652—24no—
58C2×D652C2312yes—
59C2×D652C2311yescharacteristic
60C2×D652C2311yescharacteristic
61C2×C22⋊D163—21no—
62C2×C22⋊D163—21no—
63C22×Dic163—21no—
64C2×C4×D163—21no—
65C2×C4×D163—21no—
66Dic652—24no—
67Dic652—22no—
68#14579688—24no—
69#14579688—24no—
70#14579688—24no—
71#14579688—24no—
72#14579688—22no—
73#14579688—22no—
74D1304—24no—
75D1304—22no—
76C4⋊Dic163—24no—
77(C2×D163)⋊C4—24no—
78(C2×D163)⋊C4—24no—
79C22×C4×C163C2311yescharacteristic
80M16×C163D414yescomplemented
81C2×C8×C163D414yescomplemented
82C2×C8×C163C2311yescharacteristic
83C2×C8×C163C2312yes—
84C2×C8×C163C2311yescharacteristic
85C4⋊C4×C163D412yes—
86C22⋊C4×C163D412yes—
87C42×C163C2312yes—
88C22×D163—21no—
89C22×D163—22no—
90C22×D163—21no—
91C22×D163—42no—
92C22×D163—82no—
93Dic326—22no—
94Dic326—21no—
95Dic326—22no—
96Dic326—21no—
97Dic326—21no—
98Dic326—21no—
99Dic326—42no—
100D652—21no—
101D652—22no—
102D652—21no—
103D652—21no—
104D652—22no—
105D652—21no—
106D652—42no—
107D652—84no—
108C22⋊D163—42no—
109C22⋊D163—42no—
110C2×Dic163—21no—
111C2×Dic163—22no—
112C2×Dic163—21no—
113C2×Dic163—42no—
114C4×D163—42no—
115C4×D163—42no—
116C23×C163C2×D411yescharacteristic
117C2×C4×C163C2×D414yes—
118C2×C4×C163C2×D411yescharacteristic
119C2×C4×C163C2×D412yes—
120C2×C4×C163C2×D412yes—
121C2×C4×C163C2411yescharacteristic
122C8×C163C2×D412yes—
123C8×C163C2×D412yes—
124C8×C163C2×D414yes—
125C22×C163—22no—
126C22×C163#12712yes—
127C22×C163C22×D411yescharacteristic
128C2×D163—41no—
129C2×D163—41no—
130C2×D163—42no—
131C2×D163—42no—
132C2×D163—44no—
133C2×D163—42no—
134C2×D163—84no—
135C4×C163—24no—
136C4×C163#12712yes—
137C4×C163C22×D411yescharacteristic
138C4×C163C22×D411yescharacteristic
139Dic163—41no—
140Dic163—41no—
141Dic163—42no—
142C2×C163—22no—
143C2×C163C2×#12711yescharacteristic
144C2×C163#57712yes—
145D163—81no—
146D163—81no—
147D163—82no—
148C163#315411yescharacteristic, complemented, Sylow
149#3154—1631nomaximal, Sylow
150C2×#137—1631no—
151C2×AGL(1,ℤ/8)—1631no—
152C2×#135—1631no—
153C22×SD16—1631no—
154#536—1631no—
155#535—1631no—
156#497—1632no—
157#496—1632no—
158#495—1632no—
159#492—1632no—
160#405D16311yescharacteristic, complemented
161C2×#41—1631no—
162C2×#41—1631no—
163C22×Q8—1631no—
164C22×D4—1631no—
165#137—3264no—
166AGL(1,ℤ/8)—3264no—
167#135—3264no—
168C2×Q16—1631no—
169C2×Q16—1632no—
170C2×SD16—3264no—
171C2×SD16—1632no—
172C2×SD16—1632no—
173C2×SD16—1631no—
174C2×SD16—1631no—
175C2×SD16—1632no—
176C2×SD16—1632no—
177C2×D8—1631no—
178C2×D8—1632no—
179C2×M16C2×D16311yescharacteristic
180C22×C8C2×D16311yescharacteristic
181C4⋊Q8—1632no—
182#127—1632no—
183#124—1632no—
184#124—1632no—
185#123—3262no—
186C22⋊Q8—3262no—
187#121—3262no—
188#120—3262no—
189#117C2×D16311yescharacteristic
190#97C2×D16312yescomplemented
191C4×C8C2×D16312yescomplemented
192C24—3261no—
193#41—3262no—
194#41—3262no—
195#41—3262no—
196#41—3262no—
197C2×Q8—3261no—
198C2×Q8—3262no—
199C2×Q8—3262no—
200C2×Q8—1631no—
201C2×Q8—1632no—
202C2×Q8—1631no—
203C2×D4—3261no—
204C2×D4—3261no—
205C2×D4—3264no—
206C2×D4—3262no—
207C2×D4—3262no—
208C2×D4—3261no—
209C2×D4—1631no—
210C2×D4—1632no—
211C2×D4—1631no—
212C22×C4—3261no—
213C22×C4—3261no—
214C22×C4—3261no—
215C22×C4C22×D16311yescharacteristic
216Q16—3262no—
217Q16—3264no—
218SD16—3264no—
219SD16—3264no—
220SD16—3262no—
221SD16—3262no—
222SD16—3264no—
223SD16—3264no—
224D8—3262no—
225D8—3264no—
226M16D65214yescomplemented
227C2×C8D65214yescomplemented
228C2×C8C22×D16311yescharacteristic
229C2×C8C22×D16312yes—
230C2×C8C22×D16311yescharacteristic
231C4⋊C4—3264no—
232C4⋊C4D65212yes—
233C22⋊C4—3264no—
234C22⋊C4—3264no—
235C22⋊C4D65212yes—
236C42C22×D16312yes—
237C23—13042no—
238C23—3261no—
239C23—3262no—
240C23—3261no—
241C23—6522no—
242C23C2×D65211yescharacteristic
243Q8—3262no—
244Q8—3261no—
245Q8—3262no—
246Q8—3261no—
247Q8—3261no—
248Q8—3261no—
249Q8—6522no—
250D4—13044no—
251D4—3261no—
252D4—3262no—
253D4—3261no—
254D4—3261no—
255D4—3262no—
256D4—3261no—
257D4—6522no—
258D4—6522no—
259D4—6522no—
260C2×C4—3261no—
261C2×C4—3262no—
262C2×C4—3261no—
263C2×C4—6522no—
264C2×C4—6522no—
265C2×C4—6522no—
266C2×C4C2×D65214yes—
267C2×C4C2×D65212yes—
268C2×C4C2×D65212yes—
269C2×C4C2×D65211yescharacteristic
270C2×C4C23×D16311yescharacteristic
271C8C2×D65212yes—
272C8C2×D65212yes—
273C8C2×D65214yes—
274C22—22no—
275C22—13044no—
276C22—6521no—
277C22—6521no—
278C22—6522no—
279C22—6522no—
280C22—6524no—
281C22—6522no—
282C22#1910473212yes—
283C22C22×D65211yescharacteristic
284C4—24no—
285C4—6521no—
286C4—6521no—
287C4—6522no—
288C4#1910473212yes—
289C4C22×D65211yescharacteristic
290C4C22×D65211yescharacteristic
291C2—22no—
292C2—13042no—
293C2—13041no—
294C2—13041no—
295C2C2×#1910473211yescharacteristic
296C2#2378247712yes—
subgroupG/H
centerC22C22×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#405×C163C2
socleC22×C163C22×D4
hypercenter#405D163
hypocenterC163#3154
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
1. C2×#19104768C2semidirect
2. C2×#19104763C2semidirect
3. C2×#19104757C2semidirect
4. C22×#14579688C2semidirect
5. #23782241C2semidirect
6. #23782234C2semidirect
7. #23782059C2semidirect
8. #23782054C2semidirect
9. #23782049C2semidirect
10. #23782038C2semidirect
11. #405×C163C2semidirect
19. C2×Dic652C22semidirect
20. C2×Dic652C22semidirect
25. C2×#14579688C22semidirect
26. C2×#14579688C22semidirect
27. C2×#14579688C22semidirect
29. C2×D1304C22semidirect
30. #19104735C22semidirect
33. #19104723C22semidirect
41. #97×C163C22semidirect
42. C4×C8×C163C22semidirect
80. M16×C163D4semidirect
81. C2×C8×C163D4semidirect
148. C163#3154semidirect
160. #405D163semidirect
190. #97C2×D163semidirect
191. C4×C8C2×D163semidirect
226. M16D652semidirect
227. C2×C8D652semidirect

Maximal subgroups

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

subgroupindex
C2×#191047682
C2×#191047632
C2×#191047572
C22×#145796882
#237822412
#237822342
#237820592
#237820542
#237820492
#237820382
#405×C1632
#3154163