Finite Group Databasebeta

← #35223390

Subgroups — #35223390

Order 20864.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D163C211yesmaximal, characteristic
2#23782141C211yesmaximal, characteristic, complemented
3#23782137C211yesmaximal, characteristic
4#23781768C211yesmaximal, characteristic, complemented
5#23781765C211yesmaximal, characteristic, complemented
6#23781763C211yesmaximal, characteristic, complemented
7#23781762C211yesmaximal, characteristic, complemented
8C4×#14579688C211yesmaximal, characteristic, complemented
9SD16×Dic163C211yesmaximal, characteristic, complemented
10#23781752C211yesmaximal, characteristic, complemented
11#23781700C211yesmaximal, characteristic, complemented
12#23781672C211yesmaximal, characteristic, complemented
13#23781668C211yesmaximal, characteristic
14#23781619C211yesmaximal, characteristic, complemented
15C4×SD16×C163C211yesmaximal, characteristic, complemented
16C2×Q8×D163C2211yescharacteristic
17C2×D4×D163C2211yescharacteristic
18C2×#14579688C2211yescharacteristic
19C2×#14579687C2211yescharacteristic
20C2×#14579686C2211yescharacteristic
21SD16×D163C418yescomplemented
22C2×#14579682C2211yescharacteristic
23C2×C8×D163C2211yescharacteristic
24Q8×Dic163C2211yescharacteristic, complemented
25C4×Dic326—21no—
26C4×Dic326C2211yescharacteristic, complemented
27#19104700—21no—
28#19104700C2211yescharacteristic, complemented
29C4×D652C2211yescharacteristic
30D4×Dic163C2211yescharacteristic
31#19104697C2211yescharacteristic
32C4×C22⋊D163—21no—
33(C22⋊D163)⋊C4—21no—
34#19104693—21no—
35#19104693C2211yescharacteristic
36#19104693C2211yescharacteristic
37#19104692—21no—
38#19104692C2211yescharacteristic
39C2×(C2×D163)⋊C4—21no—
40C22⋊C4×D163—21no—
41#19104668C2211yescharacteristic, complemented
42#19104668C2211yescharacteristic, complemented
43#19104667C2211yescharacteristic, complemented
44#19104660C2211yescharacteristic, complemented
45#19104659C2211yescharacteristic, complemented
46#19104658C2211yescharacteristic, complemented
47#19104657C2211yescharacteristic
48#19104656C2211yescharacteristic
49#19104655C2211yescharacteristic
50#19104646C2211yescharacteristic, complemented
51#19104645C2211yescharacteristic, complemented
52C8×Dic163C2211yescharacteristic, complemented
53C2×SD16×C163C2211yescharacteristic
54C4×Q8×C163C2211yescharacteristic, complemented
55C4×D4×C163C2211yescharacteristic
56#106×C163C2211yescharacteristic, complemented
57#103×C163C2211yescharacteristic, complemented
58#102×C163C2211yescharacteristic
59C4×C8×C163C2211yescharacteristic, complemented
60C23×D163—21no—
61C2×Dic326—21no—
62C2×Dic326C2311yescharacteristic
63Q8×D163—22no—
64Q8×D163C2×C414yescomplemented
65C2×D652C2311yescharacteristic
66C2×C22⋊D163—21no—
67D4×D163—22no—
68D4×D163C2×C414yes—
69C22×Dic163—21no—
70C2×C4×D163—21no—
71C2×C4×D163—21no—
72C2×C4×D163—21no—
73C2×C4×D163C2311yescharacteristic
74#14579688C2×C414yescomplemented
75#14579687C2×C414yescomplemented
76#14579686C2×C414yescomplemented
77#14579682—22no—
78C2×C163⋊C8C2311yescharacteristic
79C2×C163⋊C8C2311yescharacteristic
80C8×D163C2×C414yescomplemented
81C4⋊Dic163—21no—
82C4⋊Dic163C2311yescharacteristic
83C4⋊Dic163C2311yescharacteristic
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4D411yescharacteristic
87Dic163⋊C4D411yescharacteristic
88C22⋊Dic163—21no—
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4D411yescharacteristic
94(C2×D163)⋊C4D411yescharacteristic
95C4×Dic163—21no—
96C4×Dic163—21no—
97C4×Dic163—21no—
98C4×Dic163—21no—
99C4×Dic163—21no—
100C4×Dic163C2311yescharacteristic
101C2×Q8×C163C2311yescharacteristic
102C2×D4×C163C2311yescharacteristic
103C22×C4×C163—21no—
104SD16×C163C2×C414yescomplemented
105C2×C8×C163C2311yescharacteristic
106C2×C8×C163C2311yescharacteristic
107C4⋊C4×C163—21no—
108C4⋊C4×C163C2311yescharacteristic
109C4⋊C4×C163C2311yescharacteristic
110C22⋊C4×C163—21no—
111C42×C163—21no—
112C42×C163C2311yescharacteristic
113C22×D163—21no—
114C22×D163—24no—
115C22×D163—81no—
116C22×D163—81no—
117C22×D163C2×D411yescharacteristic
118Dic326—21no—
119Dic326—24no—
120Dic326C22×C412yes—
121D652—21no—
122D652C22×C412yes—
123C22⋊D163—24no—
124C2×Dic163—21no—
125C2×Dic163—21no—
126C2×Dic163—21no—
127C2×Dic163—81no—
128C2×Dic163C2×D411yescharacteristic
129C2×Dic163C2×D411yescharacteristic
130C2×Dic163C2×D411yescharacteristic
131C4×D163—24no—
132C4×D163—81no—
133C4×D163—81no—
134C4×D163C22×C412yes—
135C4×D163#4112yes—
136C163⋊C8—21no—
137C163⋊C8C22×C412yes—
138C23×C163—21no—
139Q8×C163—21no—
140Q8×C163C22×C412yes—
141D4×C163—21no—
142D4×C163C22×C412yes—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163—81no—
147C2×C4×C163C2×D411yescharacteristic
148C2×C4×C163C2×D411yescharacteristic
149C2×C4×C163C2411yescharacteristic
150C8×C163—21no—
151C8×C163C22×C412yes—
152C22×C163—22no—
153C22×C163—41no—
154C22×C163—41no—
155C22×C163C22×D411yescharacteristic
156C2×D163—21no—
157C2×D163—21no—
158C2×D163—22no—
159C2×D163—41no—
160C2×D163—41no—
161C2×D163—84no—
162C2×D163C4×D412yes—
163C4×C163—22no—
164C4×C163—21no—
165C4×C163—21no—
166C4×C163—41no—
167C4×C163—41no—
168C4×C163C23×C411yescharacteristic
169C4×C163C2×#4111yescharacteristic
170Dic163—22no—
171Dic163—21no—
172Dic163—21no—
173Dic163—41no—
174Dic163—41no—
175Dic163C4×D412yes—
176C2×C163—42no—
177C2×C163C2×C4×D411yescharacteristic
178C2×C163#57711yescharacteristic
179C2×C163#57711yescharacteristic
180D163—22no—
181D163—42no—
182C163#294911yescharacteristic, complemented, Sylow
183#2949—1631nomaximal, Sylow
184C22×SD16—1631no—
185#519—1631no—
186#518—1631no—
187#440—1631no—
188#440—1631no—
189#440—1631no—
190#440—1631no—
191C4×SD16—1631no—
192C4×SD16—1631no—
193C4×SD16—1631no—
194C4×SD16D16311yescharacteristic, complemented
195#427—1631no—
196#419—1631no—
197#418—1631no—
198#405—1631no—
199C22×Q8—1631no—
200C22×D4—1631no—
201C2×SD16—1631no—
202C2×SD16—1631no—
203C2×SD16—1638no—
204C2×SD16—1631no—
205C2×SD16C2×D16311yescharacteristic
206C2×M16—1631no—
207C22×C8—1631no—
208C4×Q8—3261no—
209C4×Q8—3261no—
210C4×Q8—1631no—
211C4×Q8—1631no—
212C4×Q8—1631no—
213C4×Q8C2×D16311yescharacteristic, complemented
214C4×D4—3261no—
215C4×D4—3261no—
216C4×D4—1631no—
217C4×D4—1631no—
218C4×D4—1631no—
219C4×D4C2×D16311yescharacteristic
220#117—3261no—
221#117—3261no—
222#117—1631no—
223#117—1631no—
224#117—1631no—
225C2×C22⋊C4—3261no—
226C2×C22⋊C4—3261no—
227#107—1631no—
228#107—1631no—
229#106—1631no—
230#106C2×D16311yescharacteristic, complemented
231#103—1631no—
232#103—1631no—
233#103—1631no—
234#103C2×D16311yescharacteristic, complemented
235#102—1631no—
236#102—1631no—
237#102—1631no—
238#102C2×D16311yescharacteristic
239#97—1631no—
240#97—1631no—
241C4×C8—1631no—
242C4×C8C2×D16311yescharacteristic, complemented
243C24—3261no—
244C2×Q8—3261no—
245C2×Q8—3262no—
246C2×Q8—1634no—
247C2×Q8—1631no—
248C2×Q8C22×D16311yescharacteristic
249C2×D4—3262no—
250C2×D4—3261no—
251C2×D4—1634no—
252C2×D4—1631no—
253C2×D4C22×D16311yescharacteristic
254C22×C4—21no—
255C22×C4—3261no—
256C22×C4—3261no—
257C22×C4—3261no—
258C22×C4—3261no—
259C22×C4—1631no—
260SD16—1634no—
261SD16—1634no—
262SD16—1634no—
263SD16C4×D16314yescomplemented
264M16—3262no—
265C2×C8—1631no—
266C2×C8—1631no—
267C2×C8—1634no—
268C2×C8C22×D16311yescharacteristic
269C2×C8C22×D16311yescharacteristic
270C4⋊C4—21no—
271C4⋊C4—3261no—
272C4⋊C4—3261no—
273C4⋊C4—3261no—
274C4⋊C4—1631no—
275C4⋊C4—1631no—
276C4⋊C4—1631no—
277C4⋊C4—1631no—
278C4⋊C4C22×D16311yescharacteristic
279C4⋊C4C22×D16311yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—3261no—
284C22⋊C4—3261no—
285C22⋊C4—3261no—
286C22⋊C4—1631no—
287C22⋊C4—1631no—
288C42—21no—
289C42—3261no—
290C42—3261no—
291C42—3261no—
292C42—3261no—
293C42—3261no—
294C42—1631no—
295C42C22×D16311yescharacteristic
296C23—21no—
297C23—13041no—
298C23—13041no—
299C23—3264no—
300C23—3261no—
301C23—1631no—
302Q8—21no—
303Q8—3264no—
304Q8—3261no—
305Q8—1632no—
306Q8C2×C4×D16312yes—
307D4—21no—
308D4—3264no—
309D4—3261no—
310D4—1632no—
311D4C2×C4×D16312yes—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—81no—
316C2×C4—13041no—
317C2×C4—13041no—
318C2×C4—13041no—
319C2×C4—3264no—
320C2×C4—3261no—
321C2×C4—3261no—
322C2×C4—3261no—
323C2×C4—1631no—
324C2×C4—1631no—
325C2×C4—1631no—
326C2×C4—1632no—
327C2×C4—1632no—
328C2×C4D4×D16311yescharacteristic
329C2×C4D4×D16311yescharacteristic
330C2×C4C23×D16311yescharacteristic
331C8—21no—
332C8—3261no—
333C8—1632no—
334C8C2×C4×D16312yes—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—13044no—
339C22—3261no—
340C22—3261no—
341C22—3262no—
342C22—6521no—
343C22—6521no—
344C22—1632no—
345C22C2×D4×D16311yescharacteristic
346C4—21no—
347C4—21no—
348C4—22no—
349C4—41no—
350C4—41no—
351C4—3261no—
352C4—3261no—
353C4—3262no—
354C4—6521no—
355C4—6521no—
356C4—1632no—
357C4C22×C4×D16311yescharacteristic
358C4#41×D16311yescharacteristic
359C2—42no—
360C2—3262no—
361C2—6522no—
362C2C4×D4×D16311yescharacteristic
363C2#2378247111yescharacteristic
364C2#2378247211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutatorC4×C163C23×C4
FrattiniC2×C4C23×D163
FittingC4×SD16×C163C2
socleC22×C163C22×D4
prime radicalC2×D4×D163C22
cosocleC2×C4×C163C24
hypercenterC4×SD16D163
hypocenterC163#2949
LCSG ▷ C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #23782141C2semidirect
4. #23781768C2semidirect
5. #23781765C2semidirect
6. #23781763C2semidirect
7. #23781762C2semidirect
8. C4×#14579688C2semidirect
9. SD16×Dic163C2semidirect
10. #23781752C2semidirect
11. #23781700C2semidirect
12. #23781672C2semidirect
14. #23781619C2semidirect
15. C4×SD16×C163C2semidirect
21. SD16×D163C4semidirect
24. Q8×Dic163C22semidirect
26. C4×Dic326C22semidirect
28. #19104700C22semidirect
41. #19104668C22semidirect
42. #19104668C22semidirect
43. #19104667C22semidirect
44. #19104660C22semidirect
45. #19104659C22semidirect
46. #19104658C22semidirect
50. #19104646C22semidirect
51. #19104645C22semidirect
52. C8×Dic163C22semidirect
54. C4×Q8×C163C22semidirect
56. #106×C163C22semidirect
57. #103×C163C22semidirect
59. C4×C8×C163C22semidirect
64. Q8×D163C2×C4semidirect
74. #14579688C2×C4semidirect
75. #14579687C2×C4semidirect
76. #14579686C2×C4semidirect
80. C8×D163C2×C4semidirect
104. SD16×C163C2×C4semidirect
182. C163#2949semidirect
194. C4×SD16D163semidirect
213. C4×Q8C2×D163semidirect
230. #106C2×D163semidirect
234. #103C2×D163semidirect
242. C4×C8C2×D163semidirect
263. SD16C4×D163semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1632
#237821412
#237821372
#237817682
#237817652
#237817632
#237817622
C4×#145796882
SD16×Dic1632
#237817522
#237817002
#237816722
#237816682
#237816192
C4×SD16×C1632
#2949163