Finite Group Databasebeta

← #35223365

Subgroups — #35223365

Order 20864.

506 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104759C211yesmaximal, characteristic
2#23782144C211yesmaximal, characteristic, complemented
3#23782137C211yesmaximal, characteristic, complemented
4#23781774C211yesmaximal, characteristic, complemented
5#23781767C211yesmaximal, characteristic, complemented
6#23781762C211yesmaximal, characteristic, complemented
7#23781752C211yesmaximal, characteristic, complemented
8C4×#14579687C211yesmaximal, characteristic, complemented
9D8×Dic163C211yesmaximal, characteristic, complemented
10C4×#14579684C211yesmaximal, characteristic, complemented
11#23781704C211yesmaximal, characteristic, complemented
12#23781658C211yesmaximal, characteristic, complemented
13#102×D163C211yesmaximal, characteristic, complemented
14#23781622C211yesmaximal, characteristic, complemented
15#442×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C2×D4×D163C2211yescharacteristic
18#19104759C418yescomplemented
19C2×#14579688C2211yescharacteristic
20C2×#14579687C2211yescharacteristic
21C2×#14579684C2211yescharacteristic
22C2×#14579682C2211yescharacteristic
23C2×C8×D163C2211yescharacteristic
24C4×Dic326C2211yescharacteristic, complemented
25#19104700C2211yescharacteristic, complemented
26C4×D652C2211yescharacteristic
27D4×Dic163C2211yescharacteristic
28D4×Dic163C2211yescharacteristic, complemented
29#19104697C2211yescharacteristic
30C4×C22⋊D163—21no—
31C4×C22⋊D163—21no—
32(C22⋊D163)⋊C4—21no—
33(C22⋊D163)⋊C4—21no—
34#19104693C2211yescharacteristic
35#19104692C2211yescharacteristic
36#19104691—21no—
37C2×Dic163⋊C4—21no—
38C4⋊C4×D163C2211yescharacteristic
39C2×(C2×D163)⋊C4—21no—
40C22⋊C4×D163—21no—
41#19104669C2211yescharacteristic, complemented
42#19104668C2211yescharacteristic, complemented
43#19104666C2211yescharacteristic, complemented
44#19104660C2211yescharacteristic, complemented
45#19104658C2211yescharacteristic, complemented
46#19104657C2211yescharacteristic
47#19104656C2211yescharacteristic
48#19104656C2211yescharacteristic, complemented
49#19104655C2211yescharacteristic
50#19104645C2211yescharacteristic, complemented
51C4×C163⋊C8C2211yescharacteristic, complemented
52C8×Dic163C2211yescharacteristic, complemented
53C2×D8×C163C2211yescharacteristic
54C4×D4×C163C2211yescharacteristic
55C4×D4×C163C2211yescharacteristic, complemented
56#106×C163C2211yescharacteristic, complemented
57#102×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic, complemented
59#97×C163C2211yescharacteristic, complemented
60C23×D163—21no—
61#14579698—22no—
62#14579698C2×C414yescomplemented
63C2×Dic326C2311yescharacteristic
64C2×D652C2311yescharacteristic
65C2×C22⋊D163—21no—
66C2×C22⋊D163—21no—
67D4×D163—22no—
68D4×D163C2×C414yescomplemented
69C22×Dic163—21no—
70C22×Dic163—21no—
71C22×Dic163—21no—
72C2×C4×D163—21no—
73C2×C4×D163—21no—
74C2×C4×D163—21no—
75C2×C4×D163C2311yescharacteristic
76#14579688C2×C414yescomplemented
77#14579687C2×C414yescomplemented
78#14579684C2×C414yescomplemented
79#14579682C2×C414yescomplemented
80C2×C163⋊C8C2311yescharacteristic
81C2×C163⋊C8C2311yescharacteristic
82C8×D163—22no—
83C4⋊Dic163C2311yescharacteristic
84C4⋊Dic163C2311yescharacteristic
85Dic163⋊C4—21no—
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88Dic163⋊C4D411yescharacteristic
89Dic163⋊C4D411yescharacteristic
90C22⋊Dic163—21no—
91C22⋊Dic163—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4D411yescharacteristic
96(C2×D163)⋊C4D411yescharacteristic
97C4×Dic163—21no—
98C4×Dic163—21no—
99C4×Dic163—21no—
100C4×Dic163C2311yescharacteristic
101C2×D4×C163C2311yescharacteristic
102C2×D4×C163C2311yescharacteristic
103C22×C4×C163—21no—
104C22×C4×C163—21no—
105D8×C163C2×C414yescomplemented
106C2×C8×C163C2311yescharacteristic
107C2×C8×C163C2311yescharacteristic
108C4⋊C4×C163C2311yescharacteristic
109C4⋊C4×C163C2311yescharacteristic
110C22⋊C4×C163—21no—
111C22⋊C4×C163—21no—
112C42×C163C2311yescharacteristic
113C22×D163—21no—
114C22×D163—24no—
115C22×D163—81no—
116C22×D163—81no—
117C22×D163C2×D411yescharacteristic
118Dic326—21no—
119Dic326C22×C412yes—
120D652—21no—
121D652C22×C412yes—
122C22⋊D163—24no—
123C22⋊D163—24no—
124C2×Dic163—21no—
125C2×Dic163—24no—
126C2×Dic163—21no—
127C2×Dic163—21no—
128C2×Dic163—81no—
129C2×Dic163—81no—
130C2×Dic163C2×D411yescharacteristic
131C2×Dic163C2×D411yescharacteristic
132C2×Dic163C2×D411yescharacteristic
133C4×D163—81no—
134C4×D163—81no—
135C4×D163—81no—
136C4×D163C22×C412yes—
137C4×D163#4112yes—
138C163⋊C8—21no—
139C163⋊C8C22×C412yes—
140C23×C163—21no—
141C23×C163—21no—
142D4×C163—21no—
143D4×C163—21no—
144D4×C163C22×C412yes—
145D4×C163C22×C412yes—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—81no—
149C2×C4×C163—81no—
150C2×C4×C163C2×D411yescharacteristic
151C2×C4×C163C2×D411yescharacteristic
152C2×C4×C163C2411yescharacteristic
153C8×C163—21no—
154C8×C163C22×C412yes—
155C22×C163—22no—
156C22×C163—22no—
157C22×C163—41no—
158C22×C163—41no—
159C22×C163—41no—
160C22×C163—41no—
161C22×C163C22×D411yescharacteristic
162C2×D163—21no—
163C2×D163—21no—
164C2×D163—22no—
165C2×D163—41no—
166C2×D163—41no—
167C2×D163—84no—
168C2×D163C4×D412yes—
169C4×C163—21no—
170C4×C163—21no—
171C4×C163—41no—
172C4×C163—41no—
173C4×C163C23×C411yescharacteristic
174C4×C163C2×#4111yescharacteristic
175Dic163—22no—
176Dic163—21no—
177Dic163—21no—
178Dic163—41no—
179Dic163—41no—
180Dic163C4×D412yes—
181C2×C163—42no—
182C2×C163—42no—
183C2×C163C2×C4×D411yescharacteristic
184C2×C163#57611yescharacteristic
185C2×C163#57711yescharacteristic
186D163—22no—
187D163—42no—
188C163#294611yescharacteristic, complemented, Sylow
189#2946—1631nomaximal, Sylow
190C2×AGL(1,ℤ/8)—1631no—
191#520—1631no—
192#518—1631no—
193#442—1631no—
194#442D16311yescharacteristic, complemented
195#440—1631no—
196#440—1631no—
197C4×SD16—1631no—
198C4×SD16—1631no—
199C4×D8—1631no—
200C4×D8—1631no—
201#428—1631no—
202#416—1631no—
203C2×#102—1631no—
204#405—1631no—
205C2×#41—1631no—
206C22×D4—1631no—
207AGL(1,ℤ/8)—1638no—
208C2×SD16—1631no—
209C2×SD16—1631no—
210C2×D8—1631no—
211C2×D8C2×D16311yescharacteristic
212C2×M16—1631no—
213C22×C8—1631no—
214C4×Q8—1631no—
215C4×Q8—1631no—
216C4×D4—3261no—
217C4×D4—3261no—
218C4×D4—3261no—
219C4×D4—3261no—
220C4×D4—1631no—
221C4×D4—1631no—
222C4×D4—1631no—
223C4×D4—1631no—
224C4×D4C2×D16311yescharacteristic
225C4×D4C2×D16311yescharacteristic, complemented
226#117—3261no—
227#117—1631no—
228#117—1631no—
229C2×C4⋊C4—3261no—
230C2×C4⋊C4—1631no—
231C2×C22⋊C4—3261no—
232C2×C22⋊C4—3261no—
233#107—1631no—
234#107—1631no—
235#106—1631no—
236#106C2×D16311yescharacteristic, complemented
237#103—1631no—
238#103—1631no—
239#102—1631no—
240#102—1631no—
241#102—1631no—
242#102—1631no—
243#102C2×D16311yescharacteristic
244#102C2×D16311yescharacteristic, complemented
245#97—1631no—
246#97C2×D16311yescharacteristic, complemented
247C4×C8—1631no—
248C4×C8—1631no—
249C24—3261no—
250#41—3262no—
251#41—1634no—
252C2×Q8—1631no—
253C2×D4—3261no—
254C2×D4—3262no—
255C2×D4—3261no—
256C2×D4—1634no—
257C2×D4—1631no—
258C2×D4C22×D16311yescharacteristic
259C2×D4C22×D16311yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—3261no—
263C22×C4—3261no—
264C22×C4—3261no—
265C22×C4—3261no—
266C22×C4—3261no—
267C22×C4—3261no—
268C22×C4—1631no—
269SD16—1634no—
270SD16—1634no—
271D8—1634no—
272D8C4×D16314yescomplemented
273M16—1634no—
274C2×C8—3262no—
275C2×C8—1631no—
276C2×C8—1631no—
277C2×C8C22×D16311yescharacteristic
278C2×C8C22×D16311yescharacteristic
279C4⋊C4—3261no—
280C4⋊C4—3261no—
281C4⋊C4—3261no—
282C4⋊C4—1631no—
283C4⋊C4—1631no—
284C4⋊C4—1631no—
285C4⋊C4—1631no—
286C4⋊C4C22×D16311yescharacteristic
287C4⋊C4C22×D16311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—3261no—
291C22⋊C4—3261no—
292C22⋊C4—3261no—
293C22⋊C4—3261no—
294C22⋊C4—3261no—
295C22⋊C4—1631no—
296C22⋊C4—1631no—
297C42—3261no—
298C42—3261no—
299C42—3261no—
300C42—1631no—
301C42C22×D16311yescharacteristic
302C23—21no—
303C23—21no—
304C23—13041no—
305C23—13041no—
306C23—3264no—
307C23—3261no—
308C23—1631no—
309Q8—3261no—
310Q8—1632no—
311D4—21no—
312D4—21no—
313D4—3264no—
314D4—3264no—
315D4—3261no—
316D4—1632no—
317D4C2×C4×D16312yes—
318D4C2×C4×D16312yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—13041no—
324C2×C4—13041no—
325C2×C4—13041no—
326C2×C4—13041no—
327C2×C4—13041no—
328C2×C4—3261no—
329C2×C4—3264no—
330C2×C4—3261no—
331C2×C4—3261no—
332C2×C4—1631no—
333C2×C4—1631no—
334C2×C4—1631no—
335C2×C4—1632no—
336C2×C4—1632no—
337C2×C4D4×D16311yescharacteristic
338C2×C4D4×D16311yescharacteristic
339C2×C4C23×D16311yescharacteristic
340C8—21no—
341C8—3261no—
342C8—1632no—
343C8C2×C4×D16312yes—
344C22—22no—
345C22—22no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—13044no—
351C22—3261no—
352C22—3261no—
353C22—3262no—
354C22—6521no—
355C22—6521no—
356C22—1632no—
357C22C2×D4×D16311yescharacteristic
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—3261no—
363C4—3261no—
364C4—3262no—
365C4—6521no—
366C4—6521no—
367C4—1632no—
368C4C22×C4×D16311yescharacteristic
369C4#41×D16311yescharacteristic
370C2—42no—
371C2—42no—
372C2—3262no—
373C2—6522no—
374C2C4×D4×D16311yescharacteristic
375C2#2378246911yescharacteristic
376C2#2378247511yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutatorC4×C163C23×C4
FrattiniC2×C4C23×D163
Fitting#442×C163C2
socleC22×C163C22×D4
prime radicalC2×#19104759C2
cosocleC2×C4×C163C24
hypercenter#442D163
hypocenterC163#2946
LCSG ▷ C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
2. #23782144C2semidirect
3. #23782137C2semidirect
4. #23781774C2semidirect
5. #23781767C2semidirect
6. #23781762C2semidirect
7. #23781752C2semidirect
8. C4×#14579687C2semidirect
9. D8×Dic163C2semidirect
10. C4×#14579684C2semidirect
11. #23781704C2semidirect
12. #23781658C2semidirect
13. #102×D163C2semidirect
14. #23781622C2semidirect
15. #442×C163C2semidirect
18. #19104759C4semidirect
24. C4×Dic326C22semidirect
25. #19104700C22semidirect
28. D4×Dic163C22semidirect
41. #19104669C22semidirect
42. #19104668C22semidirect
43. #19104666C22semidirect
44. #19104660C22semidirect
45. #19104658C22semidirect
48. #19104656C22semidirect
50. #19104645C22semidirect
51. C4×C163⋊C8C22semidirect
52. C8×Dic163C22semidirect
55. C4×D4×C163C22semidirect
56. #106×C163C22semidirect
58. #102×C163C22semidirect
59. #97×C163C22semidirect
62. #14579698C2×C4semidirect
68. D4×D163C2×C4semidirect
76. #14579688C2×C4semidirect
77. #14579687C2×C4semidirect
78. #14579684C2×C4semidirect
79. #14579682C2×C4semidirect
105. D8×C163C2×C4semidirect
188. C163#2946semidirect
194. #442D163semidirect
225. C4×D4C2×D163semidirect
236. #106C2×D163semidirect
244. #102C2×D163semidirect
246. #97C2×D163semidirect
272. D8C4×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047592
#237821442
#237821372
#237817742
#237817672
#237817622
#237817522
C4×#145796872
D8×Dic1632
C4×#145796842
#237817042
#237816582
#102×D1632
#237816222
#442×C1632
#2946163