Finite Group Databasebeta

← #35224776

Subgroups — #35224776

Order 20864.

418 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782266C211yesmaximal, characteristic, complemented
2#23782137C211yesmaximal, characteristic, complemented
3#23781992C211yesmaximal, characteristic, complemented
4#23781987C211yesmaximal, characteristic, complemented
5#23781968C211yesmaximal, characteristic, complemented
6#23781963C211yesmaximal, characteristic, complemented
7#23781859C211yesmaximal, characteristic, complemented
8#23781858C211yesmaximal, characteristic, complemented
9#23781800C211yesmaximal, characteristic, complemented
10#23781797C211yesmaximal, characteristic, complemented
11#23781775C211yesmaximal, characteristic, complemented
12#23781775C211yesmaximal, characteristic, complemented
13C4×D1304C211yesmaximal, characteristic, complemented
14#23781724C211yesmaximal, characteristic, complemented
15C4×D8×C163C211yesmaximal, characteristic, complemented
16C2×D4×D163C2211yescharacteristic
17C2×D1304C2211yescharacteristic
18C2×#14579684C2211yescharacteristic
19C2×#14579684C2211yescharacteristic
20#19104731C2211yescharacteristic, complemented
21#19104726C2211yescharacteristic, complemented
22#19104718—21no—
23#19104717—21no—
24#19104710C2211yescharacteristic
25#19104709—21no—
26#19104708—21no—
27#19104706C2211yescharacteristic
28C4×D652C2211yescharacteristic
29C4×D652C2211yescharacteristic, complemented
30D4×Dic163C2211yescharacteristic
31#19104697C2211yescharacteristic
32C4×C22⋊D163—21no—
33(C22⋊D163)⋊C4—21no—
34#19104693C2211yescharacteristic
35#19104692C2211yescharacteristic
36C2×(C2×D163)⋊C4—21no—
37C22⋊C4×D163—21no—
38#19104669C2211yescharacteristic, complemented
39#19104666C2211yescharacteristic, complemented
40#19104666C2211yescharacteristic, complemented
41Dic163⋊C8C2211yescharacteristic, complemented
42Dic163⋊C8C2211yescharacteristic, complemented
43#19104657C2211yescharacteristic
44#19104657C2211yescharacteristic, complemented
45#19104656C2211yescharacteristic
46#19104656C2211yescharacteristic, complemented
47#19104655C2211yescharacteristic
48#19104655C2211yescharacteristic, complemented
49(C2×D163)⋊C8C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51#19104646C2211yescharacteristic, complemented
52C2×D8×C163C2211yescharacteristic
53C4×D4×C163C2211yescharacteristic
54C4×D4×C163C2211yescharacteristic, complemented
55#107×C163C2211yescharacteristic, complemented
56#102×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic, complemented
58C4×C8×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60C2×D652—21no—
61C2×D652C2311yescharacteristic
62C2×D652C2311yescharacteristic
63C2×C22⋊D163—21no—
64C2×C22⋊D163—21no—
65C2×C22⋊D163—21no—
66C2×C22⋊D163—21no—
67D4×D163—22no—
68D4×D163—22no—
69C22×Dic163—21no—
70C2×C4×D163—21no—
71C2×C4×D163—21no—
72C2×C4×D163—21no—
73C2×C4×D163C2311yescharacteristic
74D1304—22no—
75#14579684—22no—
76#14579684—22no—
77C2×C163⋊C8C2311yescharacteristic
78C2×C163⋊C8C2311yescharacteristic
79C4⋊Dic163—21no—
80C4⋊Dic163C2311yescharacteristic
81C4⋊Dic163C2311yescharacteristic
82Dic163⋊C4—21no—
83Dic163⋊C4D411yescharacteristic
84Dic163⋊C4D411yescharacteristic
85C22⋊Dic163—21no—
86C22⋊Dic163—21no—
87(C2×D163)⋊C4—21no—
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4D411yescharacteristic
93(C2×D163)⋊C4D411yescharacteristic
94C4×Dic163—21no—
95C4×Dic163C2311yescharacteristic
96C2×D4×C163C2311yescharacteristic
97C2×D4×C163C2311yescharacteristic
98C22×C4×C163—21no—
99C22×C4×C163—21no—
100D8×C163—22no—
101C2×C8×C163C2311yescharacteristic
102C2×C8×C163C2311yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C4⋊C4×C163C2311yescharacteristic
105C22⋊C4×C163—21no—
106C22⋊C4×C163—21no—
107C42×C163C2311yescharacteristic
108C22×D163—21no—
109C22×D163—24no—
110C22×D163—21no—
111C22×D163—81no—
112C22×D163—81no—
113C22×D163C2×D411yescharacteristic
114D652—21no—
115D652—21no—
116D652—21no—
117D652—81no—
118D652#4112yes—
119C22⋊D163—24no—
120C22⋊D163—81no—
121C22⋊D163—81no—
122C22⋊D163—81no—
123C2×Dic163—21no—
124C2×Dic163—21no—
125C2×Dic163—81no—
126C2×Dic163C2×D411yescharacteristic
127C2×Dic163C2×D411yescharacteristic
128C2×Dic163C2×D411yescharacteristic
129C4×D163—21no—
130C4×D163—21no—
131C4×D163—81no—
132C4×D163—81no—
133C4×D163—81no—
134C163⋊C8—21no—
135C163⋊C8—21no—
136C23×C163—21no—
137C23×C163—21no—
138D4×C163—21no—
139D4×C163—21no—
140D4×C163—21no—
141D4×C163#4112yes—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—81no—
145C2×C4×C163—81no—
146C2×C4×C163C2×D411yescharacteristic
147C2×C4×C163C2×D411yescharacteristic
148C2×C4×C163C2411yescharacteristic
149C8×C163—21no—
150C8×C163—21no—
151C22×C163—22no—
152C22×C163—41no—
153C22×C163—41no—
154C22×C163—41no—
155C22×C163—41no—
156C22×C163—41no—
157C22×C163C22×D411yescharacteristic
158C2×D163—21no—
159C2×D163—22no—
160C2×D163—21no—
161C2×D163—21no—
162C2×D163—41no—
163C2×D163—41no—
164C2×D163—41no—
165C2×D163—41no—
166C2×D163—41no—
167C2×D163—84no—
168C4×C163—21no—
169C4×C163—21no—
170C4×C163—41no—
171C4×C163—41no—
172C4×C163C2×#4111yescharacteristic
173C4×C163C2×#4111yescharacteristic
174Dic163—21no—
175Dic163—21no—
176Dic163—21no—
177Dic163—41no—
178Dic163—41no—
179C2×C163—42no—
180C2×C163—81no—
181C2×C163#52511yescharacteristic
182C2×C163#57611yescharacteristic
183C2×C163#57611yescharacteristic
184D163—41no—
185D163—42no—
186D163—81no—
187C163#312211yescharacteristic, complemented, Sylow
188#3122—1631nomaximal, Sylow
189#540—1631no—
190#518—1631no—
191#482—1631no—
192#482—1631no—
193#478—1631no—
194#478—1631no—
195#459—1631no—
196#459—1631no—
197#447—1631no—
198#447—1631no—
199#442—1631no—
200#442—1631no—
201C4×D8—1631no—
202C4×D8D16311yescharacteristic, complemented
203#433—1631no—
204C22×D4—1631no—
205C2×D8—1631no—
206C2×D8—1631no—
207C2×D8—1631no—
208C2×D8C2×D16311yescharacteristic
209#127—1631no—
210#125—1631no—
211#123—3261no—
212#123—3261no—
213#121—3261no—
214#121—3261no—
215#121—1631no—
216#121—1631no—
217C4×D4—3261no—
218C4×D4—3261no—
219C4×D4—1631no—
220C4×D4—1631no—
221C4×D4—1631no—
222C4×D4—1631no—
223C4×D4C2×D16311yescharacteristic
224C4×D4C2×D16311yescharacteristic, complemented
225#117—1631no—
226#117—1631no—
227C2×C22⋊C4—3261no—
228C2×C22⋊C4—3261no—
229#107—1631no—
230#107C2×D16311yescharacteristic, complemented
231#106—1631no—
232#106—1631no—
233C4⋊C8—1631no—
234C4⋊C8—1631no—
235#102—1631no—
236#102—1631no—
237#102—1631no—
238#102—1631no—
239#102—1631no—
240#102—1631no—
241#102C2×D16311yescharacteristic
242#102C2×D16311yescharacteristic, complemented
243C22⋊C8—1631no—
244C22⋊C8—1631no—
245#97—1631no—
246C4×C8C2×D16311yescharacteristic, complemented
247C24—3261no—
248C2×D4—3261no—
249C2×D4—3262no—
250C2×D4—3261no—
251C2×D4—3262no—
252C2×D4—3261no—
253C2×D4—3261no—
254C2×D4—3261no—
255C2×D4—1631no—
256C2×D4—1631no—
257C2×D4C22×D16311yescharacteristic
258C2×D4C22×D16311yescharacteristic
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—3261no—
262C22×C4—3261no—
263C22×C4—3261no—
264C22×C4—3261no—
265C22×C4—1631no—
266D8—22no—
267D8—3262no—
268D8—3262no—
269D8—3262no—
270C2×C8—1631no—
271C2×C8—1631no—
272C2×C8C22×D16311yescharacteristic
273C2×C8C22×D16311yescharacteristic
274C4⋊C4—3261no—
275C4⋊C4—3261no—
276C4⋊C4—1631no—
277C4⋊C4—1631no—
278C4⋊C4—1631no—
279C4⋊C4—1631no—
280C4⋊C4C22×D16311yescharacteristic
281C4⋊C4C22×D16311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—3261no—
285C22⋊C4—3261no—
286C22⋊C4—3261no—
287C22⋊C4—3261no—
288C22⋊C4—3261no—
289C22⋊C4—3261no—
290C22⋊C4—3261no—
291C22⋊C4—1631no—
292C22⋊C4—1631no—
293C42—3261no—
294C42—1631no—
295C42C22×D16311yescharacteristic
296C23—21no—
297C23—21no—
298C23—13041no—
299C23—13041no—
300C23—3264no—
301C23—3261no—
302C23—3261no—
303C23—1631no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—13041no—
308D4—13041no—
309D4—13041no—
310D4—13041no—
311D4—3264no—
312D4—3261no—
313D4—3261no—
314D4—3261no—
315D4—1632no—
316D4#1457969912yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—13041no—
322C2×C4—13041no—
323C2×C4—13041no—
324C2×C4—13041no—
325C2×C4—3261no—
326C2×C4—3261no—
327C2×C4—3261no—
328C2×C4—3261no—
329C2×C4—1631no—
330C2×C4—1631no—
331C2×C4—1631no—
332C2×C4D4×D16311yescharacteristic
333C2×C4D4×D16311yescharacteristic
334C2×C4C23×D16311yescharacteristic
335C8—21no—
336C8—21no—
337C8—3261no—
338C8—3261no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—13044no—
346C22—3261no—
347C22—3262no—
348C22—3261no—
349C22—3261no—
350C22—6521no—
351C22—6521no—
352C22—6521no—
353C22—6521no—
354C22—6521no—
355C22C2×D4×D16311yescharacteristic
356C4—21no—
357C4—21no—
358C4—41no—
359C4—41no—
360C4—3261no—
361C4—3261no—
362C4—3261no—
363C4—6521no—
364C4—6521no—
365C4#41×D16311yescharacteristic
366C4C2×#1457969911yescharacteristic
367C2—42no—
368C2—81no—
369C2—13041no—
370C2—6521no—
371C2—6522no—
372C2#2378217411yescharacteristic
373C2#2378246611yescharacteristic
374C2#2378246711yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC4×D8×C163C2
socleC22×C163C22×D4
hypercenterC4×D8D163
hypocenterC163#3122
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #23782266C2semidirect
2. #23782137C2semidirect
3. #23781992C2semidirect
4. #23781987C2semidirect
5. #23781968C2semidirect
6. #23781963C2semidirect
7. #23781859C2semidirect
8. #23781858C2semidirect
9. #23781800C2semidirect
10. #23781797C2semidirect
11. #23781775C2semidirect
12. #23781775C2semidirect
13. C4×D1304C2semidirect
14. #23781724C2semidirect
15. C4×D8×C163C2semidirect
20. #19104731C22semidirect
21. #19104726C22semidirect
29. C4×D652C22semidirect
38. #19104669C22semidirect
39. #19104666C22semidirect
40. #19104666C22semidirect
41. Dic163⋊C8C22semidirect
42. Dic163⋊C8C22semidirect
44. #19104657C22semidirect
46. #19104656C22semidirect
48. #19104655C22semidirect
51. #19104646C22semidirect
54. C4×D4×C163C22semidirect
55. #107×C163C22semidirect
57. #102×C163C22semidirect
58. C4×C8×C163C22semidirect
187. C163#3122semidirect
202. C4×D8D163semidirect
224. C4×D4C2×D163semidirect
230. #107C2×D163semidirect
242. #102C2×D163semidirect
246. C4×C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237822662
#237821372
#237819922
#237819872
#237819682
#237819632
#237818592
#237818582
#237818002
#237817972
#237817752
C4×D13042
#237817242
C4×D8×C1632
#3122163