Finite Group Databasebeta

← #35224773

Subgroups — #35224773

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#23782268C211yesmaximal, characteristic, complemented
2#23782140C211yesmaximal, characteristic, complemented
3#23781990C211yesmaximal, characteristic, complemented
4#23781986C211yesmaximal, characteristic, complemented
5#23781967C211yesmaximal, characteristic, complemented
6#23781963C211yesmaximal, characteristic, complemented
7#23781858C211yesmaximal, characteristic, complemented
8#23781858C211yesmaximal, characteristic, complemented
9#23781799C211yesmaximal, characteristic, complemented
10#23781777C211yesmaximal, characteristic, complemented
11#23781774C211yesmaximal, characteristic, complemented
12D8×Dic163C211yesmaximal, characteristic, complemented
13#23781745C211yesmaximal, characteristic, complemented
14#23781723C211yesmaximal, characteristic, complemented
15#447×C163C211yesmaximal, characteristic, complemented
16C2×#14579684C2211yescharacteristic
17C2×#14579684C2211yescharacteristic
18#19104731C2211yescharacteristic, complemented
19#19104726C2211yescharacteristic, complemented
20#19104719—21no—
21#19104717—21no—
22#19104709C2211yescharacteristic
23#19104707—21no—
24#19104706—21no—
25D4×Dic163—21no—
26D4×Dic163—21no—
27D4×Dic163C2211yescharacteristic
28D4×Dic163C2211yescharacteristic
29D4×Dic163C2211yescharacteristic
30D4×Dic163C2211yescharacteristic
31D4×Dic163C2211yescharacteristic, complemented
32#19104697C2211yescharacteristic, complemented
33#19104694C2211yescharacteristic
34#19104694C2211yescharacteristic
35C2×C22⋊Dic163—21no—
36C2×C22⋊Dic163—21no—
37#19104669C2211yescharacteristic, complemented
38#19104668C2211yescharacteristic, complemented
39#19104667C2211yescharacteristic, complemented
40#19104666C2211yescharacteristic, complemented
41Dic163⋊C8C2211yescharacteristic, complemented
42Dic163⋊C8C2211yescharacteristic, complemented
43#19104657C2211yescharacteristic, complemented
44#19104657C2211yescharacteristic, complemented
45#19104656C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104656C2211yescharacteristic, complemented
48#19104656C2211yescharacteristic, complemented
49C22⋊(C163⋊C8)C2211yescharacteristic
50#19104645C2211yescharacteristic, complemented
51C8×Dic163C2211yescharacteristic, complemented
52C22×D4×C163C2211yescharacteristic
53C2×D8×C163C2211yescharacteristic
54C2×D8×C163C2211yescharacteristic
55#121×C163C2211yescharacteristic
56#102×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic
59C2×D652C2311yescharacteristic
60C2×C22⋊D163—21no—
61C2×C22⋊D163—21no—
62C2×C22⋊D163—21no—
63C22×Dic163—21no—
64C22×Dic163—21no—
65C22×Dic163—21no—
66C22×Dic163—21no—
67C22×Dic163—21no—
68C2×C4×D163—21no—
69#14579684—22no—
70#14579684—22no—
71C2×C163⋊C8C2311yescharacteristic
72C2×C163⋊C8C2311yescharacteristic
73C4⋊Dic163—21no—
74C4⋊Dic163D411yescharacteristic
75C4⋊Dic163D411yescharacteristic
76C4⋊Dic163C2311yescharacteristic
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79Dic163⋊C4—21no—
80C22⋊Dic163—21no—
81C22⋊Dic163—21no—
82C22⋊Dic163—21no—
83C22⋊Dic163—21no—
84C22⋊Dic163—21no—
85C22⋊Dic163—21no—
86C22⋊Dic163D411yescharacteristic
87C22⋊Dic163D411yescharacteristic
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90C4×Dic163—21no—
91C4×Dic163C2311yescharacteristic
92C4×Dic163C2311yescharacteristic
93C24×C163—21no—
94C2×D4×C163—22no—
95C2×D4×C163—22no—
96C2×D4×C163—21no—
97C2×D4×C163—21no—
98C2×D4×C163C2311yescharacteristic
99C2×D4×C163C2311yescharacteristic
100C2×D4×C163C2311yescharacteristic
101C22×C4×C163C2311yescharacteristic
102D8×C163—22no—
103D8×C163—22no—
104C2×C8×C163C2311yescharacteristic
105C2×C8×C163C2311yescharacteristic
106C4⋊C4×C163C2311yescharacteristic
107C22⋊C4×C163—21no—
108C22×D163—21no—
109D652—21no—
110D652—21no—
111C22⋊D163—81no—
112C22⋊D163—81no—
113C22⋊D163—81no—
114C2×Dic163—21no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—81no—
118C2×Dic163—81no—
119C2×Dic163—81no—
120C2×Dic163—81no—
121C2×Dic163—81no—
122C2×Dic163C2×D411yescharacteristic
123C2×Dic163C2×D411yescharacteristic
124C2×Dic163C2×D411yescharacteristic
125C2×Dic163C2×D411yescharacteristic
126C4×D163—81no—
127C163⋊C8—21no—
128C163⋊C8—21no—
129C23×C163—24no—
130C23×C163—21no—
131C23×C163—21no—
132C23×C163—21no—
133C23×C163—81no—
134C23×C163—81no—
135C23×C163C2×D411yescharacteristic
136D4×C163—24no—
137D4×C163—21no—
138D4×C163—21no—
139D4×C163—21no—
140D4×C163—21no—
141D4×C163—81no—
142D4×C163#4112yes—
143D4×C163#4112yes—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163C2×D411yescharacteristic
148C2×C4×C163C2411yescharacteristic
149C8×C163—21no—
150C8×C163—21no—
151C22×C163—21no—
152C22×C163—22no—
153C22×C163—22no—
154C22×C163—21no—
155C22×C163—21no—
156C22×C163—41no—
157C22×C163—41no—
158C22×C163—41no—
159C22×C163—41no—
160C22×C163—41no—
161C22×C163—41no—
162C22×C163—41no—
163C22×C163—84no—
164C22×C163C22×D411yescharacteristic
165C2×D163—41no—
166C2×D163—41no—
167C2×D163—41no—
168C4×C163—21no—
169C4×C163—41no—
170C4×C163C2×#4111yescharacteristic
171C4×C163C2×#4111yescharacteristic
172Dic163—21no—
173Dic163—21no—
174Dic163—21no—
175Dic163—21no—
176Dic163—41no—
177Dic163—41no—
178Dic163—41no—
179C2×C163—41no—
180C2×C163—42no—
181C2×C163—42no—
182C2×C163—81no—
183C2×C163#52511yescharacteristic
184C2×C163#57611yescharacteristic
185C2×C163#57611yescharacteristic
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#447D16311yescharacteristic, complemented
199#442—1631no—
200#442—1631no—
201C4×D8—1631no—
202C4×D8—1631no—
203#433—1631no—
204C22×D4C2×D16311yescharacteristic
205C2×D8—1631no—
206C2×D8—1631no—
207C2×D8C2×D16311yescharacteristic
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#121C2×D16311yescharacteristic
217C4×D4—3261no—
218C4×D4—3261no—
219C4×D4—1631no—
220C4×D4—1631no—
221C4×D4—1631no—
222C4×D4—1631no—
223C4×D4—1631no—
224C4×D4—1631no—
225#117—1631no—
226#117—1631no—
227C2×C22⋊C4—3261no—
228C2×C22⋊C4—3261no—
229#107—1631no—
230#107—1631no—
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
243C22⋊C8—1631no—
244C22⋊C8C2×D16311yescharacteristic
245#97—1631no—
246C4×C8—1631no—
247C24—21no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—3261no—
253C2×D4—3261no—
254C2×D4—3261no—
255C2×D4—1631no—
256C2×D4C22×D16311yescharacteristic
257C2×D4C22×D16311yescharacteristic
258C2×D4C22×D16311yescharacteristic
259C22×C4—3261no—
260C22×C4—3261no—
261C22×C4—3261no—
262C22×C4—3261no—
263C22×C4—3261no—
264C22×C4—3261no—
265C22×C4C22×D16311yescharacteristic
266D8—22no—
267D8—22no—
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⋊C4—1631no—
281C4⋊C4C22×D16311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—3261no—
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—
295C42—1631no—
296C23—24no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—81no—
301C23—81no—
302C23—3261no—
303C23D4×D16311yescharacteristic
304D4—24no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—81no—
310D4—13041no—
311D4—13041no—
312D4—13041no—
313D4—3261no—
314D4—3261no—
315D4#1457969812yes—
316D4#1457969812yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—13041no—
321C2×C4—13041no—
322C2×C4—13041no—
323C2×C4—13041no—
324C2×C4—13041no—
325C2×C4—13041no—
326C2×C4—3261no—
327C2×C4—3261no—
328C2×C4—3261no—
329C2×C4—1631no—
330C2×C4—1631no—
331C2×C4—1631no—
332C2×C4—1631no—
333C2×C4D4×D16311yescharacteristic
334C2×C4C23×D16311yescharacteristic
335C8—21no—
336C8—21no—
337C8—3261no—
338C8—3261no—
339C22—21no—
340C22—22no—
341C22—22no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—84no—
352C22—6521no—
353C22—6521no—
354C22—6521no—
355C22C2×D4×D16311yescharacteristic
356C4—21no—
357C4—41no—
358C4—3261no—
359C4—3261no—
360C4—3261no—
361C4—3261no—
362C4—6521no—
363C4—6521no—
364C4—6521no—
365C4C2×#1457969811yescharacteristic
366C4C2×#1457969811yescharacteristic
367C2—41no—
368C2—42no—
369C2—42no—
370C2—81no—
371C2—13041no—
372C2#2378217111yescharacteristic
373C2#2378246611yescharacteristic
374C2#2378246911yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#447×C163C2
socleC22×C163C22×D4
hypercenter#447D163
hypocenterC163#3122
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #23782268C2semidirect
2. #23782140C2semidirect
3. #23781990C2semidirect
4. #23781986C2semidirect
5. #23781967C2semidirect
6. #23781963C2semidirect
7. #23781858C2semidirect
8. #23781858C2semidirect
9. #23781799C2semidirect
10. #23781777C2semidirect
11. #23781774C2semidirect
12. D8×Dic163C2semidirect
13. #23781745C2semidirect
14. #23781723C2semidirect
15. #447×C163C2semidirect
18. #19104731C22semidirect
19. #19104726C22semidirect
31. D4×Dic163C22semidirect
32. #19104697C22semidirect
37. #19104669C22semidirect
38. #19104668C22semidirect
39. #19104667C22semidirect
40. #19104666C22semidirect
41. Dic163⋊C8C22semidirect
42. Dic163⋊C8C22semidirect
43. #19104657C22semidirect
44. #19104657C22semidirect
47. #19104656C22semidirect
48. #19104656C22semidirect
50. #19104645C22semidirect
51. C8×Dic163C22semidirect
187. C163#3122semidirect
198. #447D163semidirect

Maximal subgroups

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

subgroupindex
#237822682
#237821402
#237819902
#237819862
#237819672
#237819632
#237818582
#237817992
#237817772
#237817742
D8×Dic1632
#237817452
#237817232
#447×C1632
#3122163