Finite Group Databasebeta

← #35226011

Subgroups — #35226011

Order 20864.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104759C211yesmaximal, characteristic, complemented
2C2×D8×D163C211yesmaximal, characteristic, complemented
3#23782310C211yesmaximal, characteristic, complemented
4#23782300C211yesmaximal, characteristic, complemented
5#23782052C211yesmaximal, characteristic, complemented
6#23781917C211yesmaximal, characteristic, complemented
7#23781893C211yesmaximal, characteristic, complemented
8#23781881C211yesmaximal, characteristic, complemented
9#23781817C211yesmaximal, characteristic, complemented
10#23781807C211yesmaximal, characteristic, complemented
11#23781800C211yesmaximal, characteristic, complemented
12#23781797C211yesmaximal, characteristic, complemented
13#23781775C211yesmaximal, characteristic, complemented
14#23781735C211yesmaximal, characteristic, complemented
15#459×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic, complemented
17C2×D4×D163—21no—
18C2×D4×D163C2211yescharacteristic
19C2×D4×D163C2211yescharacteristic, complemented
20C2×D4×D163C2211yescharacteristic, complemented
21#19104759—24no—
22C2×#14579688C2211yescharacteristic, complemented
23C2×#14579687C2211yescharacteristic, complemented
24C2×#14579687C2211yescharacteristic, complemented
25C2×D1304C2211yescharacteristic
26C2×#14579684C2211yescharacteristic
27C2×#14579684C2211yescharacteristic, complemented
28C2×#14579684C2211yescharacteristic, complemented
29D8×D163—24no—
30C2×#14579682C2211yescharacteristic, complemented
31C2×C8×D163C2211yescharacteristic, complemented
32#19104723C2211yescharacteristic, complemented
33#19104719—21no—
34#19104715C2211yescharacteristic, complemented
35#19104710C2211yescharacteristic, complemented
36#19104709—21no—
37#19104706—21no—
38#19104706C2211yescharacteristic
39#19104706C2211yescharacteristic, complemented
40#19104705—21no—
41#19104703—21no—
42#19104703—21no—
43C4×D652C2211yescharacteristic
44C4×D652C2211yescharacteristic, complemented
45C2×(C2×D163)⋊C4—21no—
46#19104666C2211yescharacteristic, complemented
47#19104660C2211yescharacteristic, complemented
48#19104657C2211yescharacteristic, complemented
49#19104656C2211yescharacteristic, complemented
50#19104655C2211yescharacteristic, complemented
51(C2×D163)⋊C8C2211yescharacteristic, complemented
52(C2×D163)⋊C8C2211yescharacteristic, complemented
53#19104646C2211yescharacteristic, complemented
54C2×D8×C163C2211yescharacteristic, complemented
55C2×D8×C163C2211yescharacteristic, complemented
56#127×C163C2211yescharacteristic, complemented
57C4×D4×C163C2211yescharacteristic, complemented
58C4⋊C8×C163C2211yescharacteristic, complemented
59#102×C163C2211yescharacteristic, complemented
60#102×C163C2211yescharacteristic, complemented
61C23×D163—21no—
62C23×D163—21no—
63C23×D163—21no—
64#14579698—22no—
65#14579698—41no—
66C2×Dic326C2311yescharacteristic
67C2×D652—21no—
68C2×D652D411yescharacteristic
69C2×D652C2311yescharacteristic
70C2×D652C2311yescharacteristic
71C2×C22⋊D163—21no—
72C2×C22⋊D163—21no—
73C2×C22⋊D163—21no—
74C2×C22⋊D163—21no—
75C2×C22⋊D163—21no—
76C2×C22⋊D163—21no—
77D4×D163—22no—
78D4×D163—22no—
79D4×D163—22no—
80D4×D163—41no—
81D4×D163—41no—
82D4×D163—41no—
83D4×D163—44no—
84C22×Dic163—21no—
85C2×C4×D163—21no—
86C2×C4×D163C2311yescharacteristic
87C2×C4×D163C2311yescharacteristic
88#14579688—22no—
89#14579687—22no—
90#14579687—41no—
91D1304—22no—
92#14579684—22no—
93#14579684—41no—
94#14579684D414yescomplemented
95#14579682—22no—
96C2×C163⋊C8C2311yescharacteristic
97C2×C163⋊C8C2311yescharacteristic
98C8×D163—22no—
99C4⋊Dic163—21no—
100C4⋊Dic163D411yescharacteristic
101C4⋊Dic163C2311yescharacteristic
102C22⋊Dic163—21no—
103C22⋊Dic163—21no—
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4—21no—
106(C2×D163)⋊C4—21no—
107(C2×D163)⋊C4—21no—
108(C2×D163)⋊C4—21no—
109(C2×D163)⋊C4D411yescharacteristic
110(C2×D163)⋊C4D411yescharacteristic
111C2×D4×C163—21no—
112C2×D4×C163—21no—
113C2×D4×C163C2311yescharacteristic
114C2×D4×C163C2311yescharacteristic
115C2×D4×C163C2311yescharacteristic
116C22×C4×C163—21no—
117D8×C163—22no—
118D8×C163—22no—
119C2×C8×C163C2311yescharacteristic
120C2×C8×C163C2311yescharacteristic
121C4⋊C4×C163C2311yescharacteristic
122C22⋊C4×C163—21no—
123C42×C163C2311yescharacteristic
124C22×D163—21no—
125C22×D163—21no—
126C22×D163—42no—
127C22×D163—42no—
128C22×D163—42no—
129C22×D163—42no—
130C22×D163—42no—
131C22×D163—42no—
132C22×D163—81no—
133C22×D163—81no—
134C22×D163—81no—
135C22×D163C2×D411yescharacteristic
136C22×D163C2×D411yescharacteristic
137Dic326—21no—
138Dic326—21no—
139D652—22no—
140D652—21no—
141D652—21no—
142D652—21no—
143D652—81no—
144D652C2×D412yes—
145C22⋊D163—42no—
146C22⋊D163—42no—
147C22⋊D163—42no—
148C22⋊D163—42no—
149C22⋊D163—42no—
150C22⋊D163—42no—
151C2×Dic163—21no—
152C2×Dic163—21no—
153C2×Dic163—42no—
154C2×Dic163C2×D411yescharacteristic
155C2×Dic163C2×D411yescharacteristic
156C4×D163—21no—
157C4×D163—21no—
158C4×D163—21no—
159C4×D163—21no—
160C4×D163—42no—
161C163⋊C8—21no—
162C163⋊C8C2×D412yescomplemented
163C23×C163—21no—
164C23×C163—21no—
165C23×C163—21no—
166D4×C163—21no—
167D4×C163—21no—
168D4×C163—21no—
169D4×C163—21no—
170D4×C163—21no—
171D4×C163—42no—
172D4×C163—81no—
173D4×C163C2×D412yescomplemented
174C2×C4×C163—21no—
175C2×C4×C163—81no—
176C2×C4×C163C2×D411yescharacteristic
177C2×C4×C163C2×D411yescharacteristic
178C2×C4×C163C2411yescharacteristic
179C8×C163—21no—
180C8×C163—21no—
181C22×C163—22no—
182C22×C163—41no—
183C22×C163—41no—
184C22×C163—41no—
185C22×C163—41no—
186C22×C163—41no—
187C22×C163—41no—
188C22×C163—41no—
189C22×C163—41no—
190C22×C163C22×D411yescharacteristic
191C2×D163—21no—
192C2×D163—21no—
193C2×D163—21no—
194C2×D163—22no—
195C2×D163—21no—
196C2×D163—21no—
197C2×D163—41no—
198C2×D163—41no—
199C2×D163—41no—
200C2×D163—41no—
201C2×D163—41no—
202C2×D163—84no—
203C2×D163—84no—
204C2×D163—84no—
205C2×D163AGL(1,ℤ/8)12yescomplemented
206C4×C163—21no—
207C4×C163—21no—
208C4×C163—41no—
209C4×C163C22×D411yescharacteristic
210C4×C163H(5,𝔽2)11yescharacteristic
211Dic163—21no—
212Dic163—21no—
213Dic163—41no—
214Dic163—41no—
215C2×C163—42no—
216C2×C163—81no—
217C2×C163—81no—
218C2×C163D4211yescharacteristic
219C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C163#57611yescharacteristic
221D163—22no—
222D163—41no—
223D163—42no—
224D163—81no—
225C163#326411yescharacteristic, complemented, Sylow
226#3264—1631nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1631no—
228C22×D8—1631no—
229#546—1631no—
230D42—1631no—
231#496—1631no—
232#469—1631no—
233#465—1631no—
234#463—1631no—
235#459D16311yescharacteristic, complemented
236#450—1631no—
237#449—1631no—
238#447—1631no—
239#447—1631no—
240#442—1631no—
241#435—1631no—
242C2×#41—1631no—
243C22×D4—3261no—
244C22×D4—1631no—
245C22×D4—1631no—
246C22×D4—1631no—
247AGL(1,ℤ/8)—3264no—
248C2×SD16—1631no—
249C2×SD16—1631no—
250C2×SD16—1631no—
251C2×D8—3264no—
252C2×D8—1631no—
253C2×D8—1631no—
254C2×D8—1631no—
255C2×D8—1631no—
256C2×D8C2×D16311yescharacteristic, complemented
257C2×D8C2×D16311yescharacteristic, complemented
258C2×M16—1631no—
259C22×C8—1631no—
260#127C2×D16311yescharacteristic, complemented
261#124—1631no—
262#123—3261no—
263C22⋊Q8—1631no—
264#121—3261no—
265#121—3261no—
266#121—1631no—
267#121—1631no—
268#121—1631no—
269#120—3261no—
270#120—3261no—
271#120—3261no—
272C4×D4—1631no—
273C4×D4—1631no—
274C4×D4C2×D16311yescharacteristic, complemented
275C2×C22⋊C4—3261no—
276#106—1631no—
277C4⋊C8C2×D16311yescharacteristic, complemented
278#103—1631no—
279#102—1631no—
280#102—1631no—
281#102—1631no—
282#102C2×D16311yescharacteristic, complemented
283#102C2×D16311yescharacteristic, complemented
284C22⋊C8—1631no—
285C22⋊C8—1631no—
286#97—1631no—
287C24—3261no—
288C24—3261no—
289C24—3261no—
290#41—3262no—
291#41—6521no—
292C2×Q8—1631no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—3261no—
296C2×D4—3262no—
297C2×D4—3262no—
298C2×D4—3262no—
299C2×D4—3261no—
300C2×D4—3261no—
301C2×D4—3261no—
302C2×D4—3261no—
303C2×D4—3261no—
304C2×D4—3261no—
305C2×D4—6521no—
306C2×D4—6521no—
307C2×D4—6521no—
308C2×D4—6524no—
309C2×D4—1631no—
310C2×D4—1631no—
311C2×D4—1631no—
312C2×D4C22×D16311yescharacteristic
313C2×D4C22×D16311yescharacteristic
314C2×D4C22×D16311yescharacteristic
315C22×C4—21no—
316C22×C4—3261no—
317C22×C4—3261no—
318C22×C4—1631no—
319C22×C4—1631no—
320SD16—3262no—
321SD16—3262no—
322SD16—6521no—
323D8—22no—
324D8—22no—
325D8—3262no—
326D8—3262no—
327D8—6521no—
328D8—1634no—
329M16—3262no—
330C2×C8—3262no—
331C2×C8—1631no—
332C2×C8—1631no—
333C2×C8C22×D16311yescharacteristic
334C2×C8C22×D16311yescharacteristic
335C4⋊C4—3261no—
336C4⋊C4—1631no—
337C4⋊C4—1631no—
338C4⋊C4C22×D16311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—3261no—
341C22⋊C4—3261no—
342C22⋊C4—3261no—
343C22⋊C4—3261no—
344C22⋊C4—3261no—
345C22⋊C4—3261no—
346C22⋊C4—3261no—
347C22⋊C4—1631no—
348C22⋊C4—1631no—
349C42C22×D16311yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—13041no—
354C23—13041no—
355C23—13041no—
356C23—3261no—
357C23—3261no—
358C23—6522no—
359C23—6522no—
360C23—6522no—
361C23—6522no—
362C23—6522no—
363C23—6522no—
364C23—1631no—
365C23—1631no—
366Q8—3261no—
367Q8—3261no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—42no—
374D4—81no—
375D4—13041no—
376D4—3262no—
377D4—3261no—
378D4—3261no—
379D4—3261no—
380D4—6522no—
381D4—6522no—
382D4—6522no—
383D4—6522no—
384D4—6522no—
385D4—6522no—
386D4—1632no—
387D4D4×D16312yescomplemented
388C2×C4—21no—
389C2×C4—81no—
390C2×C4—3261no—
391C2×C4—3261no—
392C2×C4—3261no—
393C2×C4—3261no—
394C2×C4—3261no—
395C2×C4—3261no—
396C2×C4—6522no—
397C2×C4—6522no—
398C2×C4—1631no—
399C2×C4—1631no—
400C2×C4D4×D16311yescharacteristic
401C2×C4D4×D16311yescharacteristic
402C2×C4C23×D16311yescharacteristic
403C8—21no—
404C8—21no—
405C8—3261no—
406C8—1632no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—13044no—
417C22—13044no—
418C22—13044no—
419C22—3261no—
420C22—3261no—
421C22—3261no—
422C22—3261no—
423C22—3261no—
424C22—3262no—
425C22—6521no—
426C22—6521no—
427C22—6521no—
428C22—6521no—
429C22—6521no—
430C22—1632no—
431C22C2×D4×D16311yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—3261no—
436C4—3261no—
437C4—6521no—
438C4—6521no—
439C4C2×D4×D16311yescharacteristic
440C4#1910478111yescharacteristic
441C2—42no—
442C2—81no—
443C2—81no—
444C2—13041no—
445C2—3262no—
446C2—6521no—
447C2—6522no—
448C2#2378229911yescharacteristic
449C2AGL(1,ℤ/8)×D16311yescharacteristic
450C2#2378246611yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#459×C163C2
socleC22×C163C22×D4
hypercenter#459D163
hypocenterC163#3264
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#19104759C2semidirect
2. C2×D8×D163C2semidirect
3. #23782310C2semidirect
4. #23782300C2semidirect
5. #23782052C2semidirect
6. #23781917C2semidirect
7. #23781893C2semidirect
8. #23781881C2semidirect
9. #23781817C2semidirect
10. #23781807C2semidirect
11. #23781800C2semidirect
12. #23781797C2semidirect
13. #23781775C2semidirect
14. #23781735C2semidirect
15. #459×C163C2semidirect
16. C2×#14579698C22semidirect
19. C2×D4×D163C22semidirect
20. C2×D4×D163C22semidirect
22. C2×#14579688C22semidirect
23. C2×#14579687C22semidirect
24. C2×#14579687C22semidirect
27. C2×#14579684C22semidirect
28. C2×#14579684C22semidirect
30. C2×#14579682C22semidirect
31. C2×C8×D163C22semidirect
32. #19104723C22semidirect
34. #19104715C22semidirect
35. #19104710C22semidirect
39. #19104706C22semidirect
44. C4×D652C22semidirect
46. #19104666C22semidirect
47. #19104660C22semidirect
48. #19104657C22semidirect
49. #19104656C22semidirect
50. #19104655C22semidirect
51. (C2×D163)⋊C8C22semidirect
52. (C2×D163)⋊C8C22semidirect
53. #19104646C22semidirect
54. C2×D8×C163C22semidirect
55. C2×D8×C163C22semidirect
56. #127×C163C22semidirect
57. C4×D4×C163C22semidirect
58. C4⋊C8×C163C22semidirect
59. #102×C163C22semidirect
60. #102×C163C22semidirect
94. #14579684D4semidirect
162. C163⋊C8C2×D4semidirect
173. D4×C163C2×D4semidirect
205. C2×D163AGL(1,ℤ/8)semidirect
225. C163#3264semidirect
235. #459D163semidirect
256. C2×D8C2×D163semidirect
257. C2×D8C2×D163semidirect
260. #127C2×D163semidirect
274. C4×D4C2×D163semidirect
277. C4⋊C8C2×D163semidirect
282. #102C2×D163semidirect
283. #102C2×D163semidirect
387. D4D4×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047592
C2×D8×D1632
#237823102
#237823002
#237820522
#237819172
#237818932
#237818812
#237818172
#237818072
#237818002
#237817972
#237817752
#237817352
#459×C1632
#3264163