Finite Group Databasebeta

← #35226021

Subgroups — #35226021

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×#19104763C211yesmaximal, characteristic, complemented
2C22×D1304C211yesmaximal, characteristic, complemented
3#23782311C211yesmaximal, characteristic, complemented
4D4×D652C211yesmaximal, characteristic, complemented
5#23782054C211yesmaximal, characteristic, complemented
6#23781918C211yesmaximal, characteristic, complemented
7#23781894C211yesmaximal, characteristic, complemented
8#23781881C211yesmaximal, characteristic, complemented
9#23781861C211yesmaximal, characteristic, complemented
10#23781818C211yesmaximal, characteristic, complemented
11#23781812C211yesmaximal, characteristic, complemented
12#23781801C211yesmaximal, characteristic, complemented
13#23781801C211yesmaximal, characteristic, complemented
14#23781778C211yesmaximal, characteristic, complemented
15#435×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic, complemented
17C22×D652C2211yescharacteristic
18C22×D652C2211yescharacteristic, complemented
19C22×D652C2211yescharacteristic, complemented
20C2×D4×D163—21no—
21#19104763—24no—
22C2×#14579688C2211yescharacteristic, complemented
23C2×#14579688C2211yescharacteristic, complemented
24C2×#14579688C2211yescharacteristic, complemented
25C2×D1304—24no—
26C2×D1304C2211yescharacteristic
27C2×D1304C2211yescharacteristic
28C2×D1304C2211yescharacteristic, complemented
29C2×D1304C2211yescharacteristic, complemented
30C2×D1304C2211yescharacteristic, complemented
31C2×D1304C2211yescharacteristic, complemented
32#19104732C2211yescharacteristic, complemented
33#19104723C2211yescharacteristic, complemented
34#19104720—21no—
35#19104714C2211yescharacteristic, complemented
36#19104710—21no—
37#19104709C2211yescharacteristic
38#19104709C2211yescharacteristic, complemented
39#19104709C2211yescharacteristic, complemented
40#19104708—21no—
41#19104705—21no—
42#19104705—21no—
43#19104703—21no—
44C4×D652C2211yescharacteristic, complemented
45C4×D652C2211yescharacteristic, complemented
46C22⋊C4×D163—21no—
47#19104667C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104657C2211yescharacteristic, complemented
50#19104657C2211yescharacteristic, complemented
51#19104657C2211yescharacteristic, complemented
52#19104657C2211yescharacteristic, complemented
53#19104657C2211yescharacteristic, complemented
54C2×M16×C163C2211yescharacteristic, complemented
55C22×C8×C163C2211yescharacteristic, complemented
56C4×D4×C163C2211yescharacteristic
57C4⋊C8×C163C2211yescharacteristic, complemented
58C22⋊C8×C163C2211yescharacteristic, complemented
59C22⋊C8×C163C2211yescharacteristic, complemented
60#97×C163C2211yescharacteristic, complemented
61C23×D163—21no—
62C23×D163—21no—
63C23×D163—21no—
64#14579699—22no—
65#14579699—41no—
66C2×Dic326C2311yescharacteristic
67C2×D652—22no—
68C2×D652—22no—
69C2×D652—22no—
70C2×D652—21no—
71C2×D652—21no—
72C2×D652—21no—
73C2×D652—21no—
74C2×D652—21no—
75C2×D652—41no—
76C2×D652—41no—
77C2×D652—41no—
78C2×D652C2311yescharacteristic
79C2×D652C2311yescharacteristic
80C2×D652C2311yescharacteristic
81C2×D652C2311yescharacteristic
82C2×D652C2311yescharacteristic
83C2×C22⋊D163—21no—
84C2×C22⋊D163—21no—
85C2×C22⋊D163—21no—
86C2×C22⋊D163—21no—
87D4×D163—44no—
88C2×C4×D163—21no—
89C2×C4×D163—21no—
90C2×C4×D163—21no—
91#14579688—22no—
92#14579688—22no—
93#14579688—41no—
94D1304—22no—
95D1304—22no—
96D1304—22no—
97D1304—22no—
98D1304—41no—
99D1304D414yescomplemented
100C4⋊Dic163C2311yescharacteristic
101C4⋊Dic163C2311yescharacteristic
102Dic163⋊C4—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)⋊C4—21no—
110(C2×D163)⋊C4—21no—
111C2×D4×C163D411yescharacteristic
112C22×C4×C163C2311yescharacteristic
113C22×C4×C163C2311yescharacteristic
114M16×C163—22no—
115C2×C8×C163—22no—
116C2×C8×C163C2311yescharacteristic
117C2×C8×C163C2311yescharacteristic
118C2×C8×C163C2311yescharacteristic
119C2×C8×C163C2311yescharacteristic
120C4⋊C4×C163D411yescharacteristic
121C22⋊C4×C163D411yescharacteristic
122C22⋊C4×C163D411yescharacteristic
123C42×C163C2311yescharacteristic
124C22×D163—21no—
125C22×D163—21no—
126C22×D163—21no—
127C22×D163—21no—
128C22×D163—21no—
129C22×D163—42no—
130C22×D163—42no—
131C22×D163—42no—
132C22×D163—42no—
133C22×D163—42no—
134C22×D163—42no—
135C22×D163—81no—
136C22×D163—81no—
137C22×D163—81no—
138Dic326—21no—
139Dic326—21no—
140D652—21no—
141D652—21no—
142D652—21no—
143D652—21no—
144D652—21no—
145D652—21no—
146D652—21no—
147D652—21no—
148D652—42no—
149D652—42no—
150D652—42no—
151D652—42no—
152D652—81no—
153D652C2×D412yes—
154D652C2×D412yescomplemented
155C22⋊D163—42no—
156C22⋊D163—42no—
157C22⋊D163—42no—
158C22⋊D163—81no—
159C2×Dic163—21no—
160C2×Dic163—21no—
161C2×Dic163—21no—
162C4×D163—42no—
163C4×D163—42no—
164C4×D163—81no—
165C23×C163C2×D411yescharacteristic
166C23×C163C2×D411yescharacteristic
167D4×C163—22no—
168C2×C4×C163—21no—
169C2×C4×C163—21no—
170C2×C4×C163—21no—
171C2×C4×C163—21no—
172C2×C4×C163C2×D411yescharacteristic
173C2×C4×C163C2×D411yescharacteristic
174C2×C4×C163C2×D411yescharacteristic
175C2×C4×C163C2×D411yescharacteristic
176C2×C4×C163C2411yescharacteristic
177C8×C163—21no—
178C8×C163—21no—
179C8×C163—21no—
180C8×C163C2×D412yescomplemented
181C22×C163—21no—
182C22×C163—21no—
183C22×C163—21no—
184C22×C163—21no—
185C22×C163—21no—
186C22×C163AGL(1,ℤ/8)12yescomplemented
187C22×C163C22×D411yescharacteristic
188C2×D163—22no—
189C2×D163—22no—
190C2×D163—41no—
191C2×D163—41no—
192C2×D163—41no—
193C2×D163—41no—
194C2×D163—41no—
195C2×D163—41no—
196C2×D163—41no—
197C2×D163—41no—
198C2×D163—41no—
199C2×D163—41no—
200C2×D163—41no—
201C2×D163—41no—
202C2×D163—41no—
203C2×D163—84no—
204C2×D163—84no—
205C2×D163—84no—
206C4×C163—21no—
207C4×C163—21no—
208C4×C163—21no—
209C4×C163—21no—
210C4×C163C22×D411yescharacteristic
211C4×C163H(5,𝔽2)11yescharacteristic
212Dic163—41no—
213Dic163—41no—
214Dic163—41no—
215C2×C163—22no—
216C2×C163—41no—
217C2×C163D4211yescharacteristic
218C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C163#57611yescharacteristic
220D163—42no—
221D163—42no—
222D163—81no—
223D163—81no—
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#459—1631no—
236#450—1631no—
237#449—1631no—
238#447—1631no—
239#447—1631no—
240#442—1631no—
241#435D16311yescharacteristic, complemented
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×D8—1631no—
257C2×D8—1631no—
258C2×M16C2×D16311yescharacteristic, complemented
259C22×C8C2×D16311yescharacteristic, complemented
260#127—1631no—
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
275C2×C22⋊C4—3261no—
276#106—1631no—
277C4⋊C8C2×D16311yescharacteristic, complemented
278#103—1631no—
279#102—1631no—
280#102—1631no—
281#102—1631no—
282#102—1631no—
283#102—1631no—
284C22⋊C8C2×D16311yescharacteristic, complemented
285C22⋊C8C2×D16311yescharacteristic, complemented
286#97C2×D16311yescharacteristic, complemented
287C24—3261no—
288C24—3261no—
289C24—3261no—
290#41—3262no—
291#41—6521no—
292C2×Q8—1631no—
293C2×D4—3262no—
294C2×D4—3262no—
295C2×D4—3262no—
296C2×D4—3261no—
297C2×D4—3261no—
298C2×D4—3261no—
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×D4—1631no—
313C2×D4—1631no—
314C2×D4D65211yescharacteristic
315C22×C4—3261no—
316C22×C4—3261no—
317C22×C4—3261no—
318C22×C4C22×D16311yescharacteristic
319C22×C4C22×D16311yescharacteristic
320SD16—3262no—
321SD16—3262no—
322SD16—6521no—
323D8—3262no—
324D8—3262no—
325D8—3262no—
326D8—3262no—
327D8—6521no—
328D8—1634no—
329M16—22no—
330C2×C8—22no—
331C2×C8C22×D16311yescharacteristic
332C2×C8C22×D16311yescharacteristic
333C2×C8C22×D16311yescharacteristic
334C2×C8C22×D16311yescharacteristic
335C4⋊C4—3261no—
336C4⋊C4—1631no—
337C4⋊C4—1631no—
338C4⋊C4D65211yescharacteristic
339C22⋊C4—3261no—
340C22⋊C4—3261no—
341C22⋊C4—3261no—
342C22⋊C4—3261no—
343C22⋊C4—3261no—
344C22⋊C4—3261no—
345C22⋊C4—3261no—
346C22⋊C4—3261no—
347C22⋊C4D65211yescharacteristic
348C22⋊C4D65211yescharacteristic
349C42C22×D16311yescharacteristic
350C23—13041no—
351C23—13041no—
352C23—13041no—
353C23—3261no—
354C23—3261no—
355C23—3261no—
356C23—3261no—
357C23—3261no—
358C23—6522no—
359C23—6522no—
360C23—6522no—
361C23—6522no—
362C23—6522no—
363C23—6522no—
364C23C2×D65211yescharacteristic
365C23C2×D65211yescharacteristic
366Q8—3261no—
367Q8—3261no—
368D4—22no—
369D4—13041no—
370D4—13041no—
371D4—3261no—
372D4—3261no—
373D4—3261no—
374D4—3261no—
375D4—3261no—
376D4—3261no—
377D4—3261no—
378D4—3261no—
379D4—6522no—
380D4—6522no—
381D4—6522no—
382D4—6522no—
383D4—6522no—
384D4—6522no—
385D4—6522no—
386D4—1632no—
387D4—1632no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—13041no—
393C2×C4—3261no—
394C2×C4—3261no—
395C2×C4—3261no—
396C2×C4—6522no—
397C2×C4—6522no—
398C2×C4C2×D65211yescharacteristic
399C2×C4C2×D65211yescharacteristic
400C2×C4C2×D65211yescharacteristic
401C2×C4C2×D65211yescharacteristic
402C2×C4C23×D16311yescharacteristic
403C8—21no—
404C8—21no—
405C8—21no—
406C8D4×D16312yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—13044no—
413C22—13044no—
414C22—13044no—
415C22—3262no—
416C22—3262no—
417C22—6521no—
418C22—6521no—
419C22—6521no—
420C22—6521no—
421C22—6521no—
422C22—6521no—
423C22—6521no—
424C22—6521no—
425C22—6521no—
426C22—6521no—
427C22—6521no—
428C22—6521no—
429C22—6521no—
430C22#1910476312yescomplemented
431C22C22×D65211yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—6521no—
437C4—6521no—
438C4—6521no—
439C4C2×D4×D16311yescharacteristic
440C4#1910478111yescharacteristic
441C2—22no—
442C2—41no—
443C2—13041no—
444C2—13041no—
445C2—13041no—
446C2—6522no—
447C2—6522no—
448C2D4×D65211yescharacteristic
449C2C2×#1910476311yescharacteristic
450C2#2378247011yescharacteristic
subgroupG/H
centerC22C22×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#435×C163C2
socleC22×C163C22×D4
hypercenter#435D163
hypocenterC163#3264
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#19104763C2semidirect
2. C22×D1304C2semidirect
3. #23782311C2semidirect
4. D4×D652C2semidirect
5. #23782054C2semidirect
6. #23781918C2semidirect
7. #23781894C2semidirect
8. #23781881C2semidirect
9. #23781861C2semidirect
10. #23781818C2semidirect
11. #23781812C2semidirect
12. #23781801C2semidirect
13. #23781801C2semidirect
14. #23781778C2semidirect
15. #435×C163C2semidirect
16. C2×#14579699C22semidirect
18. C22×D652C22semidirect
19. C22×D652C22semidirect
22. C2×#14579688C22semidirect
23. C2×#14579688C22semidirect
24. C2×#14579688C22semidirect
28. C2×D1304C22semidirect
29. C2×D1304C22semidirect
30. C2×D1304C22semidirect
31. C2×D1304C22semidirect
32. #19104732C22semidirect
33. #19104723C22semidirect
35. #19104714C22semidirect
38. #19104709C22semidirect
39. #19104709C22semidirect
44. C4×D652C22semidirect
45. C4×D652C22semidirect
47. #19104667C22semidirect
48. #19104660C22semidirect
49. #19104657C22semidirect
50. #19104657C22semidirect
51. #19104657C22semidirect
52. #19104657C22semidirect
53. #19104657C22semidirect
54. C2×M16×C163C22semidirect
55. C22×C8×C163C22semidirect
57. C4⋊C8×C163C22semidirect
58. C22⋊C8×C163C22semidirect
59. C22⋊C8×C163C22semidirect
60. #97×C163C22semidirect
99. D1304D4semidirect
154. D652C2×D4semidirect
180. C8×C163C2×D4semidirect
186. C22×C163AGL(1,ℤ/8)semidirect
225. C163#3264semidirect
241. #435D163semidirect
258. C2×M16C2×D163semidirect
259. C22×C8C2×D163semidirect
277. C4⋊C8C2×D163semidirect
284. C22⋊C8C2×D163semidirect
285. C22⋊C8C2×D163semidirect
286. #97C2×D163semidirect
406. C8D4×D163semidirect
430. C22#19104763semidirect

Maximal subgroups

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

subgroupindex
C2×#191047632
C22×D13042
#237823112
D4×D6522
#237820542
#237819182
#237818942
#237818812
#237818612
#237818182
#237818122
#237818012
#237817782
#435×C1632
#3264163