Finite Group Databasebeta

← #35226016

Subgroups — #35226016

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
2#23782309C211yesmaximal, characteristic, complemented
3D4×C22⋊D163C211yesmaximal, characteristic, complemented
4#23782053C211yesmaximal, characteristic, complemented
5#23781915C211yesmaximal, characteristic, complemented
6#23781894C211yesmaximal, characteristic, complemented
7#23781878C211yesmaximal, characteristic, complemented
8#23781858C211yesmaximal, characteristic, complemented
9#23781816C211yesmaximal, characteristic, complemented
10#23781807C211yesmaximal, characteristic, complemented
11#23781799C211yesmaximal, characteristic, complemented
12#23781797C211yesmaximal, characteristic, complemented
13#23781777C211yesmaximal, characteristic, complemented
14#23781736C211yesmaximal, characteristic, complemented
15C22×D8×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic, complemented
17C22×C22⋊D163—21no—
18C2×D4×D163C2211yescharacteristic, complemented
19#19104759—24no—
20C2×#14579688C2211yescharacteristic, complemented
21C2×#14579687C2211yescharacteristic, complemented
22C2×#14579687C2211yescharacteristic, complemented
23C2×#14579684C2211yescharacteristic, complemented
24C2×#14579684C2211yescharacteristic, complemented
25C2×#14579682C2211yescharacteristic, complemented
26#19104731C2211yescharacteristic, complemented
27#19104722C2211yescharacteristic, complemented
28#19104716—21no—
29#19104714C2211yescharacteristic, complemented
30#19104709C2211yescharacteristic
31#19104707—21no—
32#19104707—21no—
33#19104706C2211yescharacteristic, complemented
34#19104706C2211yescharacteristic, complemented
35#19104704—21no—
36#19104704—21no—
37#19104703—21no—
38D4×Dic163C2211yescharacteristic, complemented
39D4×Dic163C2211yescharacteristic, complemented
40C4×C22⋊D163C2211yescharacteristic
41C2×C22⋊Dic163—21no—
42#19104667C2211yescharacteristic, complemented
43Dic163⋊C8C2211yescharacteristic, complemented
44#19104660C2211yescharacteristic, complemented
45#19104657C2211yescharacteristic, complemented
46#19104656C2211yescharacteristic, complemented
47#19104656C2211yescharacteristic, complemented
48#19104656C2211yescharacteristic, complemented
49#19104656C2211yescharacteristic, complemented
50C22⋊(C163⋊C8)C2211yescharacteristic, complemented
51(C2×D163)⋊C8C2211yescharacteristic, complemented
52#19104645C2211yescharacteristic, complemented
53C22×D4×C163C2211yescharacteristic
54C22×D4×C163C2211yescharacteristic, complemented
55C2×D8×C163—24no—
56C2×D8×C163C2211yescharacteristic
57C2×D8×C163C2211yescharacteristic
58C2×D8×C163C2211yescharacteristic, complemented
59C2×D8×C163C2211yescharacteristic, complemented
60C22×C8×C163C2211yescharacteristic, complemented
61C23×D163—21no—
62#14579698—22no—
63#14579698—41no—
64C2×Dic326C2311yescharacteristic
65C2×D652C2311yescharacteristic
66C2×C22⋊D163—21no—
67C2×C22⋊D163—21no—
68C2×C22⋊D163—21no—
69C2×C22⋊D163—21no—
70C2×C22⋊D163—21no—
71C2×C22⋊D163—21no—
72C2×C22⋊D163—21no—
73C2×C22⋊D163—44no—
74C2×C22⋊D163D411yescharacteristic
75D4×D163—22no—
76D4×D163—41no—
77C22×Dic163—21no—
78C22×Dic163—21no—
79C22×Dic163—21no—
80C2×C4×D163C2311yescharacteristic
81#14579688—22no—
82#14579687—22no—
83#14579687—41no—
84#14579684—22no—
85#14579684—41no—
86#14579682—22no—
87C2×C163⋊C8C2311yescharacteristic
88C2×C163⋊C8C2311yescharacteristic
89C4⋊Dic163C2311yescharacteristic
90C4⋊Dic163C2311yescharacteristic
91Dic163⋊C4—21no—
92Dic163⋊C4D411yescharacteristic
93C22⋊Dic163—21no—
94C22⋊Dic163—21no—
95C22⋊Dic163—21no—
96C22⋊Dic163—21no—
97C22⋊Dic163—21no—
98C22⋊Dic163—21no—
99C22⋊Dic163—21no—
100C22⋊Dic163D411yescharacteristic
101(C2×D163)⋊C4—21no—
102(C2×D163)⋊C4D411yescharacteristic
103C4×Dic163C2311yescharacteristic
104C24×C163—21no—
105C24×C163—21no—
106C2×D4×C163—22no—
107C2×D4×C163—22no—
108C2×D4×C163—21no—
109C2×D4×C163—21no—
110C2×D4×C163—41no—
111C2×D4×C163—41no—
112C2×D4×C163C2311yescharacteristic
113C2×D4×C163C2311yescharacteristic
114C2×D4×C163C2311yescharacteristic
115C2×D4×C163C2311yescharacteristic
116C22×C4×C163C2311yescharacteristic
117D8×C163—22no—
118D8×C163—22no—
119D8×C163—22no—
120D8×C163D414yescomplemented
121C2×C8×C163—22no—
122C2×C8×C163C2311yescharacteristic
123C2×C8×C163C2311yescharacteristic
124C22×D163—21no—
125C22×D163—42no—
126C22×D163—42no—
127C22×D163—81no—
128C22×D163C2×D411yescharacteristic
129Dic326—21no—
130Dic326—21no—
131D652—21no—
132D652—21no—
133C22⋊D163—22no—
134C22⋊D163—42no—
135C22⋊D163—42no—
136C22⋊D163—42no—
137C22⋊D163—42no—
138C22⋊D163—42no—
139C22⋊D163—81no—
140C22⋊D163—81no—
141C2×Dic163—21no—
142C2×Dic163—21no—
143C2×Dic163—21no—
144C2×Dic163—42no—
145C2×Dic163—42no—
146C2×Dic163—81no—
147C2×Dic163C2×D411yescharacteristic
148C2×Dic163C2×D411yescharacteristic
149C2×Dic163C2×D411yescharacteristic
150C4×D163—21no—
151C4×D163—21no—
152C163⋊C8—21no—
153C163⋊C8—21no—
154C23×C163—21no—
155C23×C163—21no—
156C23×C163—21no—
157C23×C163—21no—
158C23×C163—42no—
159C23×C163—42no—
160C23×C163—42no—
161C23×C163—42no—
162C23×C163—81no—
163C23×C163—81no—
164C23×C163C2×D411yescharacteristic
165D4×C163—21no—
166D4×C163—21no—
167D4×C163—21no—
168D4×C163—21no—
169D4×C163—21no—
170D4×C163—21no—
171D4×C163—42no—
172D4×C163—42no—
173D4×C163C2×D412yes—
174D4×C163C2×D412yescomplemented
175C2×C4×C163—21no—
176C2×C4×C163—21no—
177C2×C4×C163C2×D411yescharacteristic
178C2×C4×C163C2411yescharacteristic
179C8×C163—21no—
180C8×C163C2×D412yescomplemented
181C22×C163—21no—
182C22×C163—21no—
183C22×C163—22no—
184C22×C163—22no—
185C22×C163—41no—
186C22×C163—41no—
187C22×C163—41no—
188C22×C163—41no—
189C22×C163—41no—
190C22×C163—41no—
191C22×C163—41no—
192C22×C163—41no—
193C22×C163—41no—
194C22×C163—41no—
195C22×C163—84no—
196C22×C163—84no—
197C22×C163AGL(1,ℤ/8)12yescomplemented
198C22×C163C22×D411yescharacteristic
199C2×D163—21no—
200C2×D163—21no—
201C2×D163—21no—
202C2×D163—41no—
203C2×D163—41no—
204C2×D163—41no—
205C2×D163—84no—
206C4×C163—21no—
207C4×C163C22×D411yescharacteristic
208C4×C163H(5,𝔽2)11yescharacteristic
209Dic163—21no—
210Dic163—21no—
211Dic163—21no—
212Dic163—41no—
213Dic163—41no—
214Dic163—41no—
215C2×C163—22no—
216C2×C163—42no—
217C2×C163—42no—
218C2×C163—81no—
219C2×C163—81no—
220C2×C163D4211yescharacteristic
221C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
222C2×C163#57611yescharacteristic
223D163—41no—
224D163—81no—
225C163#326411yescharacteristic, complemented, Sylow
226#3264—1631nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1631no—
228C22×D8D16311yescharacteristic, complemented
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#435—1631no—
242C2×#41—1631no—
243C22×D4—3261no—
244C22×D4—1631no—
245C22×D4C2×D16311yescharacteristic
246C22×D4C2×D16311yescharacteristic, complemented
247AGL(1,ℤ/8)—3264no—
248C2×SD16—1631no—
249C2×SD16—1631no—
250C2×SD16—1631no—
251C2×D8—24no—
252C2×D8—1631no—
253C2×D8—1631no—
254C2×D8C2×D16311yescharacteristic
255C2×D8C2×D16311yescharacteristic
256C2×D8C2×D16311yescharacteristic, complemented
257C2×D8C2×D16311yescharacteristic, complemented
258C2×M16—1631no—
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×D4—1631no—
275C2×C22⋊C4—3261no—
276#106—1631no—
277C4⋊C8—1631no—
278#103—1631no—
279#102—1631no—
280#102—1631no—
281#102—1631no—
282#102—1631no—
283#102—1631no—
284C22⋊C8—1631no—
285C22⋊C8—1631no—
286#97—1631no—
287C24—21no—
288C24—21no—
289C24—3261no—
290#41—3262no—
291#41—6521no—
292C2×Q8—1631no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—3261no—
300C2×D4—3262no—
301C2×D4—3261no—
302C2×D4—3261no—
303C2×D4—3261no—
304C2×D4—3261no—
305C2×D4—3261no—
306C2×D4—3261no—
307C2×D4—6521no—
308C2×D4—6524no—
309C2×D4—1631no—
310C2×D4—1631no—
311C2×D4C22×D16311yescharacteristic
312C2×D4C22×D16311yescharacteristic
313C2×D4C22×D16311yescharacteristic
314C2×D4C22×D16311yescharacteristic
315C22×C4—3261no—
316C22×C4—3261no—
317C22×C4—3261no—
318C22×C4—1631no—
319C22×C4C22×D16311yescharacteristic
320SD16—3262no—
321SD16—3262no—
322SD16—6521no—
323D8—22no—
324D8—22no—
325D8—22no—
326D8—3262no—
327D8—6521no—
328D8C22⋊D16314yescomplemented
329M16—3262no—
330C2×C8—22no—
331C2×C8—1631no—
332C2×C8—1631no—
333C2×C8C22×D16311yescharacteristic
334C2×C8C22×D16311yescharacteristic
335C4⋊C4—3261no—
336C4⋊C4—1631no—
337C4⋊C4—1631no—
338C4⋊C4—1631no—
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⋊C4—1631no—
348C22⋊C4—1631no—
349C42—1631no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—81no—
359C23—81no—
360C23—13041no—
361C23—3261no—
362C23—6522no—
363C23—6522no—
364C23—1631no—
365C23D4×D16311yescharacteristic
366Q8—3261no—
367Q8—3261no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—42no—
376D4—13041no—
377D4—13041no—
378D4—3262no—
379D4—3261no—
380D4—3261no—
381D4—6522no—
382D4—6522no—
383D4—6522no—
384D4—6522no—
385D4—6522no—
386D4C2×C22⋊D16312yes—
387D4C2×C22⋊D16312yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—13041no—
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×C4—1631no—
401C2×C4D4×D16311yescharacteristic
402C2×C4C23×D16311yescharacteristic
403C8—21no—
404C8—3261no—
405C8—3261no—
406C8C2×C22⋊D16312yescomplemented
407C22—21no—
408C22—21no—
409C22—22no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22—84no—
423C22—13044no—
424C22—3261no—
425C22—3261no—
426C22—3261no—
427C22—6521no—
428C22—6521no—
429C22—6521no—
430C22#1910475912yescomplemented
431C22C2×D4×D16311yescharacteristic
432C4—21no—
433C4—3261no—
434C4—3261no—
435C4—3261no—
436C4—6521no—
437C4—6521no—
438C4—6521no—
439C4C22×C22⋊D16311yescharacteristic
440C4#1910478011yescharacteristic
441C2—22no—
442C2—42no—
443C2—42no—
444C2—81no—
445C2—81no—
446C2—13041no—
447C2—6521no—
448C2D4×C22⋊D16311yescharacteristic
449C2C2×#1910475911yescharacteristic
450C2#2378246611yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC22×D8×C163C2
socleC22×C163C22×D4
hypercenterC22×D8D163
hypocenterC163#3264
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#19104759C2semidirect
2. #23782309C2semidirect
3. D4×C22⋊D163C2semidirect
4. #23782053C2semidirect
5. #23781915C2semidirect
6. #23781894C2semidirect
7. #23781878C2semidirect
8. #23781858C2semidirect
9. #23781816C2semidirect
10. #23781807C2semidirect
11. #23781799C2semidirect
12. #23781797C2semidirect
13. #23781777C2semidirect
14. #23781736C2semidirect
15. C22×D8×C163C2semidirect
16. C2×#14579698C22semidirect
18. C2×D4×D163C22semidirect
20. C2×#14579688C22semidirect
21. C2×#14579687C22semidirect
22. C2×#14579687C22semidirect
23. C2×#14579684C22semidirect
24. C2×#14579684C22semidirect
25. C2×#14579682C22semidirect
26. #19104731C22semidirect
27. #19104722C22semidirect
29. #19104714C22semidirect
33. #19104706C22semidirect
34. #19104706C22semidirect
38. D4×Dic163C22semidirect
39. D4×Dic163C22semidirect
42. #19104667C22semidirect
43. Dic163⋊C8C22semidirect
44. #19104660C22semidirect
45. #19104657C22semidirect
46. #19104656C22semidirect
47. #19104656C22semidirect
48. #19104656C22semidirect
49. #19104656C22semidirect
50. C22⋊(C163⋊C8)C22semidirect
51. (C2×D163)⋊C8C22semidirect
52. #19104645C22semidirect
54. C22×D4×C163C22semidirect
58. C2×D8×C163C22semidirect
59. C2×D8×C163C22semidirect
60. C22×C8×C163C22semidirect
120. D8×C163D4semidirect
174. D4×C163C2×D4semidirect
180. C8×C163C2×D4semidirect
197. C22×C163AGL(1,ℤ/8)semidirect
225. C163#3264semidirect
228. C22×D8D163semidirect
246. C22×D4C2×D163semidirect
256. C2×D8C2×D163semidirect
257. C2×D8C2×D163semidirect
259. C22×C8C2×D163semidirect
328. D8C22⋊D163semidirect
387. D4C2×C22⋊D163semidirect
406. C8C2×C22⋊D163semidirect
430. C22#19104759semidirect

Maximal subgroups

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

subgroupindex
C2×#191047592
#237823092
D4×C22⋊D1632
#237820532
#237819152
#237818942
#237818782
#237818582
#237818162
#237818072
#237817992
#237817972
#237817772
#237817362
C22×D8×C1632
#3264163