Finite Group Databasebeta

← #35224034

Subgroups — #35224034

Order 20864.

526 conjugacy classes of subgroups.

The subgroup lattice has 424 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#23782228C211yesmaximal, characteristic, complemented
4#23782137C211yesmaximal, characteristic, complemented
5#23781881C211yesmaximal, characteristic, complemented
6#23781880C211yesmaximal, characteristic, complemented
7#23781879C211yesmaximal, characteristic, complemented
8#23781878C211yesmaximal, characteristic, complemented
9#23781859C211yesmaximal, characteristic, complemented
10#23781858C211yesmaximal, characteristic, complemented
11#23781857C211yesmaximal, characteristic, complemented
12#23781690C211yesmaximal, characteristic, complemented
13#23781668C211yesmaximal, characteristic, complemented
14#23781658C211yesmaximal, characteristic, complemented
15#459×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C2×D4×D163C2211yescharacteristic
18C2×D4×D163C2211yescharacteristic
19#19104759—24no—
20C2×#14579688C2211yescharacteristic, complemented
21C2×#14579687C2211yescharacteristic, complemented
22C2×D1304C2211yescharacteristic
23C2×#14579684C2211yescharacteristic
24C2×#14579684C2211yescharacteristic
25C2×#14579684C2211yescharacteristic, complemented
26D8×D163—24no—
27C2×#14579682C2211yescharacteristic
28C2×C8×D163C2211yescharacteristic
29#19104731C2211yescharacteristic, complemented
30#19104723C2211yescharacteristic, complemented
31#19104722C2211yescharacteristic, complemented
32#19104707—21no—
33#19104707—21no—
34#19104703—21no—
35#19104703—21no—
36C4×D652C2211yescharacteristic, complemented
37D4×Dic163C2211yescharacteristic, complemented
38#19104697C2211yescharacteristic, complemented
39C4×C22⋊D163—21no—
40(C22⋊D163)⋊C4—21no—
41#19104693C2211yescharacteristic
42#19104692C2211yescharacteristic
43C2×(C2×D163)⋊C4—21no—
44C22⋊C4×D163—21no—
45C4⋊(C163⋊C8)C2211yescharacteristic, complemented
46Dic163⋊C8C2211yescharacteristic, complemented
47Dic163⋊C8C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104658C2211yescharacteristic, complemented
50#19104657C2211yescharacteristic, complemented
51#19104656C2211yescharacteristic, complemented
52#19104656C2211yescharacteristic, complemented
53#19104655C2211yescharacteristic, complemented
54C2×D8×C163C2211yescharacteristic
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—
63#14579698—22no—
64#14579698—22no—
65C2×Dic326C2311yescharacteristic
66C2×D652C2311yescharacteristic
67C2×D652C2311yescharacteristic
68C2×C22⋊D163—21no—
69C2×C22⋊D163—21no—
70C2×C22⋊D163—21no—
71C2×C22⋊D163—21no—
72C2×C22⋊D163—21no—
73D4×D163—22no—
74D4×D163—22no—
75D4×D163—22no—
76D4×D163D414yescomplemented
77C22×Dic163—21no—
78C22×Dic163—21no—
79C2×C4×D163—21no—
80C2×C4×D163—21no—
81C2×C4×D163C2311yescharacteristic
82#14579688—22no—
83#14579687—22no—
84D1304—22no—
85#14579684—22no—
86#14579684—22no—
87#14579684—22no—
88#14579682—22no—
89C2×C163⋊C8C2311yescharacteristic
90C2×C163⋊C8C2311yescharacteristic
91C8×D163—22no—
92C4⋊Dic163C2311yescharacteristic
93Dic163⋊C4D411yescharacteristic
94Dic163⋊C4D411yescharacteristic
95C22⋊Dic163—21no—
96C22⋊Dic163—21no—
97C22⋊Dic163—21no—
98(C2×D163)⋊C4—21no—
99(C2×D163)⋊C4—21no—
100(C2×D163)⋊C4—21no—
101(C2×D163)⋊C4—21no—
102(C2×D163)⋊C4—21no—
103(C2×D163)⋊C4D411yescharacteristic
104(C2×D163)⋊C4D411yescharacteristic
105C4×Dic163—21no—
106C4×Dic163C2311yescharacteristic
107C2×D4×C163—21no—
108C2×D4×C163—21no—
109C2×D4×C163C2311yescharacteristic
110C2×D4×C163C2311yescharacteristic
111C2×D4×C163C2311yescharacteristic
112C22×C4×C163—21no—
113D8×C163—22no—
114D8×C163—22no—
115C2×C8×C163C2311yescharacteristic
116C2×C8×C163C2311yescharacteristic
117C4⋊C4×C163C2311yescharacteristic
118C22⋊C4×C163—21no—
119C42×C163C2311yescharacteristic
120C22×D163—21no—
121C22×D163—24no—
122C22×D163—21no—
123C22×D163—42no—
124C22×D163—81no—
125C22×D163—81no—
126C22×D163—81no—
127C22×D163—81no—
128C22×D163C2×D411yescharacteristic
129Dic326—21no—
130Dic326—21no—
131D652—21no—
132D652—21no—
133D652—21no—
134D652C2×D412yes—
135C22⋊D163—24no—
136C22⋊D163—42no—
137C22⋊D163—42no—
138C22⋊D163—81no—
139C22⋊D163—81no—
140C2×Dic163—21no—
141C2×Dic163—21no—
142C2×Dic163—42no—
143C2×Dic163—81no—
144C2×Dic163C2×D411yescharacteristic
145C2×Dic163C2×D411yescharacteristic
146C2×Dic163C2×D411yescharacteristic
147C4×D163—81no—
148C4×D163—81no—
149C4×D163C2×D412yes—
150C4×D163#4112yes—
151C163⋊C8—21no—
152C163⋊C8—21no—
153C23×C163—21no—
154C23×C163—21no—
155C23×C163—21no—
156D4×C163—21no—
157D4×C163—21no—
158D4×C163—21no—
159D4×C163—21no—
160D4×C163—21no—
161D4×C163—81no—
162D4×C163—81no—
163D4×C163C2×D412yes—
164C2×C4×C163—21no—
165C2×C4×C163—81no—
166C2×C4×C163C2×D411yescharacteristic
167C2×C4×C163C2×D411yescharacteristic
168C2×C4×C163C2411yescharacteristic
169C8×C163—21no—
170C8×C163—21no—
171C22×C163—22no—
172C22×C163—41no—
173C22×C163—41no—
174C22×C163—41no—
175C22×C163—41no—
176C22×C163—41no—
177C22×C163—41no—
178C22×C163—41no—
179C22×C163—41no—
180C22×C163C22×D411yescharacteristic
181C2×D163—21no—
182C2×D163—21no—
183C2×D163—22no—
184C2×D163—41no—
185C2×D163—41no—
186C2×D163—41no—
187C2×D163—41no—
188C2×D163—41no—
189C2×D163—84no—
190C2×D163—84no—
191C2×D163#12112yes—
192C4×C163—21no—
193C4×C163—21no—
194C4×C163—41no—
195C4×C163C22×D411yescharacteristic
196C4×C163C2×#4111yescharacteristic
197Dic163—21no—
198Dic163—21no—
199Dic163—41no—
200Dic163—41no—
201Dic163#12112yes—
202C2×C163—42no—
203C2×C163—81no—
204C2×C163—81no—
205C2×C163C2×#12111yescharacteristic
206C2×C163#57611yescharacteristic
207C2×C163#57611yescharacteristic
208D163—22no—
209D163—42no—
210D163—81no—
211C163#304111yescharacteristic, complemented, Sylow
212#3041—1631nomaximal, Sylow
213C2×AGL(1,ℤ/8)—1631no—
214C22×D8—1631no—
215#535—1631no—
216#518—1631no—
217#463—1631no—
218#463—1631no—
219#463—1631no—
220#463—1631no—
221#459—1631no—
222#459—1631no—
223#459—1631no—
224#459D16311yescharacteristic, complemented
225#424—1631no—
226#418—1631no—
227#416—1631no—
228C2×#41—1631no—
229C22×D4—1631no—
230C22×D4—1631no—
231AGL(1,ℤ/8)—3264no—
232C2×SD16—1631no—
233C2×SD16—1631no—
234C2×D8—3264no—
235C2×D8—1631no—
236C2×D8—1631no—
237C2×D8—1631no—
238C2×D8—1631no—
239C2×D8C2×D16311yescharacteristic
240C2×D8C2×D16311yescharacteristic, complemented
241C2×M16—1631no—
242C22×C8—1631no—
243#127—1631no—
244#127C2×D16311yescharacteristic, complemented
245#124—1631no—
246#124—1631no—
247#121—3261no—
248#121—3261no—
249#120—3261no—
250#120—3261no—
251C4×D4—3261no—
252C4×D4—3261no—
253C4×D4—1631no—
254C4×D4—1631no—
255C4×D4—1631no—
256C4×D4C2×D16311yescharacteristic, complemented
257#117—1631no—
258#117—1631no—
259C2×C22⋊C4—3261no—
260C2×C22⋊C4—3261no—
261C4⋊C8—1631no—
262C4⋊C8—1631no—
263C4⋊C8—1631no—
264C4⋊C8C2×D16311yescharacteristic, complemented
265#103—1631no—
266#103—1631no—
267#102—1631no—
268#102—1631no—
269#102—1631no—
270#102—1631no—
271#102C2×D16311yescharacteristic, complemented
272#102C2×D16311yescharacteristic, complemented
273C24—3261no—
274C24—3261no—
275#41—3262no—
276#41—3262no—
277C2×Q8—1631no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—3261no—
281C2×D4—3262no—
282C2×D4—3262no—
283C2×D4—3262no—
284C2×D4—3261no—
285C2×D4—3261no—
286C2×D4—3261no—
287C2×D4—3261no—
288C2×D4—1634no—
289C2×D4—1631no—
290C2×D4—1631no—
291C2×D4C22×D16311yescharacteristic
292C2×D4C22×D16311yescharacteristic
293C2×D4C22×D16311yescharacteristic
294C22×C4—21no—
295C22×C4—3261no—
296C22×C4—3261no—
297C22×C4—3261no—
298C22×C4—3261no—
299C22×C4—1631no—
300SD16—3262no—
301SD16—3262no—
302D8—22no—
303D8—22no—
304D8—3262no—
305D8—3262no—
306D8—3262no—
307D8—3262no—
308M16—3262no—
309C2×C8—3262no—
310C2×C8—1631no—
311C2×C8—1631no—
312C2×C8C22×D16311yescharacteristic
313C2×C8C22×D16311yescharacteristic
314C4⋊C4—1631no—
315C4⋊C4—1631no—
316C4⋊C4—1631no—
317C4⋊C4C22×D16311yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—3261no—
320C22⋊C4—3261no—
321C22⋊C4—3261no—
322C22⋊C4—3261no—
323C22⋊C4—3261no—
324C22⋊C4—3261no—
325C22⋊C4—3261no—
326C22⋊C4—3261no—
327C22⋊C4—1631no—
328C22⋊C4—1631no—
329C42—3261no—
330C42—1631no—
331C42C22×D16311yescharacteristic
332C23—21no—
333C23—21no—
334C23—21no—
335C23—13041no—
336C23—13041no—
337C23—13041no—
338C23—13041no—
339C23—3264no—
340C23—3261no—
341C23—3261no—
342C23—6522no—
343C23—1631no—
344Q8—3261no—
345Q8—3261no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—81no—
352D4—81no—
353D4—13041no—
354D4—13041no—
355D4—3264no—
356D4—3261no—
357D4—3261no—
358D4—3261no—
359D4—6522no—
360D4—6522no—
361D4—1632no—
362D4D4×D16312yes—
363C2×C4—21no—
364C2×C4—81no—
365C2×C4—13041no—
366C2×C4—13041no—
367C2×C4—13041no—
368C2×C4—3261no—
369C2×C4—3261no—
370C2×C4—6522no—
371C2×C4—1631no—
372C2×C4—1631no—
373C2×C4—1631no—
374C2×C4—1632no—
375C2×C4—1632no—
376C2×C4D4×D16311yescharacteristic
377C2×C4D4×D16311yescharacteristic
378C2×C4C23×D16311yescharacteristic
379C8—21no—
380C8—21no—
381C8—3261no—
382C8—3261no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—13044no—
393C22—13044no—
394C22—3261no—
395C22—3261no—
396C22—3262no—
397C22—6521no—
398C22—6521no—
399C22—6521no—
400C22—6521no—
401C22—6521no—
402C22—1632no—
403C22C2×D4×D16311yescharacteristic
404C4—21no—
405C4—21no—
406C4—41no—
407C4—3261no—
408C4—3261no—
409C4—6521no—
410C4—6521no—
411C4—1632no—
412C4C2×D4×D16311yescharacteristic
413C4#41×D16311yescharacteristic
414C2—42no—
415C2—81no—
416C2—81no—
417C2—13041no—
418C2—3262no—
419C2—6522no—
420C2#121×D16311yescharacteristic
421C2#2378246611yescharacteristic
422C2#2378246911yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#459×C163C2
socleC22×C163C22×D4
hypercenter#459D163
hypocenterC163#3041
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. #23782228C2semidirect
4. #23782137C2semidirect
5. #23781881C2semidirect
6. #23781880C2semidirect
7. #23781879C2semidirect
8. #23781878C2semidirect
9. #23781859C2semidirect
10. #23781858C2semidirect
11. #23781857C2semidirect
12. #23781690C2semidirect
13. #23781668C2semidirect
14. #23781658C2semidirect
15. #459×C163C2semidirect
20. C2×#14579688C22semidirect
21. C2×#14579687C22semidirect
25. C2×#14579684C22semidirect
29. #19104731C22semidirect
30. #19104723C22semidirect
31. #19104722C22semidirect
36. C4×D652C22semidirect
37. D4×Dic163C22semidirect
38. #19104697C22semidirect
45. C4⋊(C163⋊C8)C22semidirect
46. Dic163⋊C8C22semidirect
47. Dic163⋊C8C22semidirect
48. #19104660C22semidirect
49. #19104658C22semidirect
50. #19104657C22semidirect
51. #19104656C22semidirect
52. #19104656C22semidirect
53. #19104655C22semidirect
55. C2×D8×C163C22semidirect
56. #127×C163C22semidirect
57. C4×D4×C163C22semidirect
58. C4⋊C8×C163C22semidirect
59. #102×C163C22semidirect
60. #102×C163C22semidirect
76. D4×D163D4semidirect
211. C163#3041semidirect
224. #459D163semidirect
240. C2×D8C2×D163semidirect
244. #127C2×D163semidirect
256. C4×D4C2×D163semidirect
264. C4⋊C8C2×D163semidirect
271. #102C2×D163semidirect
272. #102C2×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
#237822282
#237821372
#237818812
#237818802
#237818792
#237818782
#237818592
#237818582
#237818572
#237816902
#237816682
#237816582
#459×C1632
#3041163