Finite Group Databasebeta

← #35226027

Subgroups — #35226027

Order 20864.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104761C211yesmaximal, characteristic, complemented
2C22×#14579687C211yesmaximal, characteristic, complemented
3#23782324C211yesmaximal, characteristic, complemented
4#23782052C211yesmaximal, characteristic, complemented
5#23781909C211yesmaximal, characteristic, complemented
6#23781899C211yesmaximal, characteristic, complemented
7#23781880C211yesmaximal, characteristic, complemented
8#23781862C211yesmaximal, characteristic, complemented
9#23781816C211yesmaximal, characteristic, complemented
10#23781808C211yesmaximal, characteristic, complemented
11#23781803C211yesmaximal, characteristic, complemented
12#23781799C211yesmaximal, characteristic, complemented
13#23781765C211yesmaximal, characteristic, complemented
14#23781737C211yesmaximal, characteristic, complemented
15D42×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic, complemented
17C22×Dic326C2211yescharacteristic, complemented
18#19104761—24no—
19C2×#14579687—24no—
20C2×#14579687C2211yescharacteristic
21C2×#14579687C2211yescharacteristic
22C2×#14579687C2211yescharacteristic, complemented
23C2×#14579687C2211yescharacteristic, complemented
24C2×#14579687C2211yescharacteristic, complemented
25C2×#14579687C2211yescharacteristic, complemented
26C2×#14579684C2211yescharacteristic
27C2×#14579684C2211yescharacteristic, complemented
28C2×#14579684C2211yescharacteristic, complemented
29C2×#14579683C2211yescharacteristic, complemented
30C22×C163⋊C8C2211yescharacteristic, complemented
31#19104723C2211yescharacteristic, complemented
32#19104721—21no—
33#19104714C2211yescharacteristic, complemented
34#19104713—21no—
35#19104709C2211yescharacteristic, complemented
36#19104707—21no—
37C4×Dic326C2211yescharacteristic, complemented
38(C22⋊D163)⋊C4—21no—
39#19104668C2211yescharacteristic, complemented
40C4⋊(C163⋊C8)C2211yescharacteristic, complemented
41#19104658C2211yescharacteristic, complemented
42#19104658C2211yescharacteristic, complemented
43#19104656C2211yescharacteristic
44#19104656C2211yescharacteristic, complemented
45#19104656C2211yescharacteristic, complemented
46#19104655C2211yescharacteristic, complemented
47C22⋊(C163⋊C8)C2211yescharacteristic, complemented
48C22⋊(C163⋊C8)C2211yescharacteristic, complemented
49#19104646C2211yescharacteristic, complemented
50C22×D4×C163—21no—
51C22×D4×C163C2211yescharacteristic
52C22×D4×C163C2211yescharacteristic
53#127×C163C2211yescharacteristic
54#121×C163—21no—
55#121×C163C2211yescharacteristic
56#121×C163C2211yescharacteristic
57#120×C163—21no—
58#120×C163—21no—
59C4×D4×C163C2211yescharacteristic
60C4×D4×C163C2211yescharacteristic
61#14579699—22no—
62#14579699—41no—
63C2×Dic326—22no—
64C2×Dic326—21no—
65C2×Dic326—41no—
66C2×Dic326C2311yescharacteristic
67C2×Dic326C2311yescharacteristic
68C2×D652C2311yescharacteristic
69C2×C22⋊D163—21no—
70C2×C22⋊D163—21no—
71C22×Dic163—21no—
72C2×C4×D163—21no—
73#14579687—22no—
74#14579687—22no—
75#14579687—22no—
76#14579687—22no—
77#14579687—41no—
78#14579687D414yescomplemented
79#14579684—22no—
80#14579684—22no—
81#14579684—41no—
82#14579683—22no—
83C2×C163⋊C8—22no—
84C2×C163⋊C8C2311yescharacteristic
85C2×C163⋊C8C2311yescharacteristic
86C2×C163⋊C8C2311yescharacteristic
87C2×C163⋊C8C2311yescharacteristic
88C4⋊Dic163C2311yescharacteristic
89Dic163⋊C4—21no—
90Dic163⋊C4—21no—
91C22⋊Dic163—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95C4×Dic163—21no—
96C24×C163—21no—
97C24×C163—21no—
98C2×D4×C163—22no—
99C2×D4×C163—22no—
100C2×D4×C163—21no—
101C2×D4×C163—21no—
102C2×D4×C163—21no—
103C2×D4×C163—21no—
104C2×D4×C163—21no—
105C2×D4×C163—21no—
106C2×D4×C163—41no—
107C2×D4×C163—41no—
108C2×D4×C163—44no—
109C2×D4×C163D411yescharacteristic
110C2×D4×C163C2311yescharacteristic
111C2×D4×C163C2311yescharacteristic
112C2×D4×C163C2311yescharacteristic
113C22×C4×C163—21no—
114C22×C4×C163C2311yescharacteristic
115C22×C4×C163C2311yescharacteristic
116C4⋊C4×C163D411yescharacteristic
117C4⋊C4×C163C2311yescharacteristic
118C22⋊C4×C163—21no—
119C22⋊C4×C163—21no—
120C22⋊C4×C163—21no—
121C22⋊C4×C163D411yescharacteristic
122C22⋊C4×C163D411yescharacteristic
123C42×C163C2311yescharacteristic
124C22×D163—21no—
125Dic326—21no—
126Dic326—21no—
127Dic326—21no—
128Dic326—42no—
129Dic326C2×D412yescomplemented
130D652—21no—
131D652—21no—
132C22⋊D163—42no—
133C22⋊D163—42no—
134C2×Dic163—21no—
135C2×Dic163—21no—
136C2×Dic163—21no—
137C2×Dic163—42no—
138C4×D163—42no—
139C163⋊C8—21no—
140C163⋊C8—21no—
141C163⋊C8—21no—
142C163⋊C8C2×D412yescomplemented
143C23×C163—21no—
144C23×C163—21no—
145C23×C163—21no—
146C23×C163—42no—
147C23×C163—42no—
148C23×C163—42no—
149C23×C163—42no—
150C23×C163—81no—
151C23×C163—81no—
152C23×C163C2×D411yescharacteristic
153C23×C163C2×D411yescharacteristic
154D4×C163—22no—
155D4×C163—21no—
156D4×C163—21no—
157D4×C163—21no—
158D4×C163—21no—
159D4×C163—21no—
160D4×C163—42no—
161D4×C163—42no—
162D4×C163—42no—
163D4×C163—42no—
164D4×C163—42no—
165D4×C163—81no—
166D4×C163C2×D412yes—
167C2×C4×C163—21no—
168C2×C4×C163—21no—
169C2×C4×C163—21no—
170C2×C4×C163—21no—
171C2×C4×C163—21no—
172C2×C4×C163—42no—
173C2×C4×C163C2×D411yescharacteristic
174C2×C4×C163C2×D411yescharacteristic
175C2×C4×C163C2×D411yescharacteristic
176C2×C4×C163C2×D411yescharacteristic
177C2×C4×C163C2411yescharacteristic
178C22×C163—21no—
179C22×C163—21no—
180C22×C163—21no—
181C22×C163—22no—
182C22×C163—21no—
183C22×C163—21no—
184C22×C163—41no—
185C22×C163—41no—
186C22×C163—41no—
187C22×C163—41no—
188C22×C163—41no—
189C22×C163—41no—
190C22×C163—41no—
191C22×C163—41no—
192C22×C163—84no—
193C22×C163—84no—
194C22×C163AGL(1,ℤ/8)12yescomplemented
195C22×C163C22×D411yescharacteristic
196C2×D163—41no—
197C2×D163—41no—
198C2×D163—41no—
199C4×C163—21no—
200C4×C163—21no—
201C4×C163—21no—
202C4×C163—21no—
203C4×C163—41no—
204C4×C163C22×D411yescharacteristic
205C4×C163H(5,𝔽2)11yescharacteristic
206Dic163—22no—
207Dic163—41no—
208Dic163—41no—
209C2×C163—22no—
210C2×C163—41no—
211C2×C163—42no—
212C2×C163—81no—
213C2×C163—81no—
214C2×C163D4211yescharacteristic
215C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
216C2×C163#57711yescharacteristic
217D163—81no—
218C163#326511yescharacteristic, complemented, Sylow
219#3265—1631nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1631no—
221C22×SD16—1631no—
222#548—1631no—
223D42D16311yescharacteristic, complemented
224#496—1631no—
225#468—1631no—
226#466—1631no—
227#463—1631no—
228#460—1631no—
229#450—1631no—
230#449—1631no—
231#448—1631no—
232#447—1631no—
233#440—1631no—
234#435—1631no—
235C2×#41—1631no—
236C22×Q8—1631no—
237C22×D4—21no—
238C22×D4C2×D16311yescharacteristic
239C22×D4C2×D16311yescharacteristic
240AGL(1,ℤ/8)—3264no—
241C2×SD16—3264no—
242C2×SD16—1631no—
243C2×SD16—1631no—
244C2×SD16—1631no—
245C2×SD16—1631no—
246C2×SD16—1631no—
247C2×SD16—1631no—
248C2×D8—1631no—
249C2×D8—1631no—
250C2×D8—1631no—
251C2×M16—1631no—
252C22×C8—1631no—
253#127C2×D16311yescharacteristic
254#124—3261no—
255#124—1631no—
256C22⋊Q8—3261no—
257C22⋊Q8—1631no—
258#121—21no—
259#121—3261no—
260#121—1631no—
261#121C2×D16311yescharacteristic
262#121C2×D16311yescharacteristic
263#120—21no—
264#120—21no—
265C4×Q8—1631no—
266C4×D4—3261no—
267C4×D4C2×D16311yescharacteristic
268C4×D4C2×D16311yescharacteristic
269#107—1631no—
270C4⋊C8—1631no—
271#103—1631no—
272#103—1631no—
273#102—1631no—
274#102—1631no—
275#102—1631no—
276#102—1631no—
277C22⋊C8—1631no—
278C22⋊C8—1631no—
279#97—1631no—
280C24—21no—
281C24—21no—
282#41—3262no—
283#41—6521no—
284C2×Q8—3262no—
285C2×Q8—3261no—
286C2×Q8—6521no—
287C2×Q8—1631no—
288C2×Q8—1631no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—44no—
300C2×D4—3261no—
301C2×D4—3261no—
302C2×D4—1631no—
303C2×D4C22⋊D16311yescharacteristic
304C2×D4C22×D16311yescharacteristic
305C2×D4C22×D16311yescharacteristic
306C2×D4C22×D16311yescharacteristic
307C22×C4—21no—
308C22×C4—3261no—
309C22×C4—3261no—
310C22×C4C22×D16311yescharacteristic
311C22×C4C22×D16311yescharacteristic
312SD16—3262no—
313SD16—3262no—
314SD16—3262no—
315SD16—3262no—
316SD16—6521no—
317SD16—1634no—
318D8—3262no—
319D8—3262no—
320D8—6521no—
321M16—3262no—
322C2×C8—3262no—
323C2×C8—1631no—
324C2×C8—1631no—
325C2×C8—1631no—
326C2×C8—1631no—
327C4⋊C4—3261no—
328C4⋊C4—3261no—
329C4⋊C4—1631no—
330C4⋊C4C22⋊D16311yescharacteristic
331C4⋊C4C22×D16311yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—3261no—
336C22⋊C4—3261no—
337C22⋊C4—3261no—
338C22⋊C4—3261no—
339C22⋊C4C22⋊D16311yescharacteristic
340C22⋊C4C22⋊D16311yescharacteristic
341C42—3261no—
342C42C22×D16311yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—81no—
351C23—81no—
352C23—3261no—
353C23C2×C22⋊D16311yescharacteristic
354C23C2×C22⋊D16311yescharacteristic
355Q8—3261no—
356Q8—3261no—
357Q8—3261no—
358Q8—6522no—
359Q8—1632no—
360D4—22no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—81no—
372D4—3261no—
373D4—3261no—
374D4—6522no—
375D4—6522no—
376D4D4×D16312yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—3261no—
384C2×C4—3261no—
385C2×C4—3261no—
386C2×C4—6522no—
387C2×C4—6522no—
388C2×C4C2×C22⋊D16311yescharacteristic
389C2×C4C2×C22⋊D16311yescharacteristic
390C2×C4C2×C22⋊D16311yescharacteristic
391C2×C4C2×C22⋊D16311yescharacteristic
392C2×C4C23×D16311yescharacteristic
393C8—3261no—
394C8—3261no—
395C8—3261no—
396C8—1632no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—84no—
412C22—84no—
413C22—6521no—
414C22—6521no—
415C22—6521no—
416C22#1910476112yescomplemented
417C22C22×C22⋊D16311yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—41no—
423C4—3262no—
424C4—6521no—
425C4—6521no—
426C4C2×D4×D16311yescharacteristic
427C4#1910478011yescharacteristic
428C2—22no—
429C2—41no—
430C2—42no—
431C2—81no—
432C2—81no—
433C2—13041no—
434C2D4×C22⋊D16311yescharacteristic
435C2C2×#1910476111yescharacteristic
436C2#2378247611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingD42×C163C2
socleC22×C163C22×D4
hypercenterD42D163
hypocenterC163#3265
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#19104761C2semidirect
2. C22×#14579687C2semidirect
3. #23782324C2semidirect
4. #23782052C2semidirect
5. #23781909C2semidirect
6. #23781899C2semidirect
7. #23781880C2semidirect
8. #23781862C2semidirect
9. #23781816C2semidirect
10. #23781808C2semidirect
11. #23781803C2semidirect
12. #23781799C2semidirect
13. #23781765C2semidirect
14. #23781737C2semidirect
15. D42×C163C2semidirect
16. C2×#14579699C22semidirect
17. C22×Dic326C22semidirect
22. C2×#14579687C22semidirect
23. C2×#14579687C22semidirect
24. C2×#14579687C22semidirect
25. C2×#14579687C22semidirect
27. C2×#14579684C22semidirect
28. C2×#14579684C22semidirect
29. C2×#14579683C22semidirect
30. C22×C163⋊C8C22semidirect
31. #19104723C22semidirect
33. #19104714C22semidirect
35. #19104709C22semidirect
37. C4×Dic326C22semidirect
39. #19104668C22semidirect
40. C4⋊(C163⋊C8)C22semidirect
41. #19104658C22semidirect
42. #19104658C22semidirect
44. #19104656C22semidirect
45. #19104656C22semidirect
46. #19104655C22semidirect
47. C22⋊(C163⋊C8)C22semidirect
48. C22⋊(C163⋊C8)C22semidirect
49. #19104646C22semidirect
78. #14579687D4semidirect
129. Dic326C2×D4semidirect
142. C163⋊C8C2×D4semidirect
194. C22×C163AGL(1,ℤ/8)semidirect
218. C163#3265semidirect
223. D42D163semidirect
416. C22#19104761semidirect

Maximal subgroups

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

subgroupindex
C2×#191047612
C22×#145796872
#237823242
#237820522
#237819092
#237818992
#237818802
#237818622
#237818162
#237818082
#237818032
#237817992
#237817652
#237817372
D42×C1632
#3265163