Finite Group Databasebeta

← #35226077

Subgroups — #35226077

Order 20864.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104760C211yesmaximal, characteristic, complemented
2C2×#19104755C211yesmaximal, characteristic, complemented
3#23782311C211yesmaximal, characteristic, complemented
4#23782052C211yesmaximal, characteristic, complemented
5#23781909C211yesmaximal, characteristic, complemented
6#23781899C211yesmaximal, characteristic, complemented
7#23781887C211yesmaximal, characteristic, complemented
8#23781857C211yesmaximal, characteristic, complemented
9#23781827C211yesmaximal, characteristic, complemented
10#23781810C211yesmaximal, characteristic, complemented
11#23781808C211yesmaximal, characteristic, complemented
12#23781798C211yesmaximal, characteristic, complemented
13#23781763C211yesmaximal, characteristic, complemented
14#23781737C211yesmaximal, characteristic, complemented
15#550×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C22×D652C2211yescharacteristic
18#19104760—24no—
19#19104755—24no—
20C2×#14579689C2211yescharacteristic, complemented
21C2×#14579687C2211yescharacteristic, complemented
22C2×#14579687C2211yescharacteristic, complemented
23C2×#14579686C2211yescharacteristic
24C2×#14579686C2211yescharacteristic
25C2×#14579684C2211yescharacteristic
26C2×#14579684C2211yescharacteristic
27C2×#14579684C2211yescharacteristic
28C2×#14579684C2211yescharacteristic
29C2×#14579683C2211yescharacteristic
30C22×C163⋊C8C2211yescharacteristic, complemented
31#19104723C2211yescharacteristic
32#19104720—21no—
33#19104714C2211yescharacteristic
34#19104709C2211yescharacteristic
35#19104708—21no—
36#19104703—21no—
37C4×D652C2211yescharacteristic
38C22⋊C4×D163—21no—
39#19104668C2211yescharacteristic, complemented
40C4⋊(C163⋊C8)C2211yescharacteristic, complemented
41#19104659C2211yescharacteristic, complemented
42#19104658C2211yescharacteristic, complemented
43#19104656C2211yescharacteristic, complemented
44#19104656C2211yescharacteristic, complemented
45#19104655C2211yescharacteristic
46#19104655C2211yescharacteristic
47C22⋊(C163⋊C8)C2211yescharacteristic
48C22⋊(C163⋊C8)C2211yescharacteristic, complemented
49#19104646C2211yescharacteristic, complemented
50C2×#41×C163C2211yescharacteristic
51C2×#41×C163C2211yescharacteristic, complemented
52#127×C163—21no—
53#127×C163C2211yescharacteristic, complemented
54#121×C163—21no—
55#121×C163—21no—
56#121×C163C2211yescharacteristic
57#121×C163C2211yescharacteristic, complemented
58C4×Q8×C163C2211yescharacteristic, complemented
59C4×D4×C163—21no—
60C4×D4×C163C2211yescharacteristic
61C23×D163—21no—
62#14579699—22no—
63#14579699—41no—
64C2×Dic326C2311yescharacteristic
65C2×D652—22no—
66C2×D652—21no—
67C2×D652—41no—
68C2×D652C2311yescharacteristic
69C2×D652C2311yescharacteristic
70C2×C22⋊D163—21no—
71C2×C22⋊D163—21no—
72C2×C4×D163—21no—
73C2×C4×D163—21no—
74#14579689—22no—
75#14579687—22no—
76#14579687—41no—
77#14579686—22no—
78#14579686D414yescomplemented
79#14579684—22no—
80#14579684—22no—
81#14579684—22no—
82#14579684—41no—
83#14579683—22no—
84C2×C163⋊C8—22no—
85C2×C163⋊C8C2311yescharacteristic
86C2×C163⋊C8C2311yescharacteristic
87C2×C163⋊C8C2311yescharacteristic
88C2×C163⋊C8C2311yescharacteristic
89C4⋊Dic163C2311yescharacteristic
90Dic163⋊C4—21no—
91C22⋊Dic163—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4—21no—
96#41×C163—22no—
97#41×C163—22no—
98#41×C163—41no—
99#41×C163—41no—
100C2×Q8×C163C2311yescharacteristic
101C2×D4×C163—21no—
102C2×D4×C163—21no—
103C2×D4×C163—21no—
104C2×D4×C163—21no—
105C2×D4×C163—21no—
106C2×D4×C163—21no—
107C2×D4×C163D411yescharacteristic
108C2×D4×C163C2311yescharacteristic
109C2×D4×C163C2311yescharacteristic
110C22×C4×C163—21no—
111C22×C4×C163—21no—
112C22×C4×C163C2311yescharacteristic
113C22×C4×C163C2311yescharacteristic
114C4⋊C4×C163—21no—
115C4⋊C4×C163D411yescharacteristic
116C4⋊C4×C163C2311yescharacteristic
117C22⋊C4×C163—21no—
118C22⋊C4×C163—21no—
119C22⋊C4×C163D411yescharacteristic
120C22⋊C4×C163D411yescharacteristic
121C42×C163—21no—
122C42×C163C2311yescharacteristic
123C22×D163—21no—
124C22×D163—21no—
125C22×D163—42no—
126C22×D163—42no—
127C22×D163—81no—
128Dic326—21no—
129Dic326—21no—
130D652—21no—
131D652—21no—
132D652—21no—
133D652—42no—
134D652C2×D412yes—
135C22⋊D163—42no—
136C22⋊D163—81no—
137C2×Dic163—21no—
138C2×Dic163—21no—
139C4×D163—42no—
140C4×D163—81no—
141C163⋊C8—21no—
142C163⋊C8—21no—
143C163⋊C8—21no—
144C163⋊C8C2×D412yes—
145C23×C163—21no—
146C23×C163—21no—
147C23×C163C2×D411yescharacteristic
148C23×C163C2×D411yescharacteristic
149Q8×C163—21no—
150Q8×C163C2×D412yes—
151D4×C163—21no—
152D4×C163—21no—
153D4×C163—21no—
154D4×C163—21no—
155D4×C163—41no—
156D4×C163—42no—
157D4×C163—42no—
158D4×C163—42no—
159D4×C163—81no—
160D4×C163—81no—
161D4×C163—81no—
162C2×C4×C163—21no—
163C2×C4×C163—21no—
164C2×C4×C163—21no—
165C2×C4×C163—21no—
166C2×C4×C163—21no—
167C2×C4×C163—21no—
168C2×C4×C163—42no—
169C2×C4×C163—42no—
170C2×C4×C163C2×D411yescharacteristic
171C2×C4×C163C2×D411yescharacteristic
172C2×C4×C163C2×D411yescharacteristic
173C2×C4×C163C2×D411yescharacteristic
174C2×C4×C163C2411yescharacteristic
175C22×C163—21no—
176C22×C163—21no—
177C22×C163—21no—
178C22×C163—21no—
179C22×C163—21no—
180C22×C163—21no—
181C22×C163—41no—
182C22×C163—41no—
183C22×C163—41no—
184C22×C163—41no—
185C22×C163—41no—
186C22×C163—41no—
187C22×C163C22×D411yescharacteristic
188C2×D163—22no—
189C2×D163—41no—
190C2×D163—41no—
191C2×D163—41no—
192C2×D163—41no—
193C2×D163—41no—
194C2×D163—84no—
195C4×C163—21no—
196C4×C163—21no—
197C4×C163—22no—
198C4×C163—21no—
199C4×C163—21no—
200C4×C163—41no—
201C4×C163C22×D411yescharacteristic
202C4×C163H(5,𝔽2)11yescharacteristic
203Dic163—41no—
204Dic163—41no—
205C2×C163—41no—
206C2×C163—41no—
207C2×C163—81no—
208C2×C163—81no—
209C2×C163D4211yescharacteristic
210C2×C163#57511yescharacteristic
211C2×C163#57611yescharacteristic
212D163—42no—
213D163—81no—
214C163#326811yescharacteristic, complemented, Sylow
215#3268—1631nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1631no—
217C2×#135—1631no—
218#550D16311yescharacteristic, complemented
219#546—1631no—
220#496—1631no—
221#468—1631no—
222#466—1631no—
223#464—1631no—
224#459—1631no—
225#452—1631no—
226#449—1631no—
227#449—1631no—
228#447—1631no—
229#440—1631no—
230#435—1631no—
231C2×#41—1631no—
232C2×#41C2×D16311yescharacteristic
233C2×#41C2×D16311yescharacteristic, complemented
234C22×D4—1631no—
235AGL(1,ℤ/8)—3264no—
236#135—3264no—
237C2×Q16—1631no—
238C2×SD16—1631no—
239C2×SD16—1631no—
240C2×SD16—1631no—
241C2×SD16—1631no—
242C2×D8—1631no—
243C2×D8—1631no—
244C2×D8—1631no—
245C2×D8—1631no—
246C2×M16—1631no—
247C22×C8—1631no—
248#127—21no—
249#127C2×D16311yescharacteristic, complemented
250#124—1631no—
251#123—3261no—
252C22⋊Q8—1631no—
253#121—21no—
254#121—21no—
255#121—3261no—
256#121—1631no—
257#121C2×D16311yescharacteristic
258#121C2×D16311yescharacteristic, complemented
259#120—3261no—
260C4×Q8C2×D16311yescharacteristic, complemented
261C4×D4—21no—
262C4×D4—1631no—
263C4×D4C2×D16311yescharacteristic
264C2×C22⋊C4—3261no—
265#107—1631no—
266C4⋊C8—1631no—
267#103—1631no—
268#103—1631no—
269#102—1631no—
270#102—1631no—
271#102—1631no—
272#102—1631no—
273C22⋊C8—1631no—
274C22⋊C8—1631no—
275#97—1631no—
276C24—3261no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—3262no—
282#41—6521no—
283C2×Q8—1631no—
284C2×Q8C22×D16311yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—3262no—
292C2×D4—3261no—
293C2×D4—3261no—
294C2×D4—3261no—
295C2×D4—6521no—
296C2×D4—1631no—
297C2×D4—1631no—
298C2×D4C22⋊D16311yescharacteristic
299C2×D4C22×D16311yescharacteristic
300C2×D4C22×D16311yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—3261no—
304C22×C4—3261no—
305C22×C4C22×D16311yescharacteristic
306C22×C4C22×D16311yescharacteristic
307Q16—3262no—
308SD16—3262no—
309SD16—3262no—
310SD16—6521no—
311SD16—1634no—
312D8—3262no—
313D8—3262no—
314D8—3262no—
315D8—6521no—
316M16—3262no—
317C2×C8—3262no—
318C2×C8—1631no—
319C2×C8—1631no—
320C2×C8—1631no—
321C2×C8—1631no—
322C4⋊C4—21no—
323C4⋊C4—3261no—
324C4⋊C4—1631no—
325C4⋊C4C22⋊D16311yescharacteristic
326C4⋊C4C22×D16311yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—3261no—
330C22⋊C4—3261no—
331C22⋊C4—3261no—
332C22⋊C4—3261no—
333C22⋊C4—3261no—
334C22⋊C4C22⋊D16311yescharacteristic
335C22⋊C4C22⋊D16311yescharacteristic
336C42—21no—
337C42C22×D16311yescharacteristic
338C23—21no—
339C23—21no—
340C23—13041no—
341C23—3261no—
342C23—3261no—
343C23—6522no—
344C23—6522no—
345C23C2×C22⋊D16311yescharacteristic
346C23C2×C22⋊D16311yescharacteristic
347Q8—21no—
348Q8—3261no—
349Q8—3261no—
350Q8D4×D16312yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—41no—
356D4—42no—
357D4—42no—
358D4—42no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—13041no—
363D4—3261no—
364D4—3261no—
365D4—3261no—
366D4—6522no—
367D4—6522no—
368D4—1632no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—13041no—
378C2×C4—3261no—
379C2×C4—3261no—
380C2×C4—6522no—
381C2×C4C2×C22⋊D16311yescharacteristic
382C2×C4C2×C22⋊D16311yescharacteristic
383C2×C4C2×C22⋊D16311yescharacteristic
384C2×C4C2×C22⋊D16311yescharacteristic
385C2×C4C23×D16311yescharacteristic
386C8—3261no—
387C8—3261no—
388C8—3261no—
389C8—1632no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—13044no—
403C22—3262no—
404C22—6521no—
405C22—6521no—
406C22—6521no—
407C22—6521no—
408C22—6521no—
409C22C22×C22⋊D16311yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—22no—
415C4—41no—
416C4—6521no—
417C4—6521no—
418C4C2×D4×D16311yescharacteristic
419C4#1910478011yescharacteristic
420C2—41no—
421C2—41no—
422C2—81no—
423C2—81no—
424C2—13041no—
425C2—6522no—
426C2D4×C22⋊D16311yescharacteristic
427C2#2378246311yescharacteristic
428C2#2378246811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#550×C163C2
socleC22×C163C22×D4
hypercenter#550D163
hypocenterC163#3268
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#19104760C2semidirect
2. C2×#19104755C2semidirect
3. #23782311C2semidirect
4. #23782052C2semidirect
5. #23781909C2semidirect
6. #23781899C2semidirect
7. #23781887C2semidirect
8. #23781857C2semidirect
9. #23781827C2semidirect
10. #23781810C2semidirect
11. #23781808C2semidirect
12. #23781798C2semidirect
13. #23781763C2semidirect
14. #23781737C2semidirect
15. #550×C163C2semidirect
20. C2×#14579689C22semidirect
21. C2×#14579687C22semidirect
22. C2×#14579687C22semidirect
30. C22×C163⋊C8C22semidirect
39. #19104668C22semidirect
40. C4⋊(C163⋊C8)C22semidirect
41. #19104659C22semidirect
42. #19104658C22semidirect
43. #19104656C22semidirect
44. #19104656C22semidirect
48. C22⋊(C163⋊C8)C22semidirect
49. #19104646C22semidirect
51. C2×#41×C163C22semidirect
53. #127×C163C22semidirect
57. #121×C163C22semidirect
58. C4×Q8×C163C22semidirect
78. #14579686D4semidirect
214. C163#3268semidirect
218. #550D163semidirect
233. C2×#41C2×D163semidirect
249. #127C2×D163semidirect
258. #121C2×D163semidirect
260. C4×Q8C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047602
C2×#191047552
#237823112
#237820522
#237819092
#237818992
#237818872
#237818572
#237818272
#237818102
#237818082
#237817982
#237817632
#237817372
#550×C1632
#3268163