Finite Group Databasebeta

← #35226029

Subgroups — #35226029

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
2C2×SD16×D163C211yesmaximal, characteristic, complemented
3#23782325C211yesmaximal, characteristic, complemented
4D4×C22⋊D163C211yesmaximal, characteristic, complemented
5#23782051C211yesmaximal, characteristic, complemented
6#23781913C211yesmaximal, characteristic, complemented
7#23781900C211yesmaximal, characteristic, complemented
8#23781882C211yesmaximal, characteristic, complemented
9#23781864C211yesmaximal, characteristic, complemented
10#23781812C211yesmaximal, characteristic, complemented
11#23781802C211yesmaximal, characteristic, complemented
12#23781800C211yesmaximal, characteristic, complemented
13#23781762C211yesmaximal, characteristic, complemented
14#23781739C211yesmaximal, characteristic, complemented
15#450×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic, complemented
17C2×Q8×D163C2211yescharacteristic, complemented
18C22×C22⋊D163—21no—
19C2×D4×D163C2211yescharacteristic
20#19104761—24no—
21C2×#14579688C2211yescharacteristic, complemented
22C2×#14579687C2211yescharacteristic
23C2×#14579687C2211yescharacteristic, complemented
24C2×#14579686C2211yescharacteristic, complemented
25SD16×D163—24no—
26C2×D1304C2211yescharacteristic, complemented
27C2×#14579684C2211yescharacteristic
28C2×#14579684C2211yescharacteristic, complemented
29C2×#14579683C2211yescharacteristic, complemented
30C2×C8×D163C2211yescharacteristic, complemented
31#19104731C2211yescharacteristic
32#19104724C2211yescharacteristic, complemented
33#19104721—21no—
34#19104713—21no—
35#19104710C2211yescharacteristic, complemented
36#19104709C2211yescharacteristic
37#19104708—21no—
38#19104707—21no—
39#19104706C2211yescharacteristic
40#19104704—21no—
41#19104703—21no—
42#19104700C2211yescharacteristic, complemented
43D4×Dic163C2211yescharacteristic
44#19104697—21no—
45C4×C22⋊D163C2211yescharacteristic
46#19104668C2211yescharacteristic, complemented
47Dic163⋊C8C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104659C2211yescharacteristic, complemented
50#19104657C2211yescharacteristic, complemented
51#19104655C2211yescharacteristic, complemented
52(C2×D163)⋊C8C2211yescharacteristic, complemented
53#19104645C2211yescharacteristic, complemented
54C22×D4×C163C2211yescharacteristic
55C2×SD16×C163C2211yescharacteristic
56C2×SD16×C163C2211yescharacteristic, complemented
57C22⋊Q8×C163C2211yescharacteristic, complemented
58#102×C163C2211yescharacteristic
59#102×C163C2211yescharacteristic, complemented
60C22⋊C8×C163C2211yescharacteristic, complemented
61C23×D163—21no—
62#14579699—22no—
63#14579699—41no—
64C2×Dic326—21no—
65C2×Dic326C2311yescharacteristic
66Q8×D163—22no—
67Q8×D163—41no—
68C2×D652—21no—
69C2×D652C2311yescharacteristic
70C2×D652C2311yescharacteristic
71C2×C22⋊D163—21no—
72C2×C22⋊D163—21no—
73C2×C22⋊D163—21no—
74C2×C22⋊D163—21no—
75C2×C22⋊D163—21no—
76C2×C22⋊D163—21no—
77C2×C22⋊D163—44no—
78C2×C22⋊D163D411yescharacteristic
79D4×D163—22no—
80D4×D163—41no—
81C22×Dic163—21no—
82C2×C4×D163—21no—
83C2×C4×D163—21no—
84C2×C4×D163C2311yescharacteristic
85#14579688—22no—
86#14579687—22no—
87#14579687D414yescomplemented
88#14579686—22no—
89D1304—41no—
90#14579684—22no—
91#14579684—22no—
92#14579683—22no—
93C2×C163⋊C8C2311yescharacteristic
94C2×C163⋊C8C2311yescharacteristic
95C8×D163—22no—
96C4⋊Dic163C2311yescharacteristic
97Dic163⋊C4—21no—
98Dic163⋊C4D411yescharacteristic
99C22⋊Dic163—21no—
100C22⋊Dic163—21no—
101C22⋊Dic163D411yescharacteristic
102(C2×D163)⋊C4—21no—
103(C2×D163)⋊C4—21no—
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4—21no—
106(C2×D163)⋊C4D411yescharacteristic
107C4×Dic163—21no—
108C4×Dic163C2311yescharacteristic
109C24×C163—21no—
110C2×Q8×C163C2311yescharacteristic
111C2×D4×C163—22no—
112C2×D4×C163—21no—
113C2×D4×C163—41no—
114C2×D4×C163C2311yescharacteristic
115C2×D4×C163C2311yescharacteristic
116C22×C4×C163C2311yescharacteristic
117SD16×C163—22no—
118SD16×C163—41no—
119C2×C8×C163C2311yescharacteristic
120C2×C8×C163C2311yescharacteristic
121C4⋊C4×C163—21no—
122C4⋊C4×C163C2311yescharacteristic
123C22⋊C4×C163—21no—
124C22×D163—21no—
125C22×D163—21no—
126C22×D163—42no—
127C22×D163—42no—
128C22×D163—81no—
129C22×D163C2×D411yescharacteristic
130Dic326—21no—
131Dic326—42no—
132Dic326C2×D412yescomplemented
133D652—21no—
134D652—21no—
135D652—21no—
136D652—21no—
137D652—42no—
138C22⋊D163—22no—
139C22⋊D163—42no—
140C22⋊D163—42no—
141C22⋊D163—42no—
142C22⋊D163—42no—
143C22⋊D163—42no—
144C22⋊D163—81no—
145C2×Dic163—21no—
146C2×Dic163—21no—
147C2×Dic163—42no—
148C2×Dic163C2×D411yescharacteristic
149C2×Dic163C2×D411yescharacteristic
150C2×Dic163C2×D411yescharacteristic
151C4×D163—21no—
152C4×D163—21no—
153C4×D163—42no—
154C4×D163—42no—
155C163⋊C8—21no—
156C163⋊C8C2×D412yescomplemented
157C23×C163—21no—
158C23×C163—21no—
159C23×C163—42no—
160C23×C163—42no—
161C23×C163—81no—
162C23×C163C2×D411yescharacteristic
163Q8×C163—21no—
164Q8×C163—21no—
165D4×C163—21no—
166D4×C163—21no—
167D4×C163—21no—
168D4×C163—42no—
169D4×C163C2×D412yes—
170C2×C4×C163—21no—
171C2×C4×C163—21no—
172C2×C4×C163—21no—
173C2×C4×C163—21no—
174C2×C4×C163C2×D411yescharacteristic
175C2×C4×C163C2411yescharacteristic
176C8×C163—21no—
177C8×C163—21no—
178C22×C163—21no—
179C22×C163—22no—
180C22×C163—21no—
181C22×C163—21no—
182C22×C163—41no—
183C22×C163—41no—
184C22×C163—41no—
185C22×C163—41no—
186C22×C163—41no—
187C22×C163—84no—
188C22×C163C22×D411yescharacteristic
189C2×D163—21no—
190C2×D163—21no—
191C2×D163—41no—
192C2×D163—41no—
193C2×D163—41no—
194C2×D163—41no—
195C2×D163—41no—
196C2×D163—41no—
197C2×D163—84no—
198C2×D163AGL(1,ℤ/8)12yescomplemented
199C4×C163—21no—
200C4×C163—41no—
201C4×C163—41no—
202C4×C163C22×D411yescharacteristic
203C4×C163H(5,𝔽2)11yescharacteristic
204Dic163—21no—
205Dic163—22no—
206Dic163—21no—
207Dic163—21no—
208Dic163—41no—
209C2×C163—42no—
210C2×C163—41no—
211C2×C163—81no—
212C2×C163D4211yescharacteristic
213C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C163#57711yescharacteristic
215D163—22no—
216D163—81no—
217D163—81no—
218C163#326511yescharacteristic, complemented, Sylow
219#3265—1631nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1631no—
221C22×SD16—1631no—
222#548—1631no—
223D42—1631no—
224#496—1631no—
225#468—1631no—
226#466—1631no—
227#463—1631no—
228#460—1631no—
229#450D16311yescharacteristic, complemented
230#449—1631no—
231#448—1631no—
232#447—1631no—
233#440—1631no—
234#435—1631no—
235C2×#41—1631no—
236C22×Q8—1631no—
237C22×D4—3261no—
238C22×D4—1631no—
239C22×D4C2×D16311yescharacteristic
240AGL(1,ℤ/8)—3264no—
241C2×SD16—3264no—
242C2×SD16—1631no—
243C2×SD16—1631no—
244C2×SD16—1631no—
245C2×SD16—1631no—
246C2×SD16C2×D16311yescharacteristic
247C2×SD16C2×D16311yescharacteristic, complemented
248C2×D8—1631no—
249C2×D8—1631no—
250C2×D8—1631no—
251C2×M16—1631no—
252C22×C8—1631no—
253#127—1631no—
254#124—3261no—
255#124—1631no—
256C22⋊Q8—3261no—
257C22⋊Q8C2×D16311yescharacteristic, complemented
258#121—3261no—
259#121—3261no—
260#121—1631no—
261#121—1631no—
262#121—1631no—
263#120—3261no—
264#120—3261no—
265C4×Q8—1631no—
266C4×D4—3261no—
267C4×D4—1631no—
268C4×D4—1631no—
269#107—1631no—
270C4⋊C8—1631no—
271#103—1631no—
272#103—1631no—
273#102—1631no—
274#102—1631no—
275#102C2×D16311yescharacteristic
276#102C2×D16311yescharacteristic, complemented
277C22⋊C8—1631no—
278C22⋊C8C2×D16311yescharacteristic, complemented
279#97—1631no—
280C24—21no—
281C24—3261no—
282#41—3262no—
283#41—6521no—
284C2×Q8—3261no—
285C2×Q8—3262no—
286C2×Q8—6521no—
287C2×Q8—1631no—
288C2×Q8C22×D16311yescharacteristic
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—3261no—
293C2×D4—3262no—
294C2×D4—3261no—
295C2×D4—3261no—
296C2×D4—3261no—
297C2×D4—3261no—
298C2×D4—3261no—
299C2×D4—3261no—
300C2×D4—6521no—
301C2×D4—6524no—
302C2×D4—1631no—
303C2×D4—1631no—
304C2×D4—1631no—
305C2×D4C22×D16311yescharacteristic
306C2×D4C22×D16311yescharacteristic
307C22×C4—3261no—
308C22×C4—3261no—
309C22×C4—3261no—
310C22×C4—1631no—
311C22×C4C22×D16311yescharacteristic
312SD16—22no—
313SD16—41no—
314SD16—3262no—
315SD16—3262no—
316SD16—3262no—
317SD16—1634no—
318D8—3262no—
319D8—3262no—
320D8—6521no—
321M16—3262no—
322C2×C8—3262no—
323C2×C8—1631no—
324C2×C8—1631no—
325C2×C8C22×D16311yescharacteristic
326C2×C8C22×D16311yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—3261no—
329C4⋊C4—1631no—
330C4⋊C4—1631no—
331C4⋊C4C22×D16311yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—3261no—
334C22⋊C4—3261no—
335C22⋊C4—3261no—
336C22⋊C4—3261no—
337C22⋊C4—3261no—
338C22⋊C4—3261no—
339C22⋊C4—1631no—
340C22⋊C4—1631no—
341C42—3261no—
342C42—1631no—
343C23—21no—
344C23—21no—
345C23—42no—
346C23—42no—
347C23—81no—
348C23—13041no—
349C23—3261no—
350C23—3261no—
351C23—6522no—
352C23—6522no—
353C23—1631no—
354C23D4×D16311yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—3261no—
358Q8—6522no—
359Q8—1632no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—13041no—
365D4—3262no—
366D4—3261no—
367D4—3261no—
368D4—3261no—
369D4—3261no—
370D4—6522no—
371D4—6522no—
372D4—6522no—
373D4—6522no—
374D4—6522no—
375D4—6522no—
376D4D4×D16312yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—3261no—
382C2×C4—3261no—
383C2×C4—3261no—
384C2×C4—3261no—
385C2×C4—6522no—
386C2×C4—6522no—
387C2×C4—6522no—
388C2×C4—1631no—
389C2×C4—1631no—
390C2×C4—1631no—
391C2×C4D4×D16311yescharacteristic
392C2×C4C23×D16311yescharacteristic
393C8—21no—
394C8—21no—
395C8—3261no—
396C8—1632no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—84no—
407C22—13044no—
408C22—3261no—
409C22—3261no—
410C22—6521no—
411C22—6521no—
412C22—6521no—
413C22—6521no—
414C22—6521no—
415C22—6521no—
416C22—1632no—
417C22C2×D4×D16311yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—3261no—
422C4—3261no—
423C4—3261no—
424C4—3262no—
425C4—6521no—
426C4C2×D4×D16311yescharacteristic
427C4#1910478011yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—13041no—
432C2—13041no—
433C2—3262no—
434C2#2378229711yescharacteristic
435C2AGL(1,ℤ/8)×D16311yescharacteristic
436C2#2378247111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#450×C163C2
socleC22×C163C22×D4
hypercenter#450D163
hypocenterC163#3265
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#19104761C2semidirect
2. C2×SD16×D163C2semidirect
3. #23782325C2semidirect
4. D4×C22⋊D163C2semidirect
5. #23782051C2semidirect
6. #23781913C2semidirect
7. #23781900C2semidirect
8. #23781882C2semidirect
9. #23781864C2semidirect
10. #23781812C2semidirect
11. #23781802C2semidirect
12. #23781800C2semidirect
13. #23781762C2semidirect
14. #23781739C2semidirect
15. #450×C163C2semidirect
16. C2×#14579699C22semidirect
17. C2×Q8×D163C22semidirect
21. C2×#14579688C22semidirect
23. C2×#14579687C22semidirect
24. C2×#14579686C22semidirect
26. C2×D1304C22semidirect
28. C2×#14579684C22semidirect
29. C2×#14579683C22semidirect
30. C2×C8×D163C22semidirect
32. #19104724C22semidirect
35. #19104710C22semidirect
42. #19104700C22semidirect
46. #19104668C22semidirect
47. Dic163⋊C8C22semidirect
48. #19104660C22semidirect
49. #19104659C22semidirect
50. #19104657C22semidirect
51. #19104655C22semidirect
52. (C2×D163)⋊C8C22semidirect
53. #19104645C22semidirect
56. C2×SD16×C163C22semidirect
57. C22⋊Q8×C163C22semidirect
59. #102×C163C22semidirect
60. C22⋊C8×C163C22semidirect
87. #14579687D4semidirect
132. Dic326C2×D4semidirect
156. C163⋊C8C2×D4semidirect
198. C2×D163AGL(1,ℤ/8)semidirect
218. C163#3265semidirect
229. #450D163semidirect
247. C2×SD16C2×D163semidirect
257. C22⋊Q8C2×D163semidirect
276. #102C2×D163semidirect
278. C22⋊C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047612
C2×SD16×D1632
#237823252
D4×C22⋊D1632
#237820512
#237819132
#237819002
#237818822
#237818642
#237818122
#237818022
#237818002
#237817622
#237817392
#450×C1632
#3265163