Finite Group Databasebeta

← #35224262

Subgroups — #35224262

Order 20864.

466 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782461C211yesmaximal, characteristic, complemented
2C2×#19104759C211yesmaximal, characteristic, complemented
3AGL(1,ℤ/8)×D163C211yesmaximal, characteristic, complemented
4#23782430C211yesmaximal, characteristic, complemented
5#23781934C211yesmaximal, characteristic, complemented
6#23781933C211yesmaximal, characteristic, complemented
7#23781932C211yesmaximal, characteristic, complemented
8#23781929C211yesmaximal, characteristic, complemented
9#23781928C211yesmaximal, characteristic, complemented
10#23781927C211yesmaximal, characteristic, complemented
11#23781926C211yesmaximal, characteristic, complemented
12#23781712C211yesmaximal, characteristic, complemented
13#23781646C211yesmaximal, characteristic, complemented
14#100×D163C211yesmaximal, characteristic, complemented
15#472×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17#41×D163C2211yescharacteristic
18C2×D4×D163C2211yescharacteristic
19#19104768C2211yescharacteristic, complemented
20#19104765C2211yescharacteristic, complemented
21#19104764—21no—
22#19104763C2211yescharacteristic
23#19104762—21no—
24#19104761C2211yescharacteristic
25#19104761C2211yescharacteristic, complemented
26#19104760C2211yescharacteristic
27#19104759—21no—
28#19104759—21no—
29#19104759—21no—
30#19104759—21no—
31#19104759—21no—
32#19104759—21no—
33#19104756—21no—
34#19104754—21no—
35C2×#14579688C2211yescharacteristic, complemented
36C2×#14579687C2211yescharacteristic, complemented
37SD16×D163—21no—
38C2×#14579684C2211yescharacteristic, complemented
39D8×D163—21no—
40#19104743—21no—
41#19104743C2211yescharacteristic, complemented
42#19104742C2211yescharacteristic, complemented
43#19104741—21no—
44#19104741C2211yescharacteristic, complemented
45C2×#14579682—21no—
46C2×#14579682C2211yescharacteristic
47M16×D163—21no—
48M16×D163C2211yescharacteristic
49M16×D163C2211yescharacteristic
50M16×D163C2211yescharacteristic
51#19104671C2211yescharacteristic, complemented
52#19104670C2211yescharacteristic, complemented
53#19104670C2211yescharacteristic, complemented
54#19104653C2211yescharacteristic, complemented
55#19104653C2211yescharacteristic, complemented
56#19104652C2211yescharacteristic, complemented
57#19104652C2211yescharacteristic, complemented
58#19104651C2211yescharacteristic, complemented
59#19104651C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C163C2211yescharacteristic
61AGL(1,ℤ/8)×C163C2211yescharacteristic, complemented
62C2×D8×C163C2211yescharacteristic, complemented
63#131×C163C2211yescharacteristic, complemented
64#108×C163C2211yescharacteristic, complemented
65#100×C163C2211yescharacteristic, complemented
66#100×C163C2211yescharacteristic, complemented
67C23×D163—21no—
68#14579700D411yescharacteristic
69#14579699—21no—
70#14579699C2311yescharacteristic
71#14579698—21no—
72#14579698—21no—
73#14579698—21no—
74#14579698—21no—
75#14579698—21no—
76#14579698D411yescharacteristic
77C2×Dic326C2311yescharacteristic
78Q8×D163D411yescharacteristic
79C2×D652C2311yescharacteristic
80C2×C22⋊D163—21no—
81C2×C22⋊D163—21no—
82D4×D163—21no—
83D4×D163—21no—
84D4×D163—21no—
85D4×D163—21no—
86D4×D163—21no—
87D4×D163D411yescharacteristic
88C22×Dic163—21no—
89C2×C4×D163—21no—
90C2×C4×D163C2311yescharacteristic
91Dic652—21no—
92#14579689—21no—
93#14579688—21no—
94#14579688—21no—
95#14579688—21no—
96#14579688—21no—
97#14579687—21no—
98#14579687—21no—
99#14579687—21no—
100#14579687—21no—
101#14579687—21no—
102#14579686—21no—
103D1304—21no—
104#14579684—21no—
105#14579684—21no—
106#14579684—21no—
107#14579684—21no—
108#14579684—21no—
109#14579683—21no—
110#14579683C2311yescharacteristic
111#14579683C2311yescharacteristic
112#14579683C2311yescharacteristic
113#14579682—21no—
114#14579682—21no—
115#14579682—21no—
116#14579682—21no—
117#14579682—21no—
118#14579682D411yescharacteristic
119#14579682D411yescharacteristic
120#14579682D411yescharacteristic
121#14579682D411yescharacteristic
122C2×C163⋊C8—21no—
123C2×C163⋊C8C2311yescharacteristic
124C8×D163—21no—
125C8×D163—21no—
126C8×D163—21no—
127C8×D163—21no—
128C8×D163—21no—
129#41×C163C2311yescharacteristic
130C2×D4×C163C2311yescharacteristic
131C2×D4×C163C2311yescharacteristic
132SD16×C163—21no—
133SD16×C163—21no—
134D8×C163—21no—
135D8×C163—21no—
136D8×C163—21no—
137D8×C163—21no—
138M16×C163—21no—
139M16×C163C2311yescharacteristic
140M16×C163C2311yescharacteristic
141M16×C163C2311yescharacteristic
142C2×C8×C163—21no—
143C2×C8×C163C2311yescharacteristic
144C22×D163—21no—
145C22×D163—21no—
146C22×D163—41no—
147C22×D163—41no—
148C22×D163—82no—
149C22×D163#4111yescharacteristic
150Dic326—21no—
151Dic326—21no—
152Dic326—21no—
153Dic326C2×D411yescharacteristic
154D652—21no—
155D652—21no—
156D652—21no—
157D652C2×D411yescharacteristic
158C22⋊D163—21no—
159C22⋊D163—21no—
160C22⋊D163—21no—
161C22⋊D163—41no—
162C22⋊D163—41no—
163C22⋊D163—41no—
164C22⋊D163—41no—
165C2×Dic163—21no—
166C2×Dic163—21no—
167C2×Dic163—41no—
168C2×Dic163—41no—
169C2×Dic163#4111yescharacteristic
170C4×D163—21no—
171C4×D163—21no—
172C4×D163—21no—
173C4×D163C2×D411yescharacteristic
174C4×D163C2×D411yescharacteristic
175C4×D163C2×D411yescharacteristic
176C4×D163C2×D411yescharacteristic
177C163⋊C8—21no—
178C163⋊C8—21no—
179C163⋊C8—21no—
180C163⋊C8C2×D411yescharacteristic
181C163⋊C8C2×D411yescharacteristic
182C23×C163—21no—
183C23×C163—21no—
184Q8×C163C2×D411yescharacteristic
185D4×C163—21no—
186D4×C163—21no—
187D4×C163—21no—
188D4×C163—21no—
189D4×C163—21no—
190D4×C163C2×D411yescharacteristic
191C2×C4×C163—21no—
192C2×C4×C163C2411yescharacteristic
193C8×C163—21no—
194C8×C163—21no—
195C8×C163—21no—
196C8×C163C2×D411yescharacteristic
197C8×C163C2×D411yescharacteristic
198C22×C163—21no—
199C22×C163—41no—
200C22×C163—41no—
201C22×C163—81no—
202C22×C163—81no—
203C22×C163C2×#4111yescharacteristic
204C2×D163—21no—
205C2×D163—41no—
206C2×D163—41no—
207C2×D163—81no—
208C2×D163—82no—
209C2×D163—82no—
210C2×D163—81no—
211C2×D163#12111yescharacteristic
212C2×D163#12111yescharacteristic
213C4×C163—21no—
214C4×C163C22×D411yescharacteristic
215C4×C163C22×D411yescharacteristic
216Dic163—21no—
217Dic163—41no—
218Dic163#12111yescharacteristic
219Dic163#12111yescharacteristic
220C2×C163—21no—
221C2×C163—41no—
222C2×C163—81no—
223C2×C163—81no—
224C2×C163C2×#12111yescharacteristic
225D163—21no—
226D163—21no—
227D163—41no—
228D163—81no—
229C163#306811yescharacteristic, complemented, Sylow
230#3068—1631nomaximal, Sylow
231#575—1631no—
232C2×AGL(1,ℤ/8)—1631no—
233C2×AGL(1,ℤ/8)—1631no—
234#568—1631no—
235#472—1631no—
236#472—1631no—
237#472—1631no—
238#472D16311yescharacteristic, complemented
239#471—1631no—
240#471—1631no—
241#471—1631no—
242#471—1631no—
243#430—1631no—
244#413—1631no—
245C2×#100—1631no—
246C2×#41—1631no—
247C2×#41—1631no—
248C22×D4—1631no—
249#137—3261no—
250#137—1631no—
251#137—1631no—
252AGL(1,ℤ/8)—3261no—
253AGL(1,ℤ/8)—3261no—
254AGL(1,ℤ/8)—3261no—
255AGL(1,ℤ/8)—3261no—
256AGL(1,ℤ/8)—3261no—
257AGL(1,ℤ/8)—3261no—
258AGL(1,ℤ/8)—3261no—
259AGL(1,ℤ/8)—1631no—
260AGL(1,ℤ/8)—1631no—
261AGL(1,ℤ/8)—1631no—
262AGL(1,ℤ/8)—1631no—
263AGL(1,ℤ/8)C2×D16311yescharacteristic
264AGL(1,ℤ/8)C2×D16311yescharacteristic, complemented
265#135—3261no—
266#135—3261no—
267C2×SD16—3261no—
268C2×SD16—1631no—
269C2×SD16—1631no—
270C2×D8—3261no—
271C2×D8—1631no—
272C2×D8C2×D16311yescharacteristic, complemented
273#131—3261no—
274#131—3261no—
275#131—1631no—
276#131—1631no—
277#131—1631no—
278#131C2×D16311yescharacteristic, complemented
279C2×M16—3261no—
280C2×M16—3261no—
281C2×M16—1631no—
282C2×M16—1631no—
283C2×M16—1631no—
284C2×M16—1631no—
285#108—1631no—
286#108—1631no—
287#108—1631no—
288#108C2×D16311yescharacteristic, complemented
289#101—1631no—
290#101—1631no—
291#100—1631no—
292#100—1631no—
293#100—1631no—
294#100—1631no—
295#100C2×D16311yescharacteristic, complemented
296#100C2×D16311yescharacteristic, complemented
297C24—3261no—
298#41—3261no—
299#41—3261no—
300#41—3261no—
301#41—3261no—
302#41—3261no—
303#41—3261no—
304#41—1631no—
305#41—1631no—
306#41—1631no—
307#41C22×D16311yescharacteristic
308C2×Q8—1631no—
309C2×Q8—1631no—
310C2×D4—3261no—
311C2×D4—3261no—
312C2×D4—3261no—
313C2×D4—3261no—
314C2×D4—3261no—
315C2×D4—3261no—
316C2×D4—3261no—
317C2×D4—1631no—
318C2×D4—1631no—
319C2×D4C22×D16311yescharacteristic
320C2×D4C22×D16311yescharacteristic
321C22×C4—3261no—
322C22×C4—3261no—
323C22×C4—1631no—
324Q16—3261no—
325Q16—3261no—
326SD16—21no—
327SD16—21no—
328SD16—3261no—
329SD16—3261no—
330SD16—3261no—
331SD16—3261no—
332SD16—3261no—
333SD16—3261no—
334SD16—3261no—
335SD16—3261no—
336SD16—3261no—
337SD16—3261no—
338D8—21no—
339D8—21no—
340D8—21no—
341D8—21no—
342D8—3261no—
343D8—3261no—
344D8—3261no—
345D8—3261no—
346D8—3261no—
347D8—3261no—
348M16—21no—
349M16—3261no—
350M16—3261no—
351M16—3261no—
352M16—3261no—
353M16—3261no—
354M16—3261no—
355M16—1631no—
356M16—1631no—
357M16—1631no—
358M16—1631no—
359M16—1631no—
360M16—1631no—
361M16—1631no—
362M16C22×D16311yescharacteristic
363M16C22×D16311yescharacteristic
364M16C22×D16311yescharacteristic
365C2×C8—21no—
366C2×C8—3261no—
367C2×C8—3261no—
368C2×C8—3261no—
369C2×C8—3261no—
370C2×C8—3261no—
371C2×C8—3261no—
372C2×C8—1631no—
373C2×C8C22×D16311yescharacteristic
374C23—21no—
375C23—21no—
376C23—13042no—
377C23—3261no—
378C23—3261no—
379C23—6521no—
380C23—6521no—
381C23—1631no—
382Q8—3261no—
383Q8—3261no—
384Q8—3261no—
385Q8—1631no—
386Q8D4×D16311yescharacteristic
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—3261no—
393D4—3261no—
394D4—3261no—
395D4—3261no—
396D4—3261no—
397D4—3261no—
398D4—6521no—
399D4—6521no—
400D4—6521no—
401D4—6521no—
402D4—1631no—
403D4D4×D16311yescharacteristic
404C2×C4—21no—
405C2×C4—3261no—
406C2×C4—3261no—
407C2×C4—3261no—
408C2×C4—3261no—
409C2×C4—3261no—
410C2×C4—6521no—
411C2×C4—6521no—
412C2×C4—1631no—
413C2×C4—1631no—
414C2×C4—1631no—
415C2×C4—1631no—
416C2×C4—1631no—
417C2×C4C23×D16311yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—3261no—
422C8—3261no—
423C8—3261no—
424C8—1631no—
425C8—1631no—
426C8D4×D16311yescharacteristic
427C8D4×D16311yescharacteristic
428C22—21no—
429C22—41no—
430C22—41no—
431C22—81no—
432C22—81no—
433C22—13041no—
434C22—13041no—
435C22—13042no—
436C22—13042no—
437C22—3261no—
438C22—6521no—
439C22—6521no—
440C22—1631no—
441C22—1631no—
442C22#41×D16311yescharacteristic
443C4—21no—
444C4—3261no—
445C4—6521no—
446C4—1631no—
447C4—1631no—
448C4C2×D4×D16311yescharacteristic
449C4C2×D4×D16311yescharacteristic
450C2—21no—
451C2—41no—
452C2—81no—
453C2—81no—
454C2—13041no—
455C2—3261no—
456C2—3261no—
457C2—6521no—
458C2#121×D16311yescharacteristic
subgroupG/H
centerC2#121×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#472×C163C2
socleC2×C163C2×#121
hypercenter#472D163
hypocenterC163#3068
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #23782461C2semidirect
2. C2×#19104759C2semidirect
3. AGL(1,ℤ/8)×D163C2semidirect
4. #23782430C2semidirect
5. #23781934C2semidirect
6. #23781933C2semidirect
7. #23781932C2semidirect
8. #23781929C2semidirect
9. #23781928C2semidirect
10. #23781927C2semidirect
11. #23781926C2semidirect
12. #23781712C2semidirect
13. #23781646C2semidirect
14. #100×D163C2semidirect
15. #472×C163C2semidirect
19. #19104768C22semidirect
20. #19104765C22semidirect
25. #19104761C22semidirect
35. C2×#14579688C22semidirect
36. C2×#14579687C22semidirect
38. C2×#14579684C22semidirect
41. #19104743C22semidirect
42. #19104742C22semidirect
44. #19104741C22semidirect
51. #19104671C22semidirect
52. #19104670C22semidirect
53. #19104670C22semidirect
54. #19104653C22semidirect
55. #19104653C22semidirect
56. #19104652C22semidirect
57. #19104652C22semidirect
58. #19104651C22semidirect
59. #19104651C22semidirect
61. AGL(1,ℤ/8)×C163C22semidirect
62. C2×D8×C163C22semidirect
63. #131×C163C22semidirect
64. #108×C163C22semidirect
65. #100×C163C22semidirect
66. #100×C163C22semidirect
229. C163#3068semidirect
238. #472D163semidirect
264. AGL(1,ℤ/8)C2×D163semidirect
272. C2×D8C2×D163semidirect
278. #131C2×D163semidirect
288. #108C2×D163semidirect
295. #100C2×D163semidirect
296. #100C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824612
C2×#191047592
AGL(1,ℤ/8)×D1632
#237824302
#237819342
#237819332
#237819322
#237819292
#237819282
#237819272
#237819262
#237817122
#237816462
#100×D1632
#472×C1632
#3068163