Finite Group Databasebeta

← #35226320

Subgroups — #35226320

Order 20864.

454 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782481C211yesmaximal, characteristic, complemented
2#23782476C211yesmaximal, characteristic, complemented
3#23782474C211yesmaximal, characteristic, complemented
4#23782468C211yesmaximal, characteristic, complemented
5#23782047C211yesmaximal, characteristic, complemented
6#23781938C211yesmaximal, characteristic, complemented
7#23781934C211yesmaximal, characteristic, complemented
8#23781930C211yesmaximal, characteristic, complemented
9#23781927C211yesmaximal, characteristic, complemented
10#23781844C211yesmaximal, characteristic, complemented
11#23781839C211yesmaximal, characteristic, complemented
12#23781837C211yesmaximal, characteristic, complemented
13#23781779C211yesmaximal, characteristic, complemented
14#23781779C211yesmaximal, characteristic, complemented
15#455×C163C211yesmaximal, characteristic, complemented
16#19104783C2211yescharacteristic, complemented
17#19104782C2211yescharacteristic, complemented
18#19104781C2211yescharacteristic
19#19104768C2211yescharacteristic, complemented
20#19104767—21no—
21#19104767—21no—
22#19104765—21no—
23#19104765C2211yescharacteristic
24#19104764—21no—
25#19104763C2211yescharacteristic, complemented
26#19104762—21no—
27#19104761—21no—
28#19104761—21no—
29#19104761C2211yescharacteristic
30#19104761C2211yescharacteristic, complemented
31#19104760—21no—
32#19104760C2211yescharacteristic
33#19104758—21no—
34#19104756—21no—
35#19104755—21no—
36#19104755—21no—
37#19104755C2211yescharacteristic
38#19104755C2211yescharacteristic
39C2×#14579689C2211yescharacteristic, complemented
40Q16×D163—21no—
41C2×#14579687—21no—
42C2×#14579687C2211yescharacteristic, complemented
43C2×#14579686C2211yescharacteristic, complemented
44SD16×D163—21no—
45C2×#14579684—21no—
46C2×#14579684C2211yescharacteristic
47#19104743C2211yescharacteristic, complemented
48#19104743C2211yescharacteristic, complemented
49#19104742C2211yescharacteristic
50#19104742C2211yescharacteristic, complemented
51#19104723C2211yescharacteristic, complemented
52#19104670C2211yescharacteristic, complemented
53#19104670C2211yescharacteristic, complemented
54#19104663C2211yescharacteristic, complemented
55#19104663C2211yescharacteristic, complemented
56#19104654C2211yescharacteristic, complemented
57#19104653C2211yescharacteristic, complemented
58#19104651C2211yescharacteristic, complemented
59C4×C163⋊C8C2211yescharacteristic, complemented
60H(5,𝔽2)×C163C2211yescharacteristic
61#137×C163C2211yescharacteristic
62#137×C163C2211yescharacteristic, complemented
63#124×C163C2211yescharacteristic, complemented
64#104×C163C2211yescharacteristic
65#104×C163C2211yescharacteristic, complemented
66#100×C163C2211yescharacteristic, complemented
67#14579700—21no—
68#14579700—21no—
69#14579700—21no—
70#14579699—21no—
71#14579699—21no—
72#14579699—41no—
73#14579699C2311yescharacteristic
74#14579699C2311yescharacteristic
75#14579698—21no—
76#14579698—41no—
77C2×Dic326—21no—
78C2×Dic326C2311yescharacteristic
79Q8×D163—21no—
80Q8×D163—21no—
81Q8×D163—21no—
82C2×D652—21no—
83C2×D652C2311yescharacteristic
84D4×D163—21no—
85D4×D163—41no—
86Dic652—21no—
87#14579689—21no—
88#14579689—21no—
89#14579689—21no—
90#14579689D411yescharacteristic
91#14579689D411yescharacteristic
92#14579688—21no—
93#14579688—21no—
94#14579687—21no—
95#14579687—21no—
96#14579687—21no—
97#14579687—21no—
98#14579687—21no—
99#14579687D411yescharacteristic
100#14579687D411yescharacteristic
101#14579686—21no—
102#14579686—21no—
103#14579686—21no—
104#14579686D411yescharacteristic
105#14579686D411yescharacteristic
106D1304—21no—
107#14579684—21no—
108#14579684—21no—
109#14579684—21no—
110#14579684—21no—
111#14579684—21no—
112#14579684D411yescharacteristic
113#14579684D411yescharacteristic
114#14579683—21no—
115#14579683—21no—
116#14579683C2311yescharacteristic
117#14579683C2311yescharacteristic
118#14579682—21no—
119#14579682—21no—
120C2×C163⋊C8—21no—
121C2×C163⋊C8—21no—
122C2×C163⋊C8C2311yescharacteristic
123C2×C163⋊C8C2311yescharacteristic
124C8×D163—21no—
125C8×D163—21no—
126(C2×D163)⋊C4—41no—
127#41×C163—21no—
128#41×C163—21no—
129#41×C163C2311yescharacteristic
130#41×C163C2311yescharacteristic
131C2×Q8×C163C2311yescharacteristic
132C2×D4×C163—21no—
133C2×D4×C163—21no—
134C2×D4×C163—41no—
135C2×D4×C163C2311yescharacteristic
136Q16×C163—21no—
137Q16×C163—21no—
138SD16×C163—21no—
139SD16×C163—21no—
140M16×C163C2311yescharacteristic
141M16×C163C2311yescharacteristic
142C22⋊C4×C163—41no—
143C42×C163C2311yescharacteristic
144C22×D163—21no—
145C22×D163—41no—
146Dic326—21no—
147Dic326—21no—
148Dic326—41no—
149Dic326—41no—
150Dic326C2×D411yescharacteristic
151Dic326C2×D411yescharacteristic
152D652—21no—
153D652—21no—
154D652—41no—
155D652—41no—
156D652C2×D411yescharacteristic
157D652C2×D411yescharacteristic
158C22⋊D163—21no—
159C22⋊D163—21no—
160C22⋊D163—41no—
161C22⋊D163—41no—
162C2×Dic163—21no—
163C2×Dic163—41no—
164C4×D163—21no—
165C4×D163—21no—
166C4×D163—41no—
167C4×D163—41no—
168C4×D163—41no—
169C4×D163—41no—
170C163⋊C8—21no—
171C163⋊C8—21no—
172C163⋊C8C2×D411yescharacteristic
173C163⋊C8C2×D411yescharacteristic
174C163⋊C8C2×D411yescharacteristic
175C163⋊C8C2×D411yescharacteristic
176C23×C163—21no—
177C23×C163—41no—
178Q8×C163—21no—
179Q8×C163—21no—
180Q8×C163C2×D411yescharacteristic
181Q8×C163C2×D411yescharacteristic
182D4×C163—21no—
183D4×C163—21no—
184D4×C163—21no—
185D4×C163—21no—
186D4×C163—41no—
187D4×C163—41no—
188D4×C163C2×D411yescharacteristic
189D4×C163C2×D411yescharacteristic
190C2×C4×C163—21no—
191C2×C4×C163—21no—
192C2×C4×C163—21no—
193C2×C4×C163—21no—
194C2×C4×C163—41no—
195C2×C4×C163C2411yescharacteristic
196C8×C163—21no—
197C8×C163—21no—
198C22×C163—21no—
199C22×C163—21no—
200C22×C163—41no—
201C22×C163—81no—
202C22×C163—161no—
203C22×C163H(5,𝔽2)11yescharacteristic
204C2×D163—21no—
205C2×D163—21no—
206C2×D163—41no—
207C2×D163—81no—
208C2×D163—161no—
209C4×C163—21no—
210C4×C163—21no—
211C4×C163—41no—
212C4×C163—41no—
213C4×C163C22×D411yescharacteristic
214C4×C163C22×D411yescharacteristic
215Dic163—21no—
216Dic163—21no—
217Dic163—41no—
218C2×C163—21no—
219C2×C163—41no—
220C2×C163—41no—
221C2×C163—81no—
222C2×C163D4211yescharacteristic
223D163—41no—
224D163—41no—
225D163—81no—
226C163#329011yescharacteristic, complemented, Sylow
227#3290—1631nomaximal, Sylow
228#578—1631no—
229#577—1631no—
230#577—1631no—
231#576—1631no—
232#495—1631no—
233#473—1631no—
234#472—1631no—
235#471—1631no—
236#471—1631no—
237#455—1631no—
238#455D16311yescharacteristic, complemented
239#454—1631no—
240#454—1631no—
241#443—1631no—
242#443—1631no—
2432−1+4—1631no—
2442−1+4—1631no—
245H(5,𝔽2)—1631no—
246H(5,𝔽2)C2×D16311yescharacteristic
247#137—3261no—
248#137—3261no—
249#137—3261no—
250#137—3261no—
251#137—1631no—
252#137—1631no—
253#137C2×D16311yescharacteristic
254#137C2×D16311yescharacteristic, complemented
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—
263#135—3261no—
264#135—3261no—
265#135—3261no—
266#135—3261no—
267#135—1631no—
268#135—1631no—
269C2×Q16—3261no—
270C2×Q16—1631no—
271C2×SD16—3261no—
272C2×SD16—3261no—
273C2×SD16—1631no—
274C2×SD16—1631no—
275C2×D8—3261no—
276C2×D8—1631no—
277#131—1631no—
278#131—1631no—
279#131—1631no—
280#131—1631no—
281#124—1631no—
282#124C2×D16311yescharacteristic, complemented
283#108—1631no—
284#108—1631no—
285#104—1631no—
286#104—1631no—
287#104C2×D16311yescharacteristic
288#104C2×D16311yescharacteristic, complemented
289#101—1631no—
290#101—1631no—
291#100—1631no—
292#100C2×D16311yescharacteristic, complemented
293C4×C8—1631no—
294#41—21no—
295#41—21no—
296#41—3261no—
297#41—3261no—
298#41—3261no—
299#41—3261no—
300#41—3261no—
301#41—3261no—
302#41—6521no—
303#41—6521no—
304#41—1631no—
305#41—1631no—
306#41C22×D16311yescharacteristic
307#41C22×D16311yescharacteristic
308C2×Q8—3261no—
309C2×Q8—3261no—
310C2×Q8—3261no—
311C2×Q8—3261no—
312C2×Q8—1631no—
313C2×Q8C22×D16311yescharacteristic
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—41no—
317C2×D4—3261no—
318C2×D4—3261no—
319C2×D4—6521no—
320C2×D4—1631no—
321C2×D4C22×D16311yescharacteristic
322Q16—21no—
323Q16—21no—
324Q16—3261no—
325Q16—3261no—
326Q16—3261no—
327Q16—3261no—
328Q16—1631no—
329Q16—1631no—
330SD16—21no—
331SD16—21no—
332SD16—3261no—
333SD16—3261no—
334SD16—3261no—
335SD16—3261no—
336SD16—3261no—
337SD16—3261no—
338SD16—3261no—
339SD16—3261no—
340SD16—3261no—
341SD16—3261no—
342SD16—1631no—
343SD16—1631no—
344SD16—1631no—
345SD16—1631no—
346D8—3261no—
347D8—3261no—
348D8—3261no—
349D8—3261no—
350D8—3261no—
351D8—3261no—
352D8—1631no—
353D8—1631no—
354M16—3261no—
355M16—3261no—
356M16—3261no—
357M16—3261no—
358M16—1631no—
359M16—1631no—
360M16C22×D16311yescharacteristic
361M16C22×D16311yescharacteristic
362C2×C8—3261no—
363C2×C8—3261no—
364C2×C8—3261no—
365C2×C8—3261no—
366C2×C8—1631no—
367C2×C8—1631no—
368C22⋊C4—41no—
369C22⋊C4—6521no—
370C42C22×D16311yescharacteristic
371C23—21no—
372C23—41no—
373C23—3261no—
374C23—6521no—
375Q8—21no—
376Q8—21no—
377Q8—3261no—
378Q8—3261no—
379Q8—6521no—
380Q8—6521no—
381Q8—1631no—
382Q8—1631no—
383Q8D4×D16311yescharacteristic
384Q8D4×D16311yescharacteristic
385D4—21no—
386D4—21no—
387D4—21no—
388D4—21no—
389D4—41no—
390D4—41no—
391D4—3261no—
392D4—3261no—
393D4—3261no—
394D4—3261no—
395D4—6521no—
396D4—6521no—
397D4—6521no—
398D4—6521no—
399D4—1631no—
400D4—1631no—
401D4D4×D16311yescharacteristic
402D4D4×D16311yescharacteristic
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—41no—
408C2×C4—3261no—
409C2×C4—3261no—
410C2×C4—3261no—
411C2×C4—6521no—
412C2×C4—6521no—
413C2×C4—6521no—
414C2×C4—6521no—
415C2×C4—6521no—
416C2×C4C23×D16311yescharacteristic
417C8—21no—
418C8—21no—
419C8—3261no—
420C8—3261no—
421C8—1631no—
422C8—1631no—
423C8—1631no—
424C8—1631no—
425C22—21no—
426C22—21no—
427C22—41no—
428C22—81no—
429C22—161no—
430C22—13041no—
431C22—26081no—
432C22—3261no—
433C22—3261no—
434C22—6521no—
435C22#1910478011yescharacteristic
436C4—21no—
437C4—21no—
438C4—41no—
439C4—41no—
440C4—3261no—
441C4—3261no—
442C4—6521no—
443C4C2×D4×D16311yescharacteristic
444C4C2×D4×D16311yescharacteristic
445C2—21no—
446C2—41no—
447C2—41no—
448C2—81no—
449C2—13041no—
450C2—6521no—
451C2—6521no—
452C2#2378229711yescharacteristic
subgroupG/H
centerC2#23782297
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#455×C163C2
socleC2×C163D42
hypercenter#455D163
hypocenterC163#3290
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2 ◁ C2×C4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #23782481C2semidirect
2. #23782476C2semidirect
3. #23782474C2semidirect
4. #23782468C2semidirect
5. #23782047C2semidirect
6. #23781938C2semidirect
7. #23781934C2semidirect
8. #23781930C2semidirect
9. #23781927C2semidirect
10. #23781844C2semidirect
11. #23781839C2semidirect
12. #23781837C2semidirect
13. #23781779C2semidirect
14. #23781779C2semidirect
15. #455×C163C2semidirect
16. #19104783C22semidirect
17. #19104782C22semidirect
19. #19104768C22semidirect
25. #19104763C22semidirect
30. #19104761C22semidirect
39. C2×#14579689C22semidirect
42. C2×#14579687C22semidirect
43. C2×#14579686C22semidirect
47. #19104743C22semidirect
48. #19104743C22semidirect
50. #19104742C22semidirect
51. #19104723C22semidirect
52. #19104670C22semidirect
53. #19104670C22semidirect
54. #19104663C22semidirect
55. #19104663C22semidirect
56. #19104654C22semidirect
57. #19104653C22semidirect
58. #19104651C22semidirect
59. C4×C163⋊C8C22semidirect
62. #137×C163C22semidirect
63. #124×C163C22semidirect
65. #104×C163C22semidirect
66. #100×C163C22semidirect
226. C163#3290semidirect
238. #455D163semidirect
254. #137C2×D163semidirect
282. #124C2×D163semidirect
288. #104C2×D163semidirect
292. #100C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237824812
#237824762
#237824742
#237824682
#237820472
#237819382
#237819342
#237819302
#237819272
#237818442
#237818392
#237818372
#237817792
#455×C1632
#3290163