Finite Group Databasebeta

← #35226036

Subgroups — #35226036

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×#19104763C211yesmaximal, characteristic, complemented
2C22×#14579688C211yesmaximal, characteristic, complemented
3#23782324C211yesmaximal, characteristic, complemented
4D4×D652C211yesmaximal, characteristic, complemented
5#23782054C211yesmaximal, characteristic, complemented
6#23781911C211yesmaximal, characteristic, complemented
7#23781901C211yesmaximal, characteristic, complemented
8#23781881C211yesmaximal, characteristic, complemented
9#23781866C211yesmaximal, characteristic, complemented
10#23781818C211yesmaximal, characteristic, complemented
11#23781812C211yesmaximal, characteristic, complemented
12#23781806C211yesmaximal, characteristic, complemented
13#23781801C211yesmaximal, characteristic, complemented
14#23781767C211yesmaximal, characteristic, complemented
15#435×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic, complemented
17C22×Dic326C2211yescharacteristic, complemented
18C22×D652C2211yescharacteristic
19C22×D652C2211yescharacteristic
20C2×D4×D163—21no—
21#19104763—24no—
22C2×#14579688—24no—
23C2×#14579688C2211yescharacteristic
24C2×#14579688C2211yescharacteristic
25C2×#14579688C2211yescharacteristic, complemented
26C2×#14579688C2211yescharacteristic, complemented
27C2×#14579688C2211yescharacteristic, complemented
28C2×#14579688C2211yescharacteristic, complemented
29C2×D1304C2211yescharacteristic
30C2×D1304C2211yescharacteristic, complemented
31C2×D1304C2211yescharacteristic, complemented
32#19104732C2211yescharacteristic
33#19104723C2211yescharacteristic, complemented
34#19104721—21no—
35#19104714C2211yescharacteristic, complemented
36#19104713—21no—
37#19104710—21no—
38#19104709C2211yescharacteristic
39#19104709C2211yescharacteristic
40#19104709C2211yescharacteristic, complemented
41#19104707—21no—
42#19104705—21no—
43#19104705—21no—
44C4×Dic326C2211yescharacteristic, complemented
45C4×D652C2211yescharacteristic
46(C22⋊D163)⋊C4—21no—
47#19104669C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104660C2211yescharacteristic, complemented
50#19104657C2211yescharacteristic
51#19104657C2211yescharacteristic, complemented
52#19104657C2211yescharacteristic, complemented
53#19104657C2211yescharacteristic, complemented
54C2×M16×C163C2211yescharacteristic, complemented
55C22×C8×C163C2211yescharacteristic, complemented
56C4×D4×C163C2211yescharacteristic
57C4⋊C8×C163C2211yescharacteristic, complemented
58C22⋊C8×C163C2211yescharacteristic, complemented
59C22⋊C8×C163C2211yescharacteristic, complemented
60#97×C163C2211yescharacteristic, complemented
61C23×D163—21no—
62C23×D163—21no—
63#14579699—22no—
64#14579699—41no—
65C2×Dic326—22no—
66C2×Dic326—21no—
67C2×Dic326—41no—
68C2×Dic326C2311yescharacteristic
69C2×Dic326C2311yescharacteristic
70C2×D652—22no—
71C2×D652—22no—
72C2×D652—21no—
73C2×D652—21no—
74C2×D652—21no—
75C2×D652—21no—
76C2×D652—41no—
77C2×D652—41no—
78C2×D652C2311yescharacteristic
79C2×D652C2311yescharacteristic
80C2×D652C2311yescharacteristic
81C2×D652C2311yescharacteristic
82C2×C22⋊D163—21no—
83C2×C22⋊D163—21no—
84C2×C22⋊D163—21no—
85C2×C22⋊D163—21no—
86D4×D163—44no—
87C22×Dic163—21no—
88C2×C4×D163—21no—
89C2×C4×D163—21no—
90#14579688—22no—
91#14579688—22no—
92#14579688—22no—
93#14579688—22no—
94#14579688—41no—
95#14579688D414yescomplemented
96D1304—22no—
97D1304—22no—
98D1304—41no—
99C4⋊Dic163C2311yescharacteristic
100C4⋊Dic163C2311yescharacteristic
101Dic163⋊C4—21no—
102Dic163⋊C4—21no—
103C22⋊Dic163—21no—
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4—21no—
106(C2×D163)⋊C4—21no—
107(C2×D163)⋊C4—21no—
108(C2×D163)⋊C4—21no—
109(C2×D163)⋊C4—21no—
110C4×Dic163—21no—
111C2×D4×C163D411yescharacteristic
112C22×C4×C163C2311yescharacteristic
113C22×C4×C163C2311yescharacteristic
114M16×C163—22no—
115C2×C8×C163—22no—
116C2×C8×C163C2311yescharacteristic
117C2×C8×C163C2311yescharacteristic
118C2×C8×C163C2311yescharacteristic
119C2×C8×C163C2311yescharacteristic
120C4⋊C4×C163D411yescharacteristic
121C22⋊C4×C163D411yescharacteristic
122C22⋊C4×C163D411yescharacteristic
123C42×C163C2311yescharacteristic
124C22×D163—21no—
125C22×D163—21no—
126C22×D163—21no—
127C22×D163—21no—
128C22×D163—42no—
129C22×D163—42no—
130C22×D163—42no—
131C22×D163—42no—
132C22×D163—81no—
133C22×D163—81no—
134Dic326—21no—
135Dic326—21no—
136Dic326—21no—
137Dic326—42no—
138Dic326C2×D412yescomplemented
139D652—21no—
140D652—21no—
141D652—21no—
142D652—21no—
143D652—21no—
144D652—21no—
145D652—21no—
146D652—42no—
147D652—42no—
148D652—42no—
149D652—81no—
150D652C2×D412yes—
151C22⋊D163—42no—
152C22⋊D163—42no—
153C22⋊D163—42no—
154C22⋊D163—42no—
155C2×Dic163—21no—
156C2×Dic163—21no—
157C2×Dic163—21no—
158C2×Dic163—21no—
159C2×Dic163—42no—
160C4×D163—42no—
161C4×D163—42no—
162C23×C163C2×D411yescharacteristic
163C23×C163C2×D411yescharacteristic
164D4×C163—22no—
165C2×C4×C163—21no—
166C2×C4×C163—21no—
167C2×C4×C163—21no—
168C2×C4×C163—21no—
169C2×C4×C163C2×D411yescharacteristic
170C2×C4×C163C2×D411yescharacteristic
171C2×C4×C163C2×D411yescharacteristic
172C2×C4×C163C2×D411yescharacteristic
173C2×C4×C163C2411yescharacteristic
174C8×C163—21no—
175C8×C163—21no—
176C8×C163—21no—
177C8×C163C2×D412yescomplemented
178C22×C163—21no—
179C22×C163—21no—
180C22×C163—21no—
181C22×C163—21no—
182C22×C163—21no—
183C22×C163AGL(1,ℤ/8)12yescomplemented
184C22×C163C22×D411yescharacteristic
185C2×D163—22no—
186C2×D163—41no—
187C2×D163—41no—
188C2×D163—41no—
189C2×D163—41no—
190C2×D163—41no—
191C2×D163—41no—
192C2×D163—41no—
193C2×D163—41no—
194C2×D163—41no—
195C2×D163—41no—
196C2×D163—41no—
197C2×D163—84no—
198C2×D163—84no—
199C4×C163—21no—
200C4×C163—21no—
201C4×C163—21no—
202C4×C163—21no—
203C4×C163C22×D411yescharacteristic
204C4×C163H(5,𝔽2)11yescharacteristic
205Dic163—22no—
206Dic163—41no—
207Dic163—41no—
208Dic163—41no—
209C2×C163—22no—
210C2×C163—41no—
211C2×C163D4211yescharacteristic
212C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C163#57711yescharacteristic
214D163—42no—
215D163—81no—
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#450—1631no—
230#449—1631no—
231#448—1631no—
232#447—1631no—
233#440—1631no—
234#435D16311yescharacteristic, complemented
235C2×#41—1631no—
236C22×Q8—1631no—
237C22×D4—3261no—
238C22×D4—1631no—
239C22×D4—1631no—
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×M16C2×D16311yescharacteristic, complemented
252C22×C8C2×D16311yescharacteristic, complemented
253#127—1631no—
254#124—3261no—
255#124—1631no—
256C22⋊Q8—3261no—
257C22⋊Q8—1631no—
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×D4C2×D16311yescharacteristic
269#107—1631no—
270C4⋊C8C2×D16311yescharacteristic, complemented
271#103—1631no—
272#103—1631no—
273#102—1631no—
274#102—1631no—
275#102—1631no—
276#102—1631no—
277C22⋊C8C2×D16311yescharacteristic, complemented
278C22⋊C8C2×D16311yescharacteristic, complemented
279#97C2×D16311yescharacteristic, complemented
280C24—3261no—
281C24—3261no—
282#41—3262no—
283#41—6521no—
284C2×Q8—3262no—
285C2×Q8—3261no—
286C2×Q8—6521no—
287C2×Q8—1631no—
288C2×Q8—1631no—
289C2×D4—3262no—
290C2×D4—3262no—
291C2×D4—3261no—
292C2×D4—3261no—
293C2×D4—3261no—
294C2×D4—3261no—
295C2×D4—3261no—
296C2×D4—3261no—
297C2×D4—3261no—
298C2×D4—3261no—
299C2×D4—6521no—
300C2×D4—6521no—
301C2×D4—6524no—
302C2×D4—1631no—
303C2×D4—1631no—
304C2×D4—1631no—
305C2×D4—1631no—
306C2×D4D65211yescharacteristic
307C22×C4—3261no—
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—22no—
322C2×C8—22no—
323C2×C8C22×D16311yescharacteristic
324C2×C8C22×D16311yescharacteristic
325C2×C8C22×D16311yescharacteristic
326C2×C8C22×D16311yescharacteristic
327C4⋊C4—3261no—
328C4⋊C4—3261no—
329C4⋊C4—1631no—
330C4⋊C4—1631no—
331C4⋊C4D65211yescharacteristic
332C22⋊C4—3261no—
333C22⋊C4—3261no—
334C22⋊C4—3261no—
335C22⋊C4—3261no—
336C22⋊C4—3261no—
337C22⋊C4—3261no—
338C22⋊C4—3261no—
339C22⋊C4D65211yescharacteristic
340C22⋊C4D65211yescharacteristic
341C42—3261no—
342C42C22×D16311yescharacteristic
343C23—13041no—
344C23—13041no—
345C23—3261no—
346C23—3261no—
347C23—3261no—
348C23—3261no—
349C23—6522no—
350C23—6522no—
351C23—6522no—
352C23—6522no—
353C23C2×D65211yescharacteristic
354C23C2×D65211yescharacteristic
355Q8—3261no—
356Q8—3261no—
357Q8—3261no—
358Q8—6522no—
359Q8—1632no—
360D4—22no—
361D4—13041no—
362D4—3261no—
363D4—3261no—
364D4—3261no—
365D4—3261no—
366D4—3261no—
367D4—3261no—
368D4—3261no—
369D4—6522no—
370D4—6522no—
371D4—6522no—
372D4—6522no—
373D4—6522no—
374D4—6522no—
375D4—6522no—
376D4—1632no—
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×C4C2×D65211yescharacteristic
389C2×C4C2×D65211yescharacteristic
390C2×C4C2×D65211yescharacteristic
391C2×C4C2×D65211yescharacteristic
392C2×C4C23×D16311yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8D4×D16312yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—13044no—
403C22—13044no—
404C22—3262no—
405C22—6521no—
406C22—6521no—
407C22—6521no—
408C22—6521no—
409C22—6521no—
410C22—6521no—
411C22—6521no—
412C22—6521no—
413C22—6521no—
414C22—6521no—
415C22—6521no—
416C22#1910476312yescomplemented
417C22C22×D65211yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—3262no—
423C4—6521no—
424C4—6521no—
425C4—6521no—
426C4C2×D4×D16311yescharacteristic
427C4#1910478111yescharacteristic
428C2—22no—
429C2—41no—
430C2—13041no—
431C2—13041no—
432C2—13041no—
433C2—6522no—
434C2D4×D65211yescharacteristic
435C2C2×#1910476311yescharacteristic
436C2#2378247711yescharacteristic
subgroupG/H
centerC22C22×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#435×C163C2
socleC22×C163C22×D4
hypercenter#435D163
hypocenterC163#3265
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#19104763C2semidirect
2. C22×#14579688C2semidirect
3. #23782324C2semidirect
4. D4×D652C2semidirect
5. #23782054C2semidirect
6. #23781911C2semidirect
7. #23781901C2semidirect
8. #23781881C2semidirect
9. #23781866C2semidirect
10. #23781818C2semidirect
11. #23781812C2semidirect
12. #23781806C2semidirect
13. #23781801C2semidirect
14. #23781767C2semidirect
15. #435×C163C2semidirect
16. C2×#14579699C22semidirect
17. C22×Dic326C22semidirect
25. C2×#14579688C22semidirect
26. C2×#14579688C22semidirect
27. C2×#14579688C22semidirect
28. C2×#14579688C22semidirect
30. C2×D1304C22semidirect
31. C2×D1304C22semidirect
33. #19104723C22semidirect
35. #19104714C22semidirect
40. #19104709C22semidirect
44. C4×Dic326C22semidirect
47. #19104669C22semidirect
48. #19104660C22semidirect
49. #19104660C22semidirect
51. #19104657C22semidirect
52. #19104657C22semidirect
53. #19104657C22semidirect
54. C2×M16×C163C22semidirect
55. C22×C8×C163C22semidirect
57. C4⋊C8×C163C22semidirect
58. C22⋊C8×C163C22semidirect
59. C22⋊C8×C163C22semidirect
60. #97×C163C22semidirect
95. #14579688D4semidirect
138. Dic326C2×D4semidirect
177. C8×C163C2×D4semidirect
183. C22×C163AGL(1,ℤ/8)semidirect
218. C163#3265semidirect
234. #435D163semidirect
251. C2×M16C2×D163semidirect
252. C22×C8C2×D163semidirect
270. C4⋊C8C2×D163semidirect
277. C22⋊C8C2×D163semidirect
278. C22⋊C8C2×D163semidirect
279. #97C2×D163semidirect
396. C8D4×D163semidirect
416. C22#19104763semidirect

Maximal subgroups

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

subgroupindex
C2×#191047632
C22×#145796882
#237823242
D4×D6522
#237820542
#237819112
#237819012
#237818812
#237818662
#237818182
#237818122
#237818062
#237818012
#237817672
#435×C1632
#3265163