Finite Group Databasebeta

← #35224150

Subgroups — #35224150

Order 20864.

548 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104762C211yesmaximal, characteristic, complemented
2C2×M16×D163C211yesmaximal, characteristic, complemented
3#23782178C211yesmaximal, characteristic, complemented
4#121×D163C211yesmaximal, characteristic, complemented
5#23781918C211yesmaximal, characteristic, complemented
6#23781916C211yesmaximal, characteristic, complemented
7#23781913C211yesmaximal, characteristic, complemented
8#23781902C211yesmaximal, characteristic, complemented
9#23781900C211yesmaximal, characteristic, complemented
10#23781897C211yesmaximal, characteristic, complemented
11#23781891C211yesmaximal, characteristic, complemented
12#23781706C211yesmaximal, characteristic, complemented
13#23781668C211yesmaximal, characteristic, complemented
14#23781665C211yesmaximal, characteristic, complemented
15#468×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×D4×D163—21no—
18C2×D4×D163C2211yescharacteristic
19C22×C4×D163C2211yescharacteristic
20#19104762—24no—
21C2×#14579686C2211yescharacteristic, complemented
22C2×D1304C2211yescharacteristic, complemented
23C2×#14579684C2211yescharacteristic, complemented
24C2×#14579683C2211yescharacteristic, complemented
25C2×#14579682C2211yescharacteristic
26C2×#14579682C2211yescharacteristic
27M16×D163—24no—
28C2×C8×D163C2211yescharacteristic, complemented
29C2×C8×D163C2211yescharacteristic, complemented
30#19104720—21no—
31#19104711C2211yescharacteristic, complemented
32#19104710C2211yescharacteristic
33#19104710C2211yescharacteristic, complemented
34#19104709C2211yescharacteristic
35#19104709C2211yescharacteristic, complemented
36#19104708—21no—
37#19104707—21no—
38#19104706C2211yescharacteristic
39#19104705—21no—
40#19104697—21no—
41(C22⋊D163)⋊C4—21no—
42#19104693C2211yescharacteristic
43C4⋊C4×D163C2211yescharacteristic
44C22⋊C4×D163—21no—
45#19104668C2211yescharacteristic, complemented
46#19104667C2211yescharacteristic, complemented
47#19104666C2211yescharacteristic, complemented
48#19104659C2211yescharacteristic, complemented
49#19104657C2211yescharacteristic, complemented
50#19104657C2211yescharacteristic, complemented
51#19104656C2211yescharacteristic, complemented
52#19104655C2211yescharacteristic, complemented
53#19104655C2211yescharacteristic, complemented
54C2×SD16×C163C2211yescharacteristic, complemented
55C2×M16×C163C2211yescharacteristic, complemented
56C22⋊Q8×C163C2211yescharacteristic, complemented
57#121×C163C2211yescharacteristic
58#107×C163C2211yescharacteristic, complemented
59#103×C163C2211yescharacteristic, complemented
60#102×C163C2211yescharacteristic, complemented
61C23×D163—21no—
62C23×D163D411yescharacteristic
63#14579700—22no—
64#14579700—22no—
65C2×D652—21no—
66C2×D652—21no—
67C2×D652—21no—
68C2×D652C2311yescharacteristic
69C2×D652C2311yescharacteristic
70C2×C22⋊D163—21no—
71C2×C22⋊D163—21no—
72C2×C22⋊D163—21no—
73C2×C22⋊D163—21no—
74D4×D163—22no—
75D4×D163—22no—
76D4×D163—44no—
77C22×Dic163D411yescharacteristic
78C2×C4×D163—22no—
79C2×C4×D163—22no—
80C2×C4×D163—21no—
81C2×C4×D163—21no—
82C2×C4×D163D411yescharacteristic
83C2×C4×D163D411yescharacteristic
84C2×C4×D163C2311yescharacteristic
85C2×C4×D163C2311yescharacteristic
86#14579686—22no—
87D1304—22no—
88#14579684—22no—
89#14579683—22no—
90#14579682—22no—
91#14579682D414yescomplemented
92C2×C163⋊C8C2311yescharacteristic
93C2×C163⋊C8C2311yescharacteristic
94C8×D163—22no—
95C8×D163—22no—
96C4⋊Dic163C2311yescharacteristic
97C4⋊Dic163C2311yescharacteristic
98Dic163⋊C4—21no—
99Dic163⋊C4—21no—
100C22⋊Dic163—21no—
101(C2×D163)⋊C4—21no—
102(C2×D163)⋊C4—21no—
103(C2×D163)⋊C4—21no—
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4—21no—
106(C2×D163)⋊C4—21no—
107C4×Dic163—21no—
108C2×Q8×C163C2311yescharacteristic
109C2×D4×C163—21no—
110C2×D4×C163C2311yescharacteristic
111C22×C4×C163C2311yescharacteristic
112SD16×C163—22no—
113M16×C163—22no—
114C2×C8×C163C2311yescharacteristic
115C2×C8×C163C2311yescharacteristic
116C4⋊C4×C163—21no—
117C4⋊C4×C163C2311yescharacteristic
118C4⋊C4×C163C2311yescharacteristic
119C22⋊C4×C163—21no—
120C22⋊C4×C163—21no—
121C22×D163—22no—
122C22×D163—22no—
123C22×D163—21no—
124C22×D163—21no—
125C22×D163—41no—
126C22×D163—42no—
127C22×D163—42no—
128C22×D163—42no—
129C22×D163C2×D411yescharacteristic
130C22×D163C2×D411yescharacteristic
131D652—21no—
132D652—21no—
133D652—21no—
134D652—21no—
135D652—42no—
136D652—42no—
137D652—42no—
138C22⋊D163—42no—
139C22⋊D163—42no—
140C22⋊D163—42no—
141C22⋊D163—42no—
142C2×Dic163—22no—
143C2×Dic163—21no—
144C2×Dic163—21no—
145C2×Dic163C2×D411yescharacteristic
146C2×Dic163C2×D411yescharacteristic
147C4×D163—22no—
148C4×D163—21no—
149C4×D163—21no—
150C4×D163—22no—
151C4×D163—42no—
152C4×D163—42no—
153C4×D163C2×D412yes—
154C4×D163#4112yes—
155C163⋊C8—21no—
156C163⋊C8C2×D412yescomplemented
157C23×C163—21no—
158C23×C163C2×D411yescharacteristic
159Q8×C163—21no—
160Q8×C163—21no—
161D4×C163—21no—
162D4×C163—21no—
163D4×C163—42no—
164C2×C4×C163—21no—
165C2×C4×C163—21no—
166C2×C4×C163—21no—
167C2×C4×C163—21no—
168C2×C4×C163—21no—
169C2×C4×C163C2×D411yescharacteristic
170C2×C4×C163C2411yescharacteristic
171C8×C163—21no—
172C8×C163C2×D412yescomplemented
173C22×C163—21no—
174C22×C163—21no—
175C22×C163—21no—
176C22×C163—41no—
177C22×C163—41no—
178C22×C163—41no—
179C22×C163C22×D411yescharacteristic
180C2×D163—21no—
181C2×D163—21no—
182C2×D163—21no—
183C2×D163—21no—
184C2×D163—44no—
185C2×D163—41no—
186C2×D163—41no—
187C2×D163—41no—
188C2×D163—41no—
189C2×D163—41no—
190C2×D163—41no—
191C2×D163—84no—
192C2×D163#12112yes—
193C2×D163AGL(1,ℤ/8)12yescomplemented
194C4×C163—21no—
195C4×C163—41no—
196C4×C163—41no—
197C4×C163—41no—
198C4×C163C22×D411yescharacteristic
199C4×C163C2×#4111yescharacteristic
200Dic163—21no—
201Dic163—41no—
202Dic163—41no—
203Dic163#12112yes—
204C2×C163—41no—
205C2×C163—81no—
206C2×C163C2×#12111yescharacteristic
207C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
208C2×C163#57511yescharacteristic
209D163—22no—
210D163—22no—
211D163—81no—
212D163—81no—
213C163#305511yescharacteristic, complemented, Sylow
214#3055—1631nomaximal, Sylow
215C2×AGL(1,ℤ/8)—1631no—
216C22×M16—1631no—
217#525—1631no—
218C2×#121—1631no—
219#469—1631no—
220#469—1631no—
221#468—1631no—
222#468D16311yescharacteristic, complemented
223#466—1631no—
224#466—1631no—
225#465—1631no—
226#465—1631no—
227#428—1631no—
228#418—1631no—
229#417—1631no—
230C2×#41—1631no—
231C22×D4—3261no—
232C22×D4—1631no—
233C23×C4—1631no—
234AGL(1,ℤ/8)—3264no—
235C2×SD16—1631no—
236C2×SD16C2×D16311yescharacteristic, complemented
237C2×D8—1631no—
238C2×D8—1631no—
239C2×M16—3264no—
240C2×M16—1631no—
241C2×M16—1631no—
242C2×M16—1631no—
243C2×M16C2×D16311yescharacteristic, complemented
244C22×C8—1631no—
245C22×C8—1631no—
246#123—3261no—
247C22⋊Q8—1631no—
248C22⋊Q8C2×D16311yescharacteristic, complemented
249#121—3261no—
250#121—3261no—
251#121—1631no—
252#121—1631no—
253#121—1631no—
254#121—1631no—
255#121—1631no—
256#121C2×D16311yescharacteristic
257#120—3261no—
258C4×D4—3261no—
259C4×D4—3261no—
260#117—1631no—
261C2×C4⋊C4—1631no—
262C2×C22⋊C4—3261no—
263#107—1631no—
264#107C2×D16311yescharacteristic, complemented
265#106—1631no—
266#106—1631no—
267#103—1631no—
268#103C2×D16311yescharacteristic, complemented
269#102—1631no—
270#102—1631no—
271#102—1631no—
272#102—1631no—
273#102—1631no—
274#102C2×D16311yescharacteristic, complemented
275C24—3261no—
276C24—1631no—
277#41—3262no—
278#41—3262no—
279C2×Q8C22×D16311yescharacteristic
280C2×D4—21no—
281C2×D4—3261no—
282C2×D4—3261no—
283C2×D4—3261no—
284C2×D4—3261no—
285C2×D4—3261no—
286C2×D4—3261no—
287C2×D4—3262no—
288C2×D4—3262no—
289C2×D4—3261no—
290C2×D4—6524no—
291C2×D4—1631no—
292C2×D4—1631no—
293C2×D4C22×D16311yescharacteristic
294C22×C4—3262no—
295C22×C4—3262no—
296C22×C4—3261no—
297C22×C4—3261no—
298C22×C4—1631no—
299C22×C4—1631no—
300C22×C4—1631no—
301C22×C4—1631no—
302C22×C4—1631no—
303C22×C4C22×D16311yescharacteristic
304SD16—22no—
305SD16—3262no—
306D8—3262no—
307D8—3262no—
308M16—22no—
309M16—3262no—
310M16—3262no—
311M16—1634no—
312C2×C8—3262no—
313C2×C8—3262no—
314C2×C8—1631no—
315C2×C8—1631no—
316C2×C8C22×D16311yescharacteristic
317C2×C8C22×D16311yescharacteristic
318C4⋊C4—21no—
319C4⋊C4—3261no—
320C4⋊C4—3261no—
321C4⋊C4—1631no—
322C4⋊C4—1631no—
323C4⋊C4C22×D16311yescharacteristic
324C4⋊C4C22×D16311yescharacteristic
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—3261no—
328C22⋊C4—3261no—
329C22⋊C4—3261no—
330C22⋊C4—3261no—
331C22⋊C4—3261no—
332C22⋊C4—3261no—
333C22⋊C4—3261no—
334C42—3261no—
335C23—21no—
336C23—3262no—
337C23—3262no—
338C23—3261no—
339C23—3261no—
340C23—6521no—
341C23—6522no—
342C23—6522no—
343C23—6522no—
344C23—1631no—
345C23—1631no—
346C23D4×D16311yescharacteristic
347Q8—21no—
348Q8—21no—
349D4—21no—
350D4—21no—
351D4—42no—
352D4—3261no—
353D4—3261no—
354D4—3261no—
355D4—3261no—
356D4—6522no—
357D4—6522no—
358D4—6522no—
359D4—6522no—
360D4—6522no—
361D4—6522no—
362D4—6522no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—3262no—
369C2×C4—3262no—
370C2×C4—3262no—
371C2×C4—3261no—
372C2×C4—3261no—
373C2×C4—3261no—
374C2×C4—3261no—
375C2×C4—6522no—
376C2×C4—6522no—
377C2×C4—1631no—
378C2×C4—1631no—
379C2×C4—1632no—
380C2×C4—1632no—
381C2×C4D4×D16311yescharacteristic
382C2×C4C23×D16311yescharacteristic
383C8—21no—
384C8—3261no—
385C8—1632no—
386C8D4×D16312yescomplemented
387C22—21no—
388C22—21no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—13044no—
394C22—3261no—
395C22—3261no—
396C22—3261no—
397C22—3261no—
398C22—6524no—
399C22—6521no—
400C22—6521no—
401C22—6521no—
402C22—6521no—
403C22—6521no—
404C22—6521no—
405C22—1632no—
406C22—1632no—
407C22C2×D4×D16311yescharacteristic
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—3261no—
413C4—6521no—
414C4—6521no—
415C4—1632no—
416C4C2×D4×D16311yescharacteristic
417C4#41×D16311yescharacteristic
418C2—41no—
419C2—81no—
420C2—13041no—
421C2—13041no—
422C2—3262no—
423C2—3262no—
424C2#121×D16311yescharacteristic
425C2AGL(1,ℤ/8)×D16311yescharacteristic
426C2#2378246411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#468×C163C2
socleC22×C163C22×D4
hypercenter#468D163
hypocenterC163#3055
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#19104762C2semidirect
2. C2×M16×D163C2semidirect
3. #23782178C2semidirect
4. #121×D163C2semidirect
5. #23781918C2semidirect
6. #23781916C2semidirect
7. #23781913C2semidirect
8. #23781902C2semidirect
9. #23781900C2semidirect
10. #23781897C2semidirect
11. #23781891C2semidirect
12. #23781706C2semidirect
13. #23781668C2semidirect
14. #23781665C2semidirect
15. #468×C163C2semidirect
21. C2×#14579686C22semidirect
22. C2×D1304C22semidirect
23. C2×#14579684C22semidirect
24. C2×#14579683C22semidirect
28. C2×C8×D163C22semidirect
29. C2×C8×D163C22semidirect
31. #19104711C22semidirect
33. #19104710C22semidirect
35. #19104709C22semidirect
45. #19104668C22semidirect
46. #19104667C22semidirect
47. #19104666C22semidirect
48. #19104659C22semidirect
49. #19104657C22semidirect
50. #19104657C22semidirect
51. #19104656C22semidirect
52. #19104655C22semidirect
53. #19104655C22semidirect
54. C2×SD16×C163C22semidirect
55. C2×M16×C163C22semidirect
56. C22⋊Q8×C163C22semidirect
58. #107×C163C22semidirect
59. #103×C163C22semidirect
60. #102×C163C22semidirect
91. #14579682D4semidirect
156. C163⋊C8C2×D4semidirect
172. C8×C163C2×D4semidirect
193. C2×D163AGL(1,ℤ/8)semidirect
213. C163#3055semidirect
222. #468D163semidirect
236. C2×SD16C2×D163semidirect
243. C2×M16C2×D163semidirect
248. C22⋊Q8C2×D163semidirect
264. #107C2×D163semidirect
268. #103C2×D163semidirect
274. #102C2×D163semidirect
386. C8D4×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047622
C2×M16×D1632
#237821782
#121×D1632
#237819182
#237819162
#237819132
#237819022
#237819002
#237818972
#237818912
#237817062
#237816682
#237816652
#468×C1632
#3055163