Finite Group Databasebeta

← #35225020

Subgroups — #35225020

Order 20864.

596 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104762C211yesmaximal, characteristic, complemented
2C2×#19104759C211yesmaximal, characteristic, complemented
3C2×#19104756C211yesmaximal, characteristic, complemented
4C2×D8×D163C211yesmaximal, characteristic, complemented
5#23782233C211yesmaximal, characteristic, complemented
6#23782228C211yesmaximal, characteristic, complemented
7#23782054C211yesmaximal, characteristic, complemented
8#23782053C211yesmaximal, characteristic, complemented
9#23782051C211yesmaximal, characteristic, complemented
10#23782047C211yesmaximal, characteristic, complemented
11#23782046C211yesmaximal, characteristic, complemented
12#23782041C211yesmaximal, characteristic, complemented
13#23782040C211yesmaximal, characteristic, complemented
14#23781622C211yesmaximal, characteristic, complemented
15#496×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579698C2211yescharacteristic
18C2×D4×D163C2211yescharacteristic
19C2×D4×D163C2211yescharacteristic
20#19104762—24no—
21#19104759—24no—
22#19104756—24no—
23C2×#14579689C2211yescharacteristic, complemented
24C2×#14579688C2211yescharacteristic, complemented
25C2×#14579688C2211yescharacteristic, complemented
26C2×#14579687C2211yescharacteristic, complemented
27C2×#14579686C2211yescharacteristic, complemented
28C2×D1304C2211yescharacteristic
29C2×D1304C2211yescharacteristic, complemented
30C2×#14579684C2211yescharacteristic
31C2×#14579684C2211yescharacteristic
32C2×#14579684C2211yescharacteristic, complemented
33C2×#14579684C2211yescharacteristic, complemented
34C2×#14579684C2211yescharacteristic, complemented
35D8×D163—24no—
36C2×#14579682C2211yescharacteristic
37C2×C8×D163C2211yescharacteristic
38#19104732C2211yescharacteristic, complemented
39#19104731C2211yescharacteristic, complemented
40#19104731C2211yescharacteristic, complemented
41#19104724C2211yescharacteristic, complemented
42#19104723C2211yescharacteristic, complemented
43#19104722C2211yescharacteristic, complemented
44#19104708—22no—
45#19104707—22no—
46#19104705—22no—
47#19104703—22no—
48#19104692C2211yescharacteristic
49#19104645C2211yescharacteristic, complemented
50C4×C163⋊C8C2211yescharacteristic, complemented
51C8×Dic163C2211yescharacteristic, complemented
52C2×SD16×C163C2211yescharacteristic, complemented
53C2×SD16×C163C2211yescharacteristic, complemented
54C2×D8×C163C2211yescharacteristic
55C2×D8×C163C2211yescharacteristic, complemented
56#127×C163C2211yescharacteristic, complemented
57#124×C163C2211yescharacteristic, complemented
58#97×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60C23×D163—21no—
61#14579700—22no—
62#14579700—22no—
63#14579698—22no—
64#14579698—22no—
65C2×Dic326C2311yescharacteristic
66C2×D652—21no—
67C2×D652—22no—
68C2×D652C2311yescharacteristic
69C2×D652C2311yescharacteristic
70C2×D652C2311yescharacteristic
71C2×C22⋊D163—21no—
72C2×C22⋊D163—21no—
73C2×C22⋊D163—21no—
74C2×C22⋊D163—22no—
75C2×C22⋊D163—22no—
76D4×D163—22no—
77D4×D163—22no—
78D4×D163—22no—
79D4×D163—22no—
80C22×Dic163—21no—
81C2×C4×D163—21no—
82C2×C4×D163C2311yescharacteristic
83#14579689—22no—
84#14579688—22no—
85#14579688—22no—
86#14579687—22no—
87#14579686—22no—
88D1304—22no—
89D1304—22no—
90#14579684—22no—
91#14579684—22no—
92#14579684—22no—
93#14579684—22no—
94#14579684—22no—
95#14579682D414yescomplemented
96C2×C163⋊C8C2311yescharacteristic
97C2×C163⋊C8C2311yescharacteristic
98C8×D163D414yescomplemented
99Dic163⋊C4D412yes—
100C22⋊Dic163—22no—
101(C2×D163)⋊C4—22no—
102(C2×D163)⋊C4—22no—
103(C2×D163)⋊C4D412yes—
104C4×Dic163C2311yescharacteristic
105C2×Q8×C163C2311yescharacteristic
106C2×D4×C163—22no—
107C2×D4×C163C2311yescharacteristic
108C2×D4×C163C2311yescharacteristic
109C2×D4×C163C2311yescharacteristic
110SD16×C163—22no—
111SD16×C163—22no—
112D8×C163—22no—
113D8×C163—22no—
114C2×C8×C163C2311yescharacteristic
115C2×C8×C163C2311yescharacteristic
116C22⋊C4×C163—22no—
117C42×C163C2311yescharacteristic
118C22×D163—21no—
119C22×D163—21no—
120C22×D163—21no—
121C22×D163—42no—
122C22×D163—42no—
123C22×D163—82no—
124C22×D163—82no—
125C22×D163C2×D411yescharacteristic
126Dic326—21no—
127Dic326—21no—
128D652—21no—
129D652—21no—
130D652—21no—
131D652—21no—
132D652—21no—
133D652—21no—
134D652—42no—
135D652—82no—
136C22⋊D163—42no—
137C22⋊D163—42no—
138C22⋊D163—42no—
139C22⋊D163—82no—
140C22⋊D163—82no—
141C2×Dic163—21no—
142C2×Dic163—42no—
143C2×Dic163C2×D412yes—
144C2×Dic163C2×D411yescharacteristic
145C4×D163—42no—
146C4×D163C2×D412yes—
147C4×D163C2×D412yes—
148C163⋊C8C2×D412yes—
149C163⋊C8C2×D412yes—
150C23×C163—21no—
151C23×C163—21no—
152C23×C163—21no—
153Q8×C163—21no—
154Q8×C163—21no—
155D4×C163—21no—
156D4×C163—21no—
157D4×C163—21no—
158D4×C163—21no—
159D4×C163—21no—
160D4×C163—21no—
161D4×C163—82no—
162C2×C4×C163—21no—
163C2×C4×C163C2×D412yes—
164C2×C4×C163C2411yescharacteristic
165C8×C163C2×D412yes—
166C8×C163C2×D412yes—
167C22×C163—41no—
168C22×C163—41no—
169C22×C163—41no—
170C22×C163—42no—
171C22×C163—42no—
172C22×C163—42no—
173C22×C163C22×D411yescharacteristic
174C2×D163—22no—
175C2×D163—41no—
176C2×D163—41no—
177C2×D163—41no—
178C2×D163—42no—
179C2×D163—42no—
180C2×D163—42no—
181C2×D163—84no—
182C2×D163—84no—
183C2×D163#12712yes—
184C4×C163—22no—
185C4×C163—41no—
186C4×C163C22×D411yescharacteristic
187C4×C163C22×D411yescharacteristic
188Dic163—22no—
189Dic163—41no—
190Dic163#12712yes—
191C2×C163—81no—
192C2×C163—81no—
193C2×C163—81no—
194C2×C163C2×#12711yescharacteristic
195C2×C163#57612yes—
196D163—22no—
197D163—81no—
198D163—81no—
199D163—81no—
200C163#315511yescharacteristic, complemented, Sylow
201#3155—1631nomaximal, Sylow
202C2×AGL(1,ℤ/8)—1631no—
203C2×AGL(1,ℤ/8)—1631no—
204C2×#135—1631no—
205C22×D8—1631no—
206#535—1631no—
207#535—1631no—
208#496—1631no—
209#496—1631no—
210#496—1631no—
211#496D16311yescharacteristic, complemented
212#495—1631no—
213#495—1631no—
214#493—1631no—
215#493—1631no—
216#405—1631no—
217C2×#41—1631no—
218C2×#41—1631no—
219C22×D4—1631no—
220C22×D4—1631no—
221AGL(1,ℤ/8)—3264no—
222AGL(1,ℤ/8)—3264no—
223#135—3264no—
224C2×Q16—1631no—
225C2×SD16—1631no—
226C2×SD16—1631no—
227C2×SD16—1631no—
228C2×SD16—1631no—
229C2×SD16C2×D16311yescharacteristic, complemented
230C2×SD16C2×D16311yescharacteristic, complemented
231C2×D8—3264no—
232C2×D8—1631no—
233C2×D8—1631no—
234C2×D8—1631no—
235C2×D8—1631no—
236C2×D8—1631no—
237C2×D8—1631no—
238C2×D8—1631no—
239C2×D8C2×D16311yescharacteristic
240C2×D8C2×D16311yescharacteristic, complemented
241C2×M16—1631no—
242C22×C8—1631no—
243#127—1631no—
244#127—1631no—
245#127—1631no—
246#127C2×D16311yescharacteristic, complemented
247#124—1631no—
248#124—1631no—
249#124—1631no—
250#124C2×D16311yescharacteristic, complemented
251#121—3262no—
252#121—3262no—
253#120—3262no—
254#120—3262no—
255#117—1631no—
256#97—1631no—
257#97C2×D16311yescharacteristic, complemented
258C4×C8—1631no—
259C4×C8—1631no—
260C24—3261no—
261C24—3261no—
262#41—3262no—
263#41—3262no—
264#41—3262no—
265#41—3262no—
266C2×Q8—1631no—
267C2×Q8C22×D16311yescharacteristic
268C2×D4—22no—
269C2×D4—3261no—
270C2×D4—3261no—
271C2×D4—3262no—
272C2×D4—3262no—
273C2×D4—3262no—
274C2×D4—3262no—
275C2×D4—3262no—
276C2×D4—3262no—
277C2×D4—3261no—
278C2×D4—3261no—
279C2×D4—3262no—
280C2×D4—1631no—
281C2×D4—1631no—
282C2×D4—1631no—
283C2×D4C22×D16311yescharacteristic
284C2×D4C22×D16311yescharacteristic
285C2×D4C22×D16311yescharacteristic
286C22×C4—3261no—
287C22×C4—3261no—
288C22×C4—1631no—
289Q16—3262no—
290SD16—22no—
291SD16—22no—
292SD16—3262no—
293SD16—3262no—
294SD16—3262no—
295SD16—3262no—
296D8—22no—
297D8—22no—
298D8—3262no—
299D8—3262no—
300D8—3262no—
301D8—3262no—
302D8—3262no—
303D8—3262no—
304D8—3262no—
305M16—1634no—
306C2×C8—1631no—
307C2×C8—1631no—
308C2×C8—1634no—
309C2×C8C22×D16311yescharacteristic
310C2×C8C22×D16311yescharacteristic
311C4⋊C4—1632no—
312C22⋊C4—22no—
313C22⋊C4—3262no—
314C22⋊C4—3262no—
315C22⋊C4—3262no—
316C22⋊C4—1632no—
317C42—1631no—
318C42C22×D16311yescharacteristic
319C23—21no—
320C23—21no—
321C23—21no—
322C23—13042no—
323C23—13042no—
324C23—3261no—
325C23—3261no—
326C23—3261no—
327C23—6522no—
328C23—6522no—
329C23—1631no—
330Q8—21no—
331Q8—21no—
332Q8—3261no—
333Q8—3261no—
334D4—21no—
335D4—21no—
336D4—21no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—82no—
341D4—13042no—
342D4—13042no—
343D4—13042no—
344D4—3261no—
345D4—3261no—
346D4—3261no—
347D4—3261no—
348D4—3261no—
349D4—3261no—
350D4—6522no—
351D4—6522no—
352D4—6522no—
353D4—6522no—
354C2×C4—21no—
355C2×C4—3261no—
356C2×C4—6522no—
357C2×C4—6522no—
358C2×C4—1632no—
359C2×C4—1631no—
360C2×C4—1632no—
361C2×C4—1632no—
362C2×C4D4×D16312yes—
363C2×C4C23×D16311yescharacteristic
364C8—1632no—
365C8—1632no—
366C8D4×D16312yes—
367C8D4×D16312yes—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—13044no—
375C22—13044no—
376C22—3262no—
377C22—6521no—
378C22—6521no—
379C22—6521no—
380C22—6522no—
381C22—6522no—
382C22—6522no—
383C22—1632no—
384C22C2×D4×D16311yescharacteristic
385C4—22no—
386C4—41no—
387C4—3262no—
388C4—6521no—
389C4—1632no—
390C4C2×D4×D16311yescharacteristic
391C4C2×D4×D16311yescharacteristic
392C2—81no—
393C2—81no—
394C2—81no—
395C2—13041no—
396C2—13041no—
397C2—13041no—
398C2—3262no—
399C2#127×D16311yescharacteristic
400C2#2378246912yes—
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#496×C163C2
socleC22×C163C22×D4
hypercenter#496D163
hypocenterC163#3155
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#19104762C2semidirect
2. C2×#19104759C2semidirect
3. C2×#19104756C2semidirect
4. C2×D8×D163C2semidirect
5. #23782233C2semidirect
6. #23782228C2semidirect
7. #23782054C2semidirect
8. #23782053C2semidirect
9. #23782051C2semidirect
10. #23782047C2semidirect
11. #23782046C2semidirect
12. #23782041C2semidirect
13. #23782040C2semidirect
14. #23781622C2semidirect
15. #496×C163C2semidirect
23. C2×#14579689C22semidirect
24. C2×#14579688C22semidirect
25. C2×#14579688C22semidirect
26. C2×#14579687C22semidirect
27. C2×#14579686C22semidirect
29. C2×D1304C22semidirect
32. C2×#14579684C22semidirect
33. C2×#14579684C22semidirect
34. C2×#14579684C22semidirect
38. #19104732C22semidirect
39. #19104731C22semidirect
40. #19104731C22semidirect
41. #19104724C22semidirect
42. #19104723C22semidirect
43. #19104722C22semidirect
49. #19104645C22semidirect
50. C4×C163⋊C8C22semidirect
51. C8×Dic163C22semidirect
52. C2×SD16×C163C22semidirect
53. C2×SD16×C163C22semidirect
55. C2×D8×C163C22semidirect
56. #127×C163C22semidirect
57. #124×C163C22semidirect
58. #97×C163C22semidirect
95. #14579682D4semidirect
98. C8×D163D4semidirect
200. C163#3155semidirect
211. #496D163semidirect
229. C2×SD16C2×D163semidirect
230. C2×SD16C2×D163semidirect
240. C2×D8C2×D163semidirect
246. #127C2×D163semidirect
250. #124C2×D163semidirect
257. #97C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047622
C2×#191047592
C2×#191047562
C2×D8×D1632
#237822332
#237822282
#237820542
#237820532
#237820512
#237820472
#237820462
#237820412
#237820402
#237816222
#496×C1632
#3155163