Finite Group Databasebeta

← #35224714

Subgroups — #35224714

Order 20864.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782139C211yesmaximal, characteristic, complemented
2#23782003C211yesmaximal, characteristic, complemented
3#23781986C211yesmaximal, characteristic, complemented
4#23781952C211yesmaximal, characteristic, complemented
5#23781942C211yesmaximal, characteristic, complemented
6#23781877C211yesmaximal, characteristic, complemented
7#23781877C211yesmaximal, characteristic, complemented
8#23781815C211yesmaximal, characteristic, complemented
9#23781798C211yesmaximal, characteristic
10#23781775C211yesmaximal, characteristic, complemented
11#23781763C211yesmaximal, characteristic, complemented
12C4×#14579686C211yesmaximal, characteristic, complemented
13C4×#14579684C211yesmaximal, characteristic, complemented
14#23781726C211yesmaximal, characteristic, complemented
15#542×C163C211yesmaximal, characteristic, complemented
16C22×D652C2211yescharacteristic
17C2×#14579686C2211yescharacteristic
18C2×#14579686C2211yescharacteristic
19C2×#14579684C2211yescharacteristic
20C2×#14579684C2211yescharacteristic
21C4×D652C2211yescharacteristic
22C4×D652C2211yescharacteristic
23C4×D652C2211yescharacteristic
24C4×D652C2211yescharacteristic
25#19104697—21no—
26#19104697—21no—
27#19104694C2211yescharacteristic
28C22⋊C4×D163—21no—
29C22⋊C4×D163—21no—
30#19104668C2211yescharacteristic, complemented
31#19104668C2211yescharacteristic, complemented
32#19104666C2211yescharacteristic, complemented
33#19104666C2211yescharacteristic, complemented
34C4⋊(C163⋊C8)C2211yescharacteristic, complemented
35C4⋊(C163⋊C8)C2211yescharacteristic, complemented
36#19104659C2211yescharacteristic, complemented
37#19104659C2211yescharacteristic, complemented
38#19104656C2211yescharacteristic, complemented
39#19104656C2211yescharacteristic, complemented
40#19104655C2211yescharacteristic
41#19104655C2211yescharacteristic
42#19104655C2211yescharacteristic
43#19104655C2211yescharacteristic
44C22⋊(C163⋊C8)C2211yescharacteristic
45C22⋊(C163⋊C8)C2211yescharacteristic
46#19104646C2211yescharacteristic, complemented
47C4×C163⋊C8C2211yescharacteristic, complemented
48C4⋊Q8×C163C2211yescharacteristic, complemented
49#126×C163—21no—
50#124×C163—21no—
51#124×C163C2211yescharacteristic, complemented
52#123×C163—21no—
53C22⋊Q8×C163—21no—
54C22⋊Q8×C163C2211yescharacteristic
55#121×C163C2211yescharacteristic
56C4×Q8×C163C2211yescharacteristic, complemented
57C4×D4×C163C2211yescharacteristic, complemented
58#117×C163C2211yescharacteristic
59C23×D163—21no—
60C2×D652—22no—
61C2×D652—22no—
62C2×D652—21no—
63C2×D652C2311yescharacteristic
64C2×D652C2311yescharacteristic
65C2×C4×D163—21no—
66C2×C4×D163—21no—
67C2×C4×D163—21no—
68C2×C4×D163—21no—
69#14579686—22no—
70#14579686—22no—
71#14579684—22no—
72#14579684—22no—
73C2×C163⋊C8C2311yescharacteristic
74C2×C163⋊C8C2311yescharacteristic
75C2×C163⋊C8C2311yescharacteristic
76C2×C163⋊C8C2311yescharacteristic
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79C22⋊Dic163—21no—
80(C2×D163)⋊C4—21no—
81(C2×D163)⋊C4—21no—
82(C2×D163)⋊C4—21no—
83(C2×D163)⋊C4—21no—
84C4×Dic163—21no—
85C2×Q8×C163—21no—
86C2×Q8×C163C2311yescharacteristic
87C2×D4×C163—21no—
88C2×D4×C163C2311yescharacteristic
89C22×C4×C163—21no—
90C22×C4×C163C2311yescharacteristic
91C4⋊C4×C163—21no—
92C4⋊C4×C163—21no—
93C4⋊C4×C163—21no—
94C4⋊C4×C163D411yescharacteristic
95C4⋊C4×C163D411yescharacteristic
96C4⋊C4×C163C2311yescharacteristic
97C4⋊C4×C163C2311yescharacteristic
98C22⋊C4×C163—21no—
99C22⋊C4×C163—21no—
100C22⋊C4×C163—21no—
101C22⋊C4×C163—21no—
102C22⋊C4×C163—21no—
103C22⋊C4×C163D411yescharacteristic
104C22⋊C4×C163D411yescharacteristic
105C42×C163—21no—
106C42×C163C2311yescharacteristic
107C42×C163C2311yescharacteristic
108C22×D163—21no—
109C22×D163—21no—
110C22×D163—24no—
111C22×D163—81no—
112C22×D163—81no—
113D652—24no—
114D652—21no—
115D652—21no—
116D652#4112yes—
117D652#4112yes—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C4×D163—81no—
121C4×D163—81no—
122C4×D163—81no—
123C4×D163—81no—
124C163⋊C8—21no—
125C163⋊C8—21no—
126C163⋊C8—21no—
127C163⋊C8—21no—
128C23×C163—21no—
129C23×C163C2×D411yescharacteristic
130Q8×C163—21no—
131Q8×C163—21no—
132Q8×C163—81no—
133D4×C163—21no—
134D4×C163—21no—
135D4×C163—81no—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—81no—
142C2×C4×C163C2×D411yescharacteristic
143C2×C4×C163C2×D411yescharacteristic
144C2×C4×C163C2×D411yescharacteristic
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2×D411yescharacteristic
147C2×C4×C163C2411yescharacteristic
148C22×C163—21no—
149C22×C163—21no—
150C22×C163—21no—
151C22×C163—41no—
152C22×C163—41no—
153C22×C163—41no—
154C22×C163C22×D411yescharacteristic
155C2×D163—22no—
156C2×D163—22no—
157C2×D163—41no—
158C2×D163—41no—
159C2×D163—41no—
160C2×D163—41no—
161C2×D163—84no—
162C4×C163—21no—
163C4×C163—21no—
164C4×C163—21no—
165C4×C163—21no—
166C4×C163—21no—
167C4×C163—41no—
168C4×C163—41no—
169C4×C163—41no—
170C4×C163C2×#4111yescharacteristic
171C4×C163C2×#4111yescharacteristic
172Dic163—41no—
173Dic163—41no—
174C2×C163—41no—
175C2×C163—81no—
176C2×C163#52511yescharacteristic
177C2×C163#57611yescharacteristic
178C2×C163#57711yescharacteristic
179D163—42no—
180D163—42no—
181C163#311811yescharacteristic, complemented, Sylow
182#3118—1631nomaximal, Sylow
183#542D16311yescharacteristic, complemented
184#518—1631no—
185#485—1631no—
186#482—1631no—
187#476—1631no—
188#474—1631no—
189#463—1631no—
190#463—1631no—
191#450—1631no—
192#447—1631no—
193#442—1631no—
194#440—1631no—
195C4×SD16—1631no—
196C4×D8—1631no—
197#433—1631no—
198C22×D4—1631no—
199C2×SD16—1631no—
200C2×SD16—1631no—
201C2×D8—1631no—
202C2×D8—1631no—
203C4⋊Q8C2×D16311yescharacteristic, complemented
204#126—21no—
205#124—21no—
206#124C2×D16311yescharacteristic, complemented
207#123—21no—
208C22⋊Q8—21no—
209C22⋊Q8C2×D16311yescharacteristic
210#121C2×D16311yescharacteristic
211C4×Q8C2×D16311yescharacteristic, complemented
212C4×D4—3261no—
213C4×D4—3261no—
214C4×D4—1631no—
215C4×D4—1631no—
216C4×D4—1631no—
217C4×D4—1631no—
218C4×D4C2×D16311yescharacteristic, complemented
219#117—1631no—
220#117C2×D16311yescharacteristic
221C2×C22⋊C4—3261no—
222C2×C22⋊C4—3261no—
223#107—1631no—
224#107—1631no—
225#106—1631no—
226#106—1631no—
227C4⋊C8—1631no—
228C4⋊C8—1631no—
229#103—1631no—
230#103—1631no—
231#102—1631no—
232#102—1631no—
233#102—1631no—
234#102—1631no—
235#102—1631no—
236#102—1631no—
237C22⋊C8—1631no—
238C22⋊C8—1631no—
239#97—1631no—
240C4×C8—1631no—
241C24—3261no—
242C2×Q8—21no—
243C2×Q8C22×D16311yescharacteristic
244C2×D4—21no—
245C2×D4—3262no—
246C2×D4—3262no—
247C2×D4—3261no—
248C2×D4—1631no—
249C2×D4—1631no—
250C2×D4C22×D16311yescharacteristic
251C22×C4—21no—
252C22×C4—3261no—
253C22×C4—3261no—
254C22×C4—3261no—
255C22×C4—3261no—
256C22×C4C22×D16311yescharacteristic
257SD16—3262no—
258SD16—3262no—
259D8—3262no—
260D8—3262no—
261C2×C8—1631no—
262C2×C8—1631no—
263C2×C8—1631no—
264C2×C8—1631no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—1631no—
269C4⋊C4—1631no—
270C4⋊C4C22⋊D16311yescharacteristic
271C4⋊C4C22⋊D16311yescharacteristic
272C4⋊C4C22×D16311yescharacteristic
273C4⋊C4C22×D16311yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—3261no—
284C22⋊C4C22⋊D16311yescharacteristic
285C22⋊C4C22⋊D16311yescharacteristic
286C42—21no—
287C42—3261no—
288C42C22×D16311yescharacteristic
289C42C22×D16311yescharacteristic
290C23—21no—
291C23—13041no—
292C23—13041no—
293C23—3264no—
294C23—3261no—
295C23—3261no—
296C23C2×C22⋊D16311yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—81no—
300D4—21no—
301D4—21no—
302D4—81no—
303D4—3264no—
304D4—3261no—
305D4—3261no—
306D4—1632no—
307D4—1632no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—81no—
314C2×C4—13041no—
315C2×C4—13041no—
316C2×C4—13041no—
317C2×C4—13041no—
318C2×C4—3261no—
319C2×C4—3261no—
320C2×C4C2×C22⋊D16311yescharacteristic
321C2×C4C2×C22⋊D16311yescharacteristic
322C2×C4C2×C22⋊D16311yescharacteristic
323C2×C4C2×C22⋊D16311yescharacteristic
324C2×C4C2×C22⋊D16311yescharacteristic
325C2×C4C23×D16311yescharacteristic
326C8—3261no—
327C8—3261no—
328C8—3261no—
329C8—3261no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—13044no—
337C22—3262no—
338C22—3262no—
339C22—6521no—
340C22—6521no—
341C22—6521no—
342C22—6521no—
343C22C22×C22⋊D16311yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—6521no—
353C4—6521no—
354C4#41×D16311yescharacteristic
355C4#41×D16311yescharacteristic
356C2—41no—
357C2—81no—
358C2—6522no—
359C2—6522no—
360C2#2378217211yescharacteristic
361C2#2378246811yescharacteristic
362C2#2378247311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#542×C163C2
socleC22×C163C22×D4
prime radical#23781798C2
hypercenter#542D163
hypocenterC163#3118
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #23782139C2semidirect
2. #23782003C2semidirect
3. #23781986C2semidirect
4. #23781952C2semidirect
5. #23781942C2semidirect
6. #23781877C2semidirect
7. #23781877C2semidirect
8. #23781815C2semidirect
10. #23781775C2semidirect
11. #23781763C2semidirect
12. C4×#14579686C2semidirect
13. C4×#14579684C2semidirect
14. #23781726C2semidirect
15. #542×C163C2semidirect
30. #19104668C22semidirect
31. #19104668C22semidirect
32. #19104666C22semidirect
33. #19104666C22semidirect
34. C4⋊(C163⋊C8)C22semidirect
35. C4⋊(C163⋊C8)C22semidirect
36. #19104659C22semidirect
37. #19104659C22semidirect
38. #19104656C22semidirect
39. #19104656C22semidirect
46. #19104646C22semidirect
47. C4×C163⋊C8C22semidirect
48. C4⋊Q8×C163C22semidirect
51. #124×C163C22semidirect
56. C4×Q8×C163C22semidirect
57. C4×D4×C163C22semidirect
181. C163#3118semidirect
183. #542D163semidirect
203. C4⋊Q8C2×D163semidirect
206. #124C2×D163semidirect
211. C4×Q8C2×D163semidirect
218. C4×D4C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237821392
#237820032
#237819862
#237819522
#237819422
#237818772
#237818152
#237817982
#237817752
#237817632
C4×#145796862
C4×#145796842
#237817262
#542×C1632
#3118163