Finite Group Databasebeta

← #35224722

Subgroups — #35224722

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#23782286C211yesmaximal, characteristic, complemented
2#23782139C211yesmaximal, characteristic, complemented
3#23782007C211yesmaximal, characteristic, complemented
4#23781990C211yesmaximal, characteristic, complemented
5#23781956C211yesmaximal, characteristic, complemented
6#23781946C211yesmaximal, characteristic, complemented
7#23781881C211yesmaximal, characteristic, complemented
8#23781881C211yesmaximal, characteristic, complemented
9#23781818C211yesmaximal, characteristic, complemented
10#23781801C211yesmaximal, characteristic
11#23781778C211yesmaximal, characteristic, complemented
12#23781767C211yesmaximal, characteristic, complemented
13C4×#14579688C211yesmaximal, characteristic, complemented
14C4×D1304C211yesmaximal, characteristic, complemented
15#433×C163C211yesmaximal, characteristic, complemented
16C22×D652C2211yescharacteristic
17C2×#14579688C2211yescharacteristic
18C2×#14579688C2211yescharacteristic
19C2×D1304C2211yescharacteristic
20C2×D1304C2211yescharacteristic
21#19104735C2211yescharacteristic, complemented
22#19104730—21no—
23#19104723C2211yescharacteristic, complemented
24#19104721—21no—
25#19104717—21no—
26#19104714C2211yescharacteristic
27#19104713—21no—
28#19104709C2211yescharacteristic
29C4×Dic326C2211yescharacteristic, complemented
30C4×D652C2211yescharacteristic
31C4×D652C2211yescharacteristic
32C4×D652C2211yescharacteristic
33C4×D652C2211yescharacteristic
34C4×D652C2211yescharacteristic, complemented
35#19104697—21no—
36#19104697—21no—
37#19104694C2211yescharacteristic
38C22⋊C4×D163—21no—
39C22⋊C4×D163—21no—
40#19104669C2211yescharacteristic, complemented
41#19104669C2211yescharacteristic, complemented
42#19104667C2211yescharacteristic, complemented
43#19104667C2211yescharacteristic, complemented
44#19104660C2211yescharacteristic, complemented
45#19104660C2211yescharacteristic, complemented
46#19104657C2211yescharacteristic
47#19104657C2211yescharacteristic
48#19104657C2211yescharacteristic
49#19104657C2211yescharacteristic
50#19104657C2211yescharacteristic, complemented
51#19104657C2211yescharacteristic, complemented
52#117×C163C2211yescharacteristic
53C4⋊C8×C163C2211yescharacteristic, complemented
54C4⋊C8×C163C2211yescharacteristic, complemented
55C22⋊C8×C163C2211yescharacteristic
56C22⋊C8×C163C2211yescharacteristic
57#97×C163C2211yescharacteristic, complemented
58C4×C8×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60C2×Dic326—21no—
61C2×Dic326C2311yescharacteristic
62C2×D652—22no—
63C2×D652—22no—
64C2×D652—21no—
65C2×D652C2311yescharacteristic
66C2×D652C2311yescharacteristic
67C2×D652C2311yescharacteristic
68C2×C22⋊D163—21no—
69C2×C4×D163—21no—
70C2×C4×D163—21no—
71C2×C4×D163—21no—
72C2×C4×D163—21no—
73C2×C4×D163—21no—
74#14579688—22no—
75#14579688—22no—
76D1304—22no—
77D1304—22no—
78C4⋊Dic163—21no—
79C4⋊Dic163C2311yescharacteristic
80C4⋊Dic163C2311yescharacteristic
81C4⋊Dic163C2311yescharacteristic
82C4⋊Dic163C2311yescharacteristic
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85C22⋊Dic163—21no—
86C22⋊Dic163—21no—
87(C2×D163)⋊C4—21no—
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95C4×Dic163—21no—
96C4×Dic163—21no—
97C22×C4×C163C2311yescharacteristic
98C2×C8×C163C2311yescharacteristic
99C2×C8×C163C2311yescharacteristic
100C2×C8×C163C2311yescharacteristic
101C2×C8×C163C2311yescharacteristic
102C4⋊C4×C163D411yescharacteristic
103C4⋊C4×C163D411yescharacteristic
104C22⋊C4×C163D411yescharacteristic
105C22⋊C4×C163D411yescharacteristic
106C42×C163C2311yescharacteristic
107C42×C163C2311yescharacteristic
108C22×D163—21no—
109C22×D163—21no—
110C22×D163—24no—
111C22×D163—21no—
112C22×D163—81no—
113C22×D163—81no—
114Dic326—21no—
115Dic326—21no—
116Dic326—81no—
117D652—24no—
118D652—21no—
119D652—21no—
120D652—21no—
121D652—21no—
122D652#4112yes—
123D652#4112yes—
124C22⋊D163—81no—
125C2×Dic163—21no—
126C2×Dic163—21no—
127C2×Dic163—21no—
128C2×Dic163—21no—
129C2×Dic163—21no—
130C4×D163—81no—
131C4×D163—81no—
132C4×D163—81no—
133C4×D163—81no—
134C4×D163—81no—
135C23×C163C2×D411yescharacteristic
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163C2×D411yescharacteristic
139C2×C4×C163C2×D411yescharacteristic
140C2×C4×C163C2×D411yescharacteristic
141C2×C4×C163C2×D411yescharacteristic
142C2×C4×C163C2×D411yescharacteristic
143C2×C4×C163C2411yescharacteristic
144C8×C163—21no—
145C8×C163—21no—
146C8×C163—21no—
147C8×C163—21no—
148C22×C163—21no—
149C22×C163—21no—
150C22×C163—21no—
151C22×C163C22×D411yescharacteristic
152C2×D163—22no—
153C2×D163—22no—
154C2×D163—41no—
155C2×D163—41no—
156C2×D163—41no—
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×C163C2×#4111yescharacteristic
168C4×C163C2×#4111yescharacteristic
169Dic163—41no—
170Dic163—41no—
171Dic163—41no—
172Dic163—41no—
173Dic163—41no—
174C2×C163—41no—
175C2×C163#52511yescharacteristic
176C2×C163#57611yescharacteristic
177C2×C163#57711yescharacteristic
178D163—42no—
179D163—42no—
180D163—81no—
181C163#311811yescharacteristic, complemented, Sylow
182#3118—1631nomaximal, Sylow
183#542—1631no—
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#433D16311yescharacteristic, complemented
198C22×D4—1631no—
199C2×SD16—1631no—
200C2×SD16—1631no—
201C2×D8—1631no—
202C2×D8—1631no—
203C4⋊Q8—1631no—
204#126—3261no—
205#124—3261no—
206#124—1631no—
207#123—3261no—
208C22⋊Q8—3261no—
209C22⋊Q8—1631no—
210#121—1631no—
211C4×Q8—1631no—
212C4×D4—3261no—
213C4×D4—3261no—
214C4×D4—1631no—
215C4×D4—1631no—
216C4×D4—1631no—
217C4×D4—1631no—
218C4×D4—1631no—
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⋊C8C2×D16311yescharacteristic, complemented
228C4⋊C8C2×D16311yescharacteristic, complemented
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⋊C8C2×D16311yescharacteristic
238C22⋊C8C2×D16311yescharacteristic
239#97C2×D16311yescharacteristic, complemented
240C4×C8C2×D16311yescharacteristic, complemented
241C24—3261no—
242C2×Q8—3261no—
243C2×Q8—1631no—
244C2×D4—3262no—
245C2×D4—3262no—
246C2×D4—3261no—
247C2×D4—3261no—
248C2×D4—1631no—
249C2×D4—1631no—
250C2×D4—1631no—
251C22×C4—3261no—
252C22×C4—3261no—
253C22×C4—3261no—
254C22×C4—3261no—
255C22×C4—3261no—
256C22×C4C22×D16311yescharacteristic
257SD16—3262no—
258SD16—3262no—
259D8—3262no—
260D8—3262no—
261C2×C8C22×D16311yescharacteristic
262C2×C8C22×D16311yescharacteristic
263C2×C8C22×D16311yescharacteristic
264C2×C8C22×D16311yescharacteristic
265C4⋊C4—3261no—
266C4⋊C4—3261no—
267C4⋊C4—3261no—
268C4⋊C4—1631no—
269C4⋊C4—1631no—
270C4⋊C4—1631no—
271C4⋊C4—1631no—
272C4⋊C4D65211yescharacteristic
273C4⋊C4D65211yescharacteristic
274C22⋊C4—3261no—
275C22⋊C4—3261no—
276C22⋊C4—3261no—
277C22⋊C4—3261no—
278C22⋊C4—3261no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—3261no—
284C22⋊C4D65211yescharacteristic
285C22⋊C4D65211yescharacteristic
286C42—3261no—
287C42—3261no—
288C42C22×D16311yescharacteristic
289C42C22×D16311yescharacteristic
290C23—13041no—
291C23—13041no—
292C23—3264no—
293C23—3261no—
294C23—3261no—
295C23—3261no—
296C23C2×D65211yescharacteristic
297Q8—13041no—
298Q8—3261no—
299Q8—3261no—
300D4—13041no—
301D4—3264no—
302D4—3261no—
303D4—3261no—
304D4—3261no—
305D4—3261no—
306D4—1632no—
307D4—1632no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—13041no—
311C2×C4—13041no—
312C2×C4—13041no—
313C2×C4—13041no—
314C2×C4—13041no—
315C2×C4—3261no—
316C2×C4—3261no—
317C2×C4—3261no—
318C2×C4—3261no—
319C2×C4—3261no—
320C2×C4C2×D65211yescharacteristic
321C2×C4C2×D65211yescharacteristic
322C2×C4C2×D65211yescharacteristic
323C2×C4C2×D65211yescharacteristic
324C2×C4C2×D65211yescharacteristic
325C2×C4C23×D16311yescharacteristic
326C8—21no—
327C8—21no—
328C8—21no—
329C8—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—13044no—
334C22—3262no—
335C22—3262no—
336C22—6521no—
337C22—6521no—
338C22—6521no—
339C22—6521no—
340C22—6521no—
341C22—6521no—
342C22—6521no—
343C22C22×D65211yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—6521no—
350C4—6521no—
351C4—6521no—
352C4—6521no—
353C4—6521no—
354C4#41×D16311yescharacteristic
355C4#41×D16311yescharacteristic
356C2—41no—
357C2—13041no—
358C2—6522no—
359C2—6522no—
360C2#2378217711yescharacteristic
361C2#2378247011yescharacteristic
362C2#2378247711yescharacteristic
subgroupG/H
centerC22C22×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#433×C163C2
socleC22×C163C22×D4
prime radical#23781801C2
hypercenter#433D163
hypocenterC163#3118
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #23782286C2semidirect
2. #23782139C2semidirect
3. #23782007C2semidirect
4. #23781990C2semidirect
5. #23781956C2semidirect
6. #23781946C2semidirect
7. #23781881C2semidirect
8. #23781881C2semidirect
9. #23781818C2semidirect
11. #23781778C2semidirect
12. #23781767C2semidirect
13. C4×#14579688C2semidirect
14. C4×D1304C2semidirect
15. #433×C163C2semidirect
21. #19104735C22semidirect
23. #19104723C22semidirect
29. C4×Dic326C22semidirect
34. C4×D652C22semidirect
40. #19104669C22semidirect
41. #19104669C22semidirect
42. #19104667C22semidirect
43. #19104667C22semidirect
44. #19104660C22semidirect
45. #19104660C22semidirect
50. #19104657C22semidirect
51. #19104657C22semidirect
53. C4⋊C8×C163C22semidirect
54. C4⋊C8×C163C22semidirect
57. #97×C163C22semidirect
58. C4×C8×C163C22semidirect
181. C163#3118semidirect
197. #433D163semidirect
227. C4⋊C8C2×D163semidirect
228. C4⋊C8C2×D163semidirect
239. #97C2×D163semidirect
240. C4×C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237822862
#237821392
#237820072
#237819902
#237819562
#237819462
#237818812
#237818182
#237818012
#237817782
#237817672
C4×#145796882
C4×D13042
#433×C1632
#3118163