Finite Group Databasebeta

← #35226490

Subgroups — #35226490

Order 20864.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782315C211yesmaximal, characteristic, complemented
2#23782303C211yesmaximal, characteristic, complemented
3#23782026C211yesmaximal, characteristic, complemented
4#23781994C211yesmaximal, characteristic, complemented
5#23781984C211yesmaximal, characteristic, complemented
6#23781953C211yesmaximal, characteristic, complemented
7#23781916C211yesmaximal, characteristic, complemented
8#23781891C211yesmaximal, characteristic, complemented
9#23781882C211yesmaximal, characteristic, complemented
10#23781797C211yesmaximal, characteristic, complemented
11#23781777C211yesmaximal, characteristic, complemented
12#23781739C211yesmaximal, characteristic, complemented
13#23781666C211yesmaximal, characteristic, complemented
14#102×D163C211yesmaximal, characteristic, complemented
15#449×C163C211yesmaximal, characteristic, complemented
16C2×#14579698—21no—
17C2×D4×D163C2211yescharacteristic
18C2×#14579684C2211yescharacteristic
19C2×#14579683C2211yescharacteristic, complemented
20C2×C8×D163C2211yescharacteristic
21#19104734C2211yescharacteristic, complemented
22#19104724C2211yescharacteristic
23#19104718—21no—
24#19104717—21no—
25#19104716—21no—
26#19104712—21no—
27#19104711C2211yescharacteristic
28#19104709C2211yescharacteristic
29#19104707—21no—
30#19104706C2211yescharacteristic
31#19104703—21no—
32D4×Dic163C2211yescharacteristic
33D4×Dic163C2211yescharacteristic, complemented
34C4×C22⋊D163C2211yescharacteristic
35C2×C4⋊Dic163C2211yescharacteristic, complemented
36C4⋊C4×D163C2211yescharacteristic
37C2×(C2×D163)⋊C4—21no—
38#19104667C2211yescharacteristic, complemented
39#19104667C2211yescharacteristic, complemented
40#19104666C2211yescharacteristic, complemented
41Dic163⋊C8C2211yescharacteristic, complemented
42#19104659C2211yescharacteristic, complemented
43#19104659C2211yescharacteristic, complemented
44#19104657C2211yescharacteristic
45#19104657C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104656C2211yescharacteristic
48#19104656C2211yescharacteristic, complemented
49#19104655C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51#19104645C2211yescharacteristic, complemented
52C2×#41×C163C2211yescharacteristic
53C2×SD16×C163C2211yescharacteristic, complemented
54C2×D8×C163C2211yescharacteristic
55#121×C163C2211yescharacteristic, complemented
56#103×C163C2211yescharacteristic, complemented
57#102×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60#14579698—44no—
61C2×Dic326—21no—
62C2×D652C2311yescharacteristic
63C2×C22⋊D163—21no—
64C2×C22⋊D163—21no—
65C2×C22⋊D163—21no—
66C2×C22⋊D163D411yescharacteristic
67D4×D163—22no—
68D4×D163—41no—
69C22×Dic163—21no—
70C22×Dic163—21no—
71C22×Dic163—21no—
72C2×C4×D163—21no—
73C2×C4×D163C2311yescharacteristic
74#14579684—41no—
75#14579683—41no—
76C2×C163⋊C8C2311yescharacteristic
77C2×C163⋊C8C2311yescharacteristic
78C8×D163—41no—
79C4⋊Dic163—21no—
80C4⋊Dic163C2311yescharacteristic
81C4⋊Dic163C2311yescharacteristic
82C4⋊Dic163C2311yescharacteristic
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4D411yescharacteristic
87C22⋊Dic163—21no—
88C22⋊Dic163—21no—
89C22⋊Dic163—21no—
90C22⋊Dic163D411yescharacteristic
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4D411yescharacteristic
96C4×Dic163C2311yescharacteristic
97#41×C163—22no—
98#41×C163—41no—
99C2×Q8×C163C2311yescharacteristic
100C2×D4×C163—21no—
101C2×D4×C163—21no—
102C2×D4×C163C2311yescharacteristic
103C2×D4×C163C2311yescharacteristic
104C22×C4×C163—21no—
105C22×C4×C163C2311yescharacteristic
106SD16×C163—41no—
107D8×C163—22no—
108C2×C8×C163C2311yescharacteristic
109C2×C8×C163C2311yescharacteristic
110C4⋊C4×C163C2311yescharacteristic
111C22⋊C4×C163—21no—
112C22×D163—21no—
113C22×D163—42no—
114C22×D163—42no—
115C22×D163—81no—
116C22×D163C2×D411yescharacteristic
117Dic326—42no—
118D652—21no—
119D652—21no—
120C22⋊D163—22no—
121C22⋊D163—42no—
122C22⋊D163—42no—
123C22⋊D163—81no—
124C2×Dic163—21no—
125C2×Dic163—21no—
126C2×Dic163—21no—
127C2×Dic163—42no—
128C2×Dic163—42no—
129C2×Dic163—81no—
130C2×Dic163C2×D411yescharacteristic
131C2×Dic163C2×D411yescharacteristic
132C2×Dic163C2×D411yescharacteristic
133C4×D163—21no—
134C4×D163—21no—
135C4×D163—42no—
136C163⋊C8—21no—
137C163⋊C8—21no—
138C23×C163—21no—
139C23×C163—21no—
140C23×C163C2×D411yescharacteristic
141Q8×C163—21no—
142Q8×C163—21no—
143D4×C163—21no—
144D4×C163—21no—
145D4×C163—21no—
146D4×C163—42no—
147D4×C163—42no—
148D4×C163#4112yescomplemented
149C2×C4×C163—21no—
150C2×C4×C163—21no—
151C2×C4×C163—21no—
152C2×C4×C163—21no—
153C2×C4×C163—42no—
154C2×C4×C163C2×D411yescharacteristic
155C2×C4×C163C2411yescharacteristic
156C8×C163—21no—
157C8×C163—21no—
158C22×C163—21no—
159C22×C163—22no—
160C22×C163—21no—
161C22×C163—21no—
162C22×C163—41no—
163C22×C163—41no—
164C22×C163—41no—
165C22×C163—41no—
166C22×C163—41no—
167C22×C163C22×D411yescharacteristic
168C2×D163—21no—
169C2×D163—21no—
170C2×D163—41no—
171C2×D163—41no—
172C2×D163—41no—
173C2×D163—84no—
174C2×D163AGL(1,ℤ/8)12yes—
175C4×C163—21no—
176C4×C163—41no—
177C4×C163—41no—
178C4×C163C2×#4111yescharacteristic
179C4×C163H(5,𝔽2)11yescharacteristic
180Dic163—21no—
181Dic163—21no—
182Dic163—21no—
183Dic163—41no—
184Dic163—41no—
185Dic163—41no—
186C2×C163—42no—
187C2×C163—41no—
188C2×C163—81no—
189C2×C163#54611yescharacteristic
190C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C163#57711yescharacteristic
192D163—22no—
193D163—81no—
194C163#330611yescharacteristic, complemented, Sylow
195#3306—1631nomaximal, Sylow
196#547—1631no—
197#546—1631no—
198#489—1631no—
199#483—1631no—
200#481—1631no—
201#476—1631no—
202#469—1631no—
203#465—1631no—
204#463—1631no—
205#449D16311yescharacteristic, complemented
206#447—1631no—
207#442—1631no—
208#435—1631no—
209#417—1631no—
210C2×#102—1631no—
211C2×#41—3261no—
212C2×#41C2×D16311yescharacteristic
213C22×D4—1631no—
214C2×SD16C2×D16311yescharacteristic, complemented
215C2×D8—1631no—
216C2×D8C2×D16311yescharacteristic
217C2×M16—1631no—
218C22×C8—1631no—
219C4⋊Q8—1631no—
220#124—1631no—
221#123—3261no—
222#123—3261no—
223#123—3261no—
224C22⋊Q8—3261no—
225C22⋊Q8—1631no—
226#121—3261no—
227#121—1631no—
228#121—1631no—
229#121C2×D16311yescharacteristic, complemented
230#120—3261no—
231C4×D4—1631no—
232C4×D4—1631no—
233C4×D4—1631no—
234C2×C4⋊C4—1631no—
235C2×C4⋊C4—1631no—
236C2×C22⋊C4—3261no—
237#106—1631no—
238#106—1631no—
239#106—1631no—
240C4⋊C8—1631no—
241#103—1631no—
242#103—1631no—
243#103C2×D16311yescharacteristic, complemented
244#102—1631no—
245#102—1631no—
246#102—1631no—
247#102—1631no—
248#102—1631no—
249#102—1631no—
250#102C2×D16311yescharacteristic
251C22⋊C8—1631no—
252C22⋊C8C2×D16311yescharacteristic, complemented
253#97—1631no—
254C24—3261no—
255#41—22no—
256#41—41no—
257#41—6524no—
258C2×Q8—3261no—
259C2×Q8C22×D16311yescharacteristic
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—3261no—
263C2×D4—3262no—
264C2×D4—3261no—
265C2×D4—3261no—
266C2×D4—6521no—
267C2×D4—1631no—
268C2×D4—1631no—
269C2×D4C22×D16311yescharacteristic
270C2×D4C22×D16311yescharacteristic
271C22×C4—21no—
272C22×C4—3261no—
273C22×C4—3261no—
274C22×C4—3261no—
275C22×C4—3261no—
276C22×C4—1631no—
277C22×C4C22×D16311yescharacteristic
278SD16—41no—
279D8—22no—
280D8—6521no—
281M16—6521no—
282C2×C8—6521no—
283C2×C8—1631no—
284C2×C8—1631no—
285C2×C8C22×D16311yescharacteristic
286C2×C8C22×D16311yescharacteristic
287C4⋊C4—3261no—
288C4⋊C4—3261no—
289C4⋊C4—3261no—
290C4⋊C4—3261no—
291C4⋊C4—1631no—
292C4⋊C4—1631no—
293C4⋊C4—1631no—
294C4⋊C4—1631no—
295C4⋊C4C22×D16311yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—3261no—
298C22⋊C4—3261no—
299C22⋊C4—3261no—
300C22⋊C4—3261no—
301C22⋊C4—3261no—
302C22⋊C4—3261no—
303C22⋊C4—3261no—
304C22⋊C4—1631no—
305C22⋊C4—1631no—
306C42—1631no—
307C23—21no—
308C23—21no—
309C23—13041no—
310C23—3261no—
311C23—6522no—
312C23—6522no—
313C23—1631no—
314C23D4×D16311yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—6522no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—42no—
323D4—13041no—
324D4—3262no—
325D4—3261no—
326D4—3261no—
327D4—6522no—
328D4—6522no—
329D4#1457969812yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—13041no—
336C2×C4—3261no—
337C2×C4—3261no—
338C2×C4—3261no—
339C2×C4—3261no—
340C2×C4—3261no—
341C2×C4—6522no—
342C2×C4—6522no—
343C2×C4—6522no—
344C2×C4—1631no—
345C2×C4—1631no—
346C2×C4—1631no—
347C2×C4D4×D16311yescharacteristic
348C2×C4C23×D16311yescharacteristic
349C8—21no—
350C8—21no—
351C8—3261no—
352C8—3261no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—13044no—
363C22—3261no—
364C22—3261no—
365C22—6521no—
366C22—6521no—
367C22—6521no—
368C22—1632no—
369C22C2×D4×D16311yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—3261no—
374C4—3261no—
375C4—3261no—
376C4—6521no—
377C4—6521no—
378C4—6521no—
379C4C2×#1457969811yescharacteristic
380C4#1910478011yescharacteristic
381C2—41no—
382C2—42no—
383C2—81no—
384C2—13041no—
385C2—3262no—
386C2#2378230211yescharacteristic
387C2AGL(1,ℤ/8)×D16311yescharacteristic
388C2#2378247211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#449×C163C2
socleC22×C163C22×D4
hypercenter#449D163
hypocenterC163#3306
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #23782315C2semidirect
2. #23782303C2semidirect
3. #23782026C2semidirect
4. #23781994C2semidirect
5. #23781984C2semidirect
6. #23781953C2semidirect
7. #23781916C2semidirect
8. #23781891C2semidirect
9. #23781882C2semidirect
10. #23781797C2semidirect
11. #23781777C2semidirect
12. #23781739C2semidirect
13. #23781666C2semidirect
14. #102×D163C2semidirect
15. #449×C163C2semidirect
19. C2×#14579683C22semidirect
21. #19104734C22semidirect
33. D4×Dic163C22semidirect
35. C2×C4⋊Dic163C22semidirect
38. #19104667C22semidirect
39. #19104667C22semidirect
40. #19104666C22semidirect
41. Dic163⋊C8C22semidirect
42. #19104659C22semidirect
43. #19104659C22semidirect
48. #19104656C22semidirect
51. #19104645C22semidirect
53. C2×SD16×C163C22semidirect
55. #121×C163C22semidirect
56. #103×C163C22semidirect
58. C22⋊C8×C163C22semidirect
148. D4×C163#41semidirect
194. C163#3306semidirect
205. #449D163semidirect
214. C2×SD16C2×D163semidirect
229. #121C2×D163semidirect
243. #103C2×D163semidirect
252. C22⋊C8C2×D163semidirect
329. D4#14579698semidirect

Maximal subgroups

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

subgroupindex
#237823152
#237823032
#237820262
#237819942
#237819842
#237819532
#237819162
#237818912
#237818822
#237817972
#237817772
#237817392
#237816662
#102×D1632
#449×C1632
#3306163