Finite Group Databasebeta

← #35224724

Subgroups — #35224724

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#23782287C211yesmaximal, characteristic, complemented
2#23782137C211yesmaximal, characteristic, complemented
3#23782004C211yesmaximal, characteristic, complemented
4#23781992C211yesmaximal, characteristic, complemented
5#23781953C211yesmaximal, characteristic, complemented
6#23781944C211yesmaximal, characteristic, complemented
7#23781883C211yesmaximal, characteristic, complemented
8#23781882C211yesmaximal, characteristic, complemented
9#23781814C211yesmaximal, characteristic, complemented
10#23781800C211yesmaximal, characteristic
11#23781778C211yesmaximal, characteristic, complemented
12C4×#14579686C211yesmaximal, characteristic, complemented
13C4×#14579684C211yesmaximal, characteristic, complemented
14#23781727C211yesmaximal, characteristic, complemented
15#440×C163C211yesmaximal, characteristic, complemented
16C2×D4×D163C2211yescharacteristic
17C2×#14579686C2211yescharacteristic
18C2×D1304C2211yescharacteristic
19C2×#14579684C2211yescharacteristic
20#19104734C2211yescharacteristic, complemented
21#19104729—21no—
22#19104724C2211yescharacteristic, complemented
23#19104723—21no—
24#19104720—21no—
25#19104713—21no—
26#19104711C2211yescharacteristic
27#19104710C2211yescharacteristic
28C4×D652C2211yescharacteristic
29C4×D652C2211yescharacteristic, complemented
30D4×Dic163C2211yescharacteristic
31#19104697C2211yescharacteristic
32C4×C22⋊D163—21no—
33(C22⋊D163)⋊C4—21no—
34#19104693C2211yescharacteristic
35#19104692C2211yescharacteristic
36C2×(C2×D163)⋊C4—21no—
37C22⋊C4×D163—21no—
38#19104668C2211yescharacteristic, complemented
39#19104667C2211yescharacteristic, complemented
40#19104666C2211yescharacteristic, complemented
41Dic163⋊C8C2211yescharacteristic, complemented
42Dic163⋊C8C2211yescharacteristic, complemented
43#19104659C2211yescharacteristic, complemented
44#19104657C2211yescharacteristic
45#19104657C2211yescharacteristic, complemented
46#19104656C2211yescharacteristic
47#19104655C2211yescharacteristic
48#19104655C2211yescharacteristic, complemented
49(C2×D163)⋊C8C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51C4×C163⋊C8C2211yescharacteristic, complemented
52C2×SD16×C163C2211yescharacteristic
53C4×Q8×C163C2211yescharacteristic, complemented
54C4×D4×C163C2211yescharacteristic
55#107×C163C2211yescharacteristic, complemented
56#103×C163C2211yescharacteristic, complemented
57#102×C163C2211yescharacteristic
58#97×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60C2×Dic326—21no—
61C2×D652—21no—
62C2×D652C2311yescharacteristic
63C2×D652C2311yescharacteristic
64C2×C22⋊D163—21no—
65D4×D163—22no—
66D4×D163—22no—
67C22×Dic163—21no—
68C2×C4×D163—21no—
69C2×C4×D163—21no—
70C2×C4×D163—21no—
71C2×C4×D163C2311yescharacteristic
72#14579686—22no—
73D1304—22no—
74#14579684—22no—
75C2×C163⋊C8C2311yescharacteristic
76C2×C163⋊C8C2311yescharacteristic
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4D411yescharacteristic
82Dic163⋊C4D411yescharacteristic
83C22⋊Dic163—21no—
84(C2×D163)⋊C4—21no—
85(C2×D163)⋊C4—21no—
86(C2×D163)⋊C4—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)⋊C4D411yescharacteristic
93(C2×D163)⋊C4D411yescharacteristic
94C4×Dic163—21no—
95C4×Dic163C2311yescharacteristic
96C2×Q8×C163C2311yescharacteristic
97C2×D4×C163C2311yescharacteristic
98C22×C4×C163—21no—
99SD16×C163—22no—
100C2×C8×C163C2311yescharacteristic
101C2×C8×C163C2311yescharacteristic
102C4⋊C4×C163—21no—
103C4⋊C4×C163C2311yescharacteristic
104C4⋊C4×C163C2311yescharacteristic
105C22⋊C4×C163—21no—
106C42×C163—21no—
107C42×C163C2311yescharacteristic
108C22×D163—21no—
109C22×D163—24no—
110C22×D163—21no—
111C22×D163—81no—
112C22×D163—81no—
113C22×D163C2×D411yescharacteristic
114Dic326—81no—
115D652—21no—
116D652—21no—
117D652—21no—
118D652—81no—
119D652#4112yes—
120C22⋊D163—24no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—81no—
124C2×Dic163C2×D411yescharacteristic
125C2×Dic163C2×D411yescharacteristic
126C2×Dic163C2×D411yescharacteristic
127C4×D163—21no—
128C4×D163—21no—
129C4×D163—81no—
130C4×D163—81no—
131C4×D163—81no—
132C163⋊C8—21no—
133C163⋊C8—21no—
134C23×C163—21no—
135Q8×C163—21no—
136Q8×C163—21no—
137D4×C163—21no—
138D4×C163#4112yes—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—81no—
143C2×C4×C163C2×D411yescharacteristic
144C2×C4×C163C2×D411yescharacteristic
145C2×C4×C163C2411yescharacteristic
146C8×C163—21no—
147C8×C163—21no—
148C22×C163—22no—
149C22×C163—41no—
150C22×C163—41no—
151C22×C163C22×D411yescharacteristic
152C2×D163—21no—
153C2×D163—22no—
154C2×D163—21no—
155C2×D163—21no—
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—41no—
165C4×C163—41no—
166C4×C163—41no—
167C4×C163C2×#4111yescharacteristic
168C4×C163C2×#4111yescharacteristic
169Dic163—21no—
170Dic163—21no—
171Dic163—21no—
172Dic163—41no—
173Dic163—41no—
174C2×C163—42no—
175C2×C163#52511yescharacteristic
176C2×C163#57611yescharacteristic
177C2×C163#57711yescharacteristic
178D163—41no—
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#440D16311yescharacteristic, complemented
195C4×SD16—1631no—
196C4×D8—1631no—
197#433—1631no—
198C22×D4—1631no—
199C2×SD16—1631no—
200C2×SD16C2×D16311yescharacteristic
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×Q8C2×D16311yescharacteristic, complemented
212C4×D4—3261no—
213C4×D4—3261no—
214C4×D4—1631no—
215C4×D4—1631no—
216C4×D4—1631no—
217C4×D4—1631no—
218C4×D4C2×D16311yescharacteristic
219#117—1631no—
220#117—1631no—
221C2×C22⋊C4—3261no—
222C2×C22⋊C4—3261no—
223#107—1631no—
224#107C2×D16311yescharacteristic, complemented
225#106—1631no—
226#106—1631no—
227C4⋊C8—1631no—
228C4⋊C8—1631no—
229#103—1631no—
230#103C2×D16311yescharacteristic, complemented
231#102—1631no—
232#102—1631no—
233#102—1631no—
234#102—1631no—
235#102—1631no—
236#102C2×D16311yescharacteristic
237C22⋊C8—1631no—
238C22⋊C8—1631no—
239#97C2×D16311yescharacteristic, complemented
240C4×C8—1631no—
241C24—3261no—
242C2×Q8—3261no—
243C2×Q8C22×D16311yescharacteristic
244C2×D4—3261no—
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×C4—1631no—
257SD16—22no—
258SD16—3262no—
259D8—3262no—
260D8—3262no—
261C2×C8—1631no—
262C2×C8—1631no—
263C2×C8C22×D16311yescharacteristic
264C2×C8C22×D16311yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—3261no—
267C4⋊C4—3261no—
268C4⋊C4—1631no—
269C4⋊C4—1631no—
270C4⋊C4—1631no—
271C4⋊C4—1631no—
272C4⋊C4C22×D16311yescharacteristic
273C4⋊C4C22×D16311yescharacteristic
274C22⋊C4—21no—
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⋊C4—1631no—
285C22⋊C4—1631no—
286C42—21no—
287C42—3261no—
288C42—1631no—
289C42C22×D16311yescharacteristic
290C23—21no—
291C23—13041no—
292C23—13041no—
293C23—3264no—
294C23—3261no—
295C23—3261no—
296C23—1631no—
297Q8—21no—
298Q8—21no—
299Q8—13041no—
300D4—21no—
301D4—13041no—
302D4—3264no—
303D4—3261no—
304D4—3261no—
305D4—3261no—
306D4—1632no—
307D4#1457969912yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—13041no—
313C2×C4—13041no—
314C2×C4—13041no—
315C2×C4—13041no—
316C2×C4—3261no—
317C2×C4—3261no—
318C2×C4—3261no—
319C2×C4—3261no—
320C2×C4—1631no—
321C2×C4—1631no—
322C2×C4—1631no—
323C2×C4D4×D16311yescharacteristic
324C2×C4D4×D16311yescharacteristic
325C2×C4C23×D16311yescharacteristic
326C8—21no—
327C8—21no—
328C8—3261no—
329C8—3261no—
330C22—22no—
331C22—41no—
332C22—41no—
333C22—13044no—
334C22—3261no—
335C22—3262no—
336C22—3261no—
337C22—3261no—
338C22—6521no—
339C22—6521no—
340C22—6521no—
341C22—6521no—
342C22—6521no—
343C22C2×D4×D16311yescharacteristic
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—3261no—
350C4—3261no—
351C4—3261no—
352C4—6521no—
353C4—6521no—
354C4#41×D16311yescharacteristic
355C4C2×#1457969911yescharacteristic
356C2—42no—
357C2—13041no—
358C2—6521no—
359C2—6522no—
360C2#2378217411yescharacteristic
361C2#2378246911yescharacteristic
362C2#2378247411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#440×C163C2
socleC22×C163C22×D4
prime radical#23781800C2
hypercenter#440D163
hypocenterC163#3118
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #23782287C2semidirect
2. #23782137C2semidirect
3. #23782004C2semidirect
4. #23781992C2semidirect
5. #23781953C2semidirect
6. #23781944C2semidirect
7. #23781883C2semidirect
8. #23781882C2semidirect
9. #23781814C2semidirect
11. #23781778C2semidirect
12. C4×#14579686C2semidirect
13. C4×#14579684C2semidirect
14. #23781727C2semidirect
15. #440×C163C2semidirect
20. #19104734C22semidirect
22. #19104724C22semidirect
29. C4×D652C22semidirect
38. #19104668C22semidirect
39. #19104667C22semidirect
40. #19104666C22semidirect
41. Dic163⋊C8C22semidirect
42. Dic163⋊C8C22semidirect
43. #19104659C22semidirect
45. #19104657C22semidirect
48. #19104655C22semidirect
51. C4×C163⋊C8C22semidirect
53. C4×Q8×C163C22semidirect
55. #107×C163C22semidirect
56. #103×C163C22semidirect
58. #97×C163C22semidirect
181. C163#3118semidirect
194. #440D163semidirect
211. C4×Q8C2×D163semidirect
224. #107C2×D163semidirect
230. #103C2×D163semidirect
239. #97C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237822872
#237821372
#237820042
#237819922
#237819532
#237819442
#237818832
#237818822
#237818142
#237818002
#237817782
C4×#145796862
C4×#145796842
#237817272
#440×C1632
#3118163