Finite Group Databasebeta

← #35224331

Subgroups — #35224331

Order 20864.

440 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782221C211yesmaximal, characteristic, complemented
2C4×D4×D163C211yesmaximal, characteristic
3#23781966C211yesmaximal, characteristic, complemented
4#23781965C211yesmaximal, characteristic, complemented
5#23781962C211yesmaximal, characteristic, complemented
6#23781954C211yesmaximal, characteristic, complemented
7#23781953C211yesmaximal, characteristic, complemented
8#23781944C211yesmaximal, characteristic, complemented
9#23781943C211yesmaximal, characteristic, complemented
10#23781706C211yesmaximal, characteristic, complemented
11#23781704C211yesmaximal, characteristic, complemented
12#23781687C211yesmaximal, characteristic, complemented
13#23781668C211yesmaximal, characteristic
14#23781668C211yesmaximal, characteristic
15#478×C163C211yesmaximal, characteristic, complemented
16C2×D4×D163C2211yescharacteristic
17C22×C4×D163—21no—
18C2×#14579682C2211yescharacteristic
19C2×#14579682C2211yescharacteristic
20#19104734C2211yescharacteristic, complemented
21#19104734C2211yescharacteristic, complemented
22#19104727C2211yescharacteristic, complemented
23#19104713—21no—
24#19104713—21no—
25#19104700—21no—
26#19104700—21no—
27C4×D652C2211yescharacteristic
28D4×Dic163C2211yescharacteristic
29#19104697C2211yescharacteristic
30C4×C22⋊D163—21no—
31(C22⋊D163)⋊C4—21no—
32#19104693C2211yescharacteristic
33#19104693C2211yescharacteristic
34C4⋊C4×D163C2211yescharacteristic
35C22⋊C4×D163—21no—
36C42×D163C2211yescharacteristic
37#19104669C2211yescharacteristic, complemented
38#19104668C2211yescharacteristic, complemented
39#19104668C2211yescharacteristic, complemented
40#19104667C2211yescharacteristic, complemented
41#19104666C2211yescharacteristic, complemented
42#19104666C2211yescharacteristic, complemented
43C4⋊(C163⋊C8)C2211yescharacteristic, complemented
44Dic163⋊C8C2211yescharacteristic, complemented
45Dic163⋊C8C2211yescharacteristic, complemented
46#19104657C2211yescharacteristic
47#19104657C2211yescharacteristic
48#19104656C2211yescharacteristic
49#19104656C2211yescharacteristic
50#19104655C2211yescharacteristic
51#19104655C2211yescharacteristic
52#125×C163C2211yescharacteristic, complemented
53C4×D4×C163C2211yescharacteristic
54#107×C163C2211yescharacteristic, complemented
55#106×C163C2211yescharacteristic, complemented
56C4⋊C8×C163C2211yescharacteristic, complemented
57#102×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59C23×D163—21no—
60C2×Dic326—21no—
61C2×Dic326—21no—
62C2×D652C2311yescharacteristic
63C2×C22⋊D163—21no—
64D4×D163—22no—
65D4×D163Q814yescomplemented
66C22×Dic163—21no—
67C2×C4×D163—21no—
68C2×C4×D163—21no—
69C2×C4×D163—21no—
70C2×C4×D163—44no—
71C2×C4×D163D411yescharacteristic
72C2×C4×D163D411yescharacteristic
73C2×C4×D163C2311yescharacteristic
74#14579682—22no—
75#14579682—22no—
76C2×C163⋊C8C2311yescharacteristic
77C2×C163⋊C8C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79C4⋊Dic163C2311yescharacteristic
80C4⋊Dic163C2311yescharacteristic
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86C22⋊Dic163—21no—
87(C2×D163)⋊C4—21no—
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91C4×Dic163—21no—
92C4×Dic163—21no—
93C4×Dic163D411yescharacteristic
94C4×Dic163D411yescharacteristic
95C4×Dic163C2311yescharacteristic
96C2×D4×C163C2311yescharacteristic
97C22×C4×C163—21no—
98C2×C8×C163C2311yescharacteristic
99C2×C8×C163C2311yescharacteristic
100C4⋊C4×C163—21no—
101C4⋊C4×C163—21no—
102C4⋊C4×C163C2311yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C4⋊C4×C163C2311yescharacteristic
105C22⋊C4×C163—21no—
106C42×C163C2311yescharacteristic
107C22×D163—21no—
108C22×D163—24no—
109C22×D163—42no—
110C22×D163—42no—
111C22×D163C2×D411yescharacteristic
112Dic326—42no—
113Dic326—42no—
114D652—21no—
115D652C2×Q812yes—
116C22⋊D163—24no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163—42no—
121C2×Dic163C2×D411yescharacteristic
122C2×Dic163C2×D411yescharacteristic
123C2×Dic163C2×D411yescharacteristic
124C4×D163—22no—
125C4×D163—41no—
126C4×D163—42no—
127C4×D163—42no—
128C4×D163—42no—
129C4×D163C2×Q812yes—
130C4×D163#4112yes—
131C163⋊C8—21no—
132C163⋊C8—21no—
133C23×C163—21no—
134D4×C163—21no—
135D4×C163C2×Q812yes—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163—42no—
140C2×C4×C163C2×D411yescharacteristic
141C2×C4×C163C2×D411yescharacteristic
142C2×C4×C163C2411yescharacteristic
143C8×C163—21no—
144C8×C163—21no—
145C22×C163—22no—
146C22×C163—41no—
147C22×C163—41no—
148C22×C163C22×D411yescharacteristic
149C2×D163—21no—
150C2×D163—21no—
151C2×D163—22no—
152C2×D163—48no—
153C2×D163—41no—
154C2×D163—41no—
155C2×D163C22⋊Q812yes—
156C4×C163—21no—
157C4×C163—21no—
158C4×C163—41no—
159C4×C163—41no—
160C4×C163—41no—
161C4×C163C22×Q811yescharacteristic
162C4×C163C2×#4111yescharacteristic
163Dic163—21no—
164Dic163—41no—
165Dic163—41no—
166Dic163—41no—
167Dic163C22⋊Q812yes—
168Dic163AGL(1,ℤ/8)12yes—
169C2×C163—42no—
170C2×C163C2×C22⋊Q811yescharacteristic
171C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
172C2×C163#57511yescharacteristic
173D163—22no—
174D163—42no—
175C163#307711yescharacteristic, complemented, Sylow
176#3077—1631nomaximal, Sylow
177#533—1631no—
178C2×C4×D4—1631no—
179#478—1631no—
180#478—1631no—
181#478—1631no—
182#478D16311yescharacteristic, complemented
183#476—1631no—
184#476—1631no—
185#474—1631no—
186#474—1631no—
187#428—1631no—
188#428—1631no—
189#423—1631no—
190#418—1631no—
191#418—1631no—
192C22×D4—1631no—
193C23×C4—3261no—
194C2×M16—1631no—
195C2×M16—1631no—
196C4⋊Q8—1631no—
197C4⋊Q8—1631no—
198#125—1631no—
199#125C2×D16311yescharacteristic, complemented
200C22⋊Q8—3261no—
201C22⋊Q8—3261no—
202C4×Q8—3261no—
203C4×Q8—3261no—
204C4×D4—3261no—
205C4×D4—3261no—
206C4×D4—1631no—
207C4×D4—1631no—
208C4×D4—1631no—
209C4×D4C2×D16311yescharacteristic
210#117—1631no—
211#117—1631no—
212C2×C4⋊C4—1631no—
213C2×C22⋊C4—3261no—
214C2×C42—1631no—
215#107—1631no—
216#107—1631no—
217#107—1631no—
218#107C2×D16311yescharacteristic, complemented
219#106—1631no—
220#106—1631no—
221#106—1631no—
222#106C2×D16311yescharacteristic, complemented
223C4⋊C8—1631no—
224C4⋊C8—1631no—
225C4⋊C8—1631no—
226C4⋊C8C2×D16311yescharacteristic, complemented
227#102—1631no—
228#102—1631no—
229#102—1631no—
230#102—1631no—
231#102—1631no—
232#102—1631no—
233#102C2×D16311yescharacteristic
234#102C2×D16311yescharacteristic
235C24—3261no—
236C2×Q8—3261no—
237C2×Q8—3261no—
238C2×D4—3262no—
239C2×D4—3261no—
240C2×D4—1634no—
241C2×D4—1631no—
242C2×D4C22×D16311yescharacteristic
243C22×C4—21no—
244C22×C4—3261no—
245C22×C4—3261no—
246C22×C4—3261no—
247C22×C4—3261no—
248C22×C4—6524no—
249C22×C4—1631no—
250C22×C4—1631no—
251C22×C4—1631no—
252M16—3262no—
253M16—3262no—
254C2×C8—1631no—
255C2×C8—1631no—
256C2×C8C22×D16311yescharacteristic
257C2×C8C22×D16311yescharacteristic
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—3261no—
261C4⋊C4—3261no—
262C4⋊C4—3261no—
263C4⋊C4—3261no—
264C4⋊C4—3261no—
265C4⋊C4—1631no—
266C4⋊C4—1631no—
267C4⋊C4—1631no—
268C4⋊C4C22×D16311yescharacteristic
269C4⋊C4C22×D16311yescharacteristic
270C4⋊C4C22×D16311yescharacteristic
271C22⋊C4—21no—
272C22⋊C4—3261no—
273C22⋊C4—3261no—
274C22⋊C4—3261no—
275C22⋊C4—3261no—
276C22⋊C4—3261no—
277C42—3261no—
278C42—3261no—
279C42—1631no—
280C42—1631no—
281C42—1631no—
282C42C22×D16311yescharacteristic
283C23—21no—
284C23—3264no—
285C23—3261no—
286C23—6522no—
287C23—6522no—
288C23—1631no—
289Q8—6522no—
290Q8—6522no—
291D4—21no—
292D4—3264no—
293D4—3261no—
294D4—1632no—
295D4Q8×D16312yes—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—3262no—
301C2×C4—3261no—
302C2×C4—3261no—
303C2×C4—3261no—
304C2×C4—6521no—
305C2×C4—6522no—
306C2×C4—6522no—
307C2×C4—6522no—
308C2×C4—6522no—
309C2×C4—1631no—
310C2×C4—1631no—
311C2×C4—1631no—
312C2×C4—1632no—
313C2×C4—1632no—
314C2×C4D4×D16311yescharacteristic
315C2×C4D4×D16311yescharacteristic
316C2×C4C23×D16311yescharacteristic
317C8—21no—
318C8—21no—
319C8—3261no—
320C8—3261no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—3261no—
325C22—3261no—
326C22—3262no—
327C22—6528no—
328C22—6521no—
329C22—6521no—
330C22—1632no—
331C22C2×D4×D16311yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—3261no—
338C4—6521no—
339C4—6521no—
340C4—6521no—
341C4—1632no—
342C4—1632no—
343C4C2×Q8×D16311yescharacteristic
344C4#41×D16311yescharacteristic
345C2—42no—
346C2—3262no—
347C2—6522no—
348C2C22⋊Q8×D16311yescharacteristic
349C2AGL(1,ℤ/8)×D16311yescharacteristic
350C2#2378246211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#478×C163C2
socleC22×C163C22×D4
prime radicalC2×D4×D163C22
hypercenter#478D163
hypocenterC163#3077
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #23782221C2semidirect
3. #23781966C2semidirect
4. #23781965C2semidirect
5. #23781962C2semidirect
6. #23781954C2semidirect
7. #23781953C2semidirect
8. #23781944C2semidirect
9. #23781943C2semidirect
10. #23781706C2semidirect
11. #23781704C2semidirect
12. #23781687C2semidirect
15. #478×C163C2semidirect
20. #19104734C22semidirect
21. #19104734C22semidirect
22. #19104727C22semidirect
37. #19104669C22semidirect
38. #19104668C22semidirect
39. #19104668C22semidirect
40. #19104667C22semidirect
41. #19104666C22semidirect
42. #19104666C22semidirect
43. C4⋊(C163⋊C8)C22semidirect
44. Dic163⋊C8C22semidirect
45. Dic163⋊C8C22semidirect
52. #125×C163C22semidirect
54. #107×C163C22semidirect
55. #106×C163C22semidirect
56. C4⋊C8×C163C22semidirect
65. D4×D163Q8semidirect
175. C163#3077semidirect
182. #478D163semidirect
199. #125C2×D163semidirect
218. #107C2×D163semidirect
222. #106C2×D163semidirect
226. C4⋊C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237822212
C4×D4×D1632
#237819662
#237819652
#237819622
#237819542
#237819532
#237819442
#237819432
#237817062
#237817042
#237816872
#237816682
#478×C1632
#3077163