Finite Group Databasebeta

← #35224786

Subgroups — #35224786

Order 20864.

394 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#23782276C211yesmaximal, characteristic, complemented
2#23782137C211yesmaximal, characteristic
3#23782005C211yesmaximal, characteristic, complemented
4#23782004C211yesmaximal, characteristic, complemented
5#23781966C211yesmaximal, characteristic, complemented
6#23781965C211yesmaximal, characteristic, complemented
7#23781867C211yesmaximal, characteristic, complemented
8#23781814C211yesmaximal, characteristic
9#23781814C211yesmaximal, characteristic
10#23781768C211yesmaximal, characteristic, complemented
11#23781766C211yesmaximal, characteristic, complemented
12SD16×Dic163C211yesmaximal, characteristic, complemented
13#23781750C211yesmaximal, characteristic, complemented
14#23781725C211yesmaximal, characteristic, complemented
15#461×C163C211yesmaximal, characteristic, complemented
16C2×D4×D163C2211yescharacteristic
17C2×#14579686C2211yescharacteristic
18C2×#14579686C2211yescharacteristic
19#19104730—21no—
20#19104730—21no—
21#19104727C2211yescharacteristic, complemented
22#19104725—21no—
23#19104725—21no—
24#19104711C2211yescharacteristic
25#19104711C2211yescharacteristic
26Q8×Dic163C2211yescharacteristic, complemented
27Q8×Dic163C2211yescharacteristic, complemented
28C4×D652C2211yescharacteristic
29D4×Dic163C2211yescharacteristic
30#19104697C2211yescharacteristic
31C4×C22⋊D163—21no—
32(C22⋊D163)⋊C4—21no—
33#19104693C2211yescharacteristic
34#19104692C2211yescharacteristic
35C2×(C2×D163)⋊C4—21no—
36C22⋊C4×D163—21no—
37#19104669C2211yescharacteristic, complemented
38#19104668C2211yescharacteristic, complemented
39#19104667C2211yescharacteristic, complemented
40#19104666C2211yescharacteristic, complemented
41C4⋊(C163⋊C8)C2211yescharacteristic, complemented
42#19104659C2211yescharacteristic, complemented
43#19104659C2211yescharacteristic, complemented
44#19104657C2211yescharacteristic
45#19104657C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104656C2211yescharacteristic
48(C2×D163)⋊C8C2211yescharacteristic
49(C2×D163)⋊C8C2211yescharacteristic
50#19104645C2211yescharacteristic, complemented
51C8×Dic163C2211yescharacteristic, complemented
52C2×SD16×C163C2211yescharacteristic
53C2×SD16×C163C2211yescharacteristic
54C4⋊Q8×C163C2211yescharacteristic, complemented
55C4×D4×C163C2211yescharacteristic
56C4⋊C8×C163C2211yescharacteristic, complemented
57#103×C163C2211yescharacteristic, complemented
58#103×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60C2×D652C2311yescharacteristic
61C2×C22⋊D163—21no—
62D4×D163—22no—
63D4×D163—22no—
64C22×Dic163—21no—
65C2×C4×D163—21no—
66C2×C4×D163—21no—
67C2×C4×D163C2311yescharacteristic
68#14579686—22no—
69#14579686—22no—
70C2×C163⋊C8C2311yescharacteristic
71C2×C163⋊C8C2311yescharacteristic
72C4⋊Dic163—21no—
73C4⋊Dic163—21no—
74C4⋊Dic163C2311yescharacteristic
75C4⋊Dic163C2311yescharacteristic
76C4⋊Dic163C2311yescharacteristic
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
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)⋊C4D411yescharacteristic
90(C2×D163)⋊C4D411yescharacteristic
91C4×Dic163—21no—
92C4×Dic163—21no—
93C4×Dic163—21no—
94C4×Dic163C2311yescharacteristic
95C2×Q8×C163C2311yescharacteristic
96C2×Q8×C163C2311yescharacteristic
97C2×D4×C163C2311yescharacteristic
98C22×C4×C163—21no—
99SD16×C163—22no—
100SD16×C163—22no—
101C2×C8×C163C2311yescharacteristic
102C2×C8×C163C2311yescharacteristic
103C4⋊C4×C163—21no—
104C4⋊C4×C163—21no—
105C4⋊C4×C163C2311yescharacteristic
106C22⋊C4×C163—21no—
107C42×C163C2311yescharacteristic
108C22×D163—21no—
109C22×D163—24no—
110C22×D163—81no—
111C22×D163—81no—
112C22×D163C2×D411yescharacteristic
113D652—21no—
114D652#4112yes—
115C22⋊D163—24no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163—81no—
120C2×Dic163C2×D411yescharacteristic
121C2×Dic163C2×D411yescharacteristic
122C2×Dic163C2×D411yescharacteristic
123C4×D163—21no—
124C4×D163—21no—
125C4×D163—81no—
126C4×D163—81no—
127C163⋊C8—21no—
128C163⋊C8—21no—
129C23×C163—21no—
130Q8×C163—21no—
131Q8×C163—21no—
132Q8×C163—21no—
133Q8×C163—21no—
134D4×C163—21no—
135D4×C163#4112yes—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163—81no—
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—22no—
151C2×D163—21no—
152C2×D163—21no—
153C2×D163—41no—
154C2×D163—41no—
155C2×D163—84no—
156C4×C163—21no—
157C4×C163—21no—
158C4×C163—41no—
159C4×C163—41no—
160C4×C163—41no—
161C4×C163C2×#4111yescharacteristic
162C4×C163C2×#4111yescharacteristic
163Dic163—21no—
164Dic163—21no—
165Dic163—21no—
166Dic163—41no—
167Dic163—41no—
168Dic163—41no—
169C2×C163—42no—
170C2×C163#52511yescharacteristic
171C2×C163#57711yescharacteristic
172C2×C163#57711yescharacteristic
173D163—41no—
174D163—42no—
175C163#312311yescharacteristic, complemented, Sylow
176#3123—1631nomaximal, Sylow
177#541—1631no—
178#518—1631no—
179#485—1631no—
180#485—1631no—
181#478—1631no—
182#478—1631no—
183#461—1631no—
184#461D16311yescharacteristic, complemented
185#450—1631no—
186#450—1631no—
187#440—1631no—
188#440—1631no—
189C4×SD16—1631no—
190C4×SD16—1631no—
191#433—1631no—
192C22×D4—1631no—
193C2×SD16—1631no—
194C2×SD16—1631no—
195C2×SD16C2×D16311yescharacteristic
196C2×SD16C2×D16311yescharacteristic
197C4⋊Q8C2×D16311yescharacteristic, complemented
198#126—3261no—
199#126—3261no—
200#125—3261no—
201#125—3261no—
202#125—1631no—
203C22⋊Q8—1631no—
204C22⋊Q8—1631no—
205C4×Q8—1631no—
206C4×Q8—1631no—
207C4×D4—3261no—
208C4×D4—3261no—
209C4×D4—1631no—
210C4×D4—1631no—
211C4×D4—1631no—
212C4×D4C2×D16311yescharacteristic
213#117—1631no—
214#117—1631no—
215C2×C22⋊C4—3261no—
216C2×C22⋊C4—3261no—
217#107—1631no—
218#107—1631no—
219#106—1631no—
220#106—1631no—
221C4⋊C8—1631no—
222C4⋊C8C2×D16311yescharacteristic, complemented
223#103—1631no—
224#103—1631no—
225#103C2×D16311yescharacteristic, complemented
226#103C2×D16311yescharacteristic, complemented
227#102—1631no—
228#102—1631no—
229#102—1631no—
230#102—1631no—
231C22⋊C8—1631no—
232C22⋊C8—1631no—
233#97—1631no—
234C4×C8—1631no—
235C24—3261no—
236C2×Q8C22×D16311yescharacteristic
237C2×Q8C22×D16311yescharacteristic
238C2×D4—3262no—
239C2×D4—3262no—
240C2×D4—3261no—
241C2×D4—1631no—
242C2×D4C22×D16311yescharacteristic
243C22×C4—21no—
244C22×C4—3261no—
245C22×C4—3261no—
246C22×C4—3261no—
247C22×C4—1631no—
248SD16—22no—
249SD16—22no—
250SD16—3262no—
251SD16—3262no—
252C2×C8—1631no—
253C2×C8—1631no—
254C2×C8C22×D16311yescharacteristic
255C2×C8C22×D16311yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—3261no—
259C4⋊C4—3261no—
260C4⋊C4—3261no—
261C4⋊C4—3261no—
262C4⋊C4—3261no—
263C4⋊C4—3261no—
264C4⋊C4—1631no—
265C4⋊C4—1631no—
266C4⋊C4—1631no—
267C4⋊C4—1631no—
268C4⋊C4—1631no—
269C4⋊C4C22×D16311yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—3261no—
272C22⋊C4—3261no—
273C22⋊C4—3261no—
274C22⋊C4—3261no—
275C22⋊C4—3261no—
276C22⋊C4—3261no—
277C22⋊C4—1631no—
278C22⋊C4—1631no—
279C42—3261no—
280C42—3261no—
281C42—3261no—
282C42—1631no—
283C42C22×D16311yescharacteristic
284C23—21no—
285C23—13041no—
286C23—13041no—
287C23—3264no—
288C23—3261no—
289C23—1631no—
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294D4—21no—
295D4—3264no—
296D4—3261no—
297D4—1632no—
298D4#1457969812yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—13041no—
304C2×C4—13041no—
305C2×C4—13041no—
306C2×C4—3261no—
307C2×C4—3261no—
308C2×C4—3261no—
309C2×C4—3261no—
310C2×C4—3261no—
311C2×C4—1631no—
312C2×C4—1631no—
313C2×C4—1631no—
314C2×C4D4×D16311yescharacteristic
315C2×C4D4×D16311yescharacteristic
316C2×C4C23×D16311yescharacteristic
317C8—21no—
318C8—21no—
319C8—3261no—
320C8—3261no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—13044no—
325C22—3261no—
326C22—3262no—
327C22—3261no—
328C22—3261no—
329C22—6521no—
330C22—6521no—
331C22C2×D4×D16311yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—3261no—
338C4—3261no—
339C4—3261no—
340C4—6521no—
341C4—6521no—
342C4—6521no—
343C4#41×D16311yescharacteristic
344C4C2×#1457969811yescharacteristic
345C2—42no—
346C2—6521no—
347C2—6522no—
348C2#2378217511yescharacteristic
349C2#2378247111yescharacteristic
350C2#2378247411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#461×C163C2
socleC22×C163C22×D4
prime radicalC2×D4×D163C22
hypercenter#461D163
hypocenterC163#3123
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #23782276C2semidirect
3. #23782005C2semidirect
4. #23782004C2semidirect
5. #23781966C2semidirect
6. #23781965C2semidirect
7. #23781867C2semidirect
10. #23781768C2semidirect
11. #23781766C2semidirect
12. SD16×Dic163C2semidirect
13. #23781750C2semidirect
14. #23781725C2semidirect
15. #461×C163C2semidirect
21. #19104727C22semidirect
26. Q8×Dic163C22semidirect
27. Q8×Dic163C22semidirect
37. #19104669C22semidirect
38. #19104668C22semidirect
39. #19104667C22semidirect
40. #19104666C22semidirect
41. C4⋊(C163⋊C8)C22semidirect
42. #19104659C22semidirect
43. #19104659C22semidirect
50. #19104645C22semidirect
51. C8×Dic163C22semidirect
54. C4⋊Q8×C163C22semidirect
56. C4⋊C8×C163C22semidirect
57. #103×C163C22semidirect
58. #103×C163C22semidirect
175. C163#3123semidirect
184. #461D163semidirect
197. C4⋊Q8C2×D163semidirect
222. C4⋊C8C2×D163semidirect
225. #103C2×D163semidirect
226. #103C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237822762
#237821372
#237820052
#237820042
#237819662
#237819652
#237818672
#237818142
#237817682
#237817662
SD16×Dic1632
#237817502
#237817252
#461×C1632
#3123163