Finite Group Databasebeta

← #35224782

Subgroups — #35224782

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#23782275C211yesmaximal, characteristic, complemented
2#23782140C211yesmaximal, characteristic
3#23782007C211yesmaximal, characteristic, complemented
4#23782003C211yesmaximal, characteristic, complemented
5#23781967C211yesmaximal, characteristic, complemented
6#23781963C211yesmaximal, characteristic, complemented
7#23781868C211yesmaximal, characteristic, complemented
8#23781868C211yesmaximal, characteristic, complemented
9#23781816C211yesmaximal, characteristic
10#23781766C211yesmaximal, characteristic, complemented
11#23781762C211yesmaximal, characteristic, complemented
12#23781756C211yesmaximal, characteristic, complemented
13SD16×Dic163C211yesmaximal, characteristic, complemented
14#23781723C211yesmaximal, characteristic, complemented
15#450×C163C211yesmaximal, characteristic
16C2×#14579687C2211yescharacteristic
17C2×#14579687C2211yescharacteristic
18#19104733C2211yescharacteristic, complemented
19#19104728—21no—
20#19104728—21no—
21#19104726—21no—
22#19104726C2211yescharacteristic, complemented
23#19104725—21no—
24#19104714C2211yescharacteristic
25Q8×Dic163C2211yescharacteristic, complemented
26#19104700C2211yescharacteristic, complemented
27D4×Dic163—21no—
28D4×Dic163—21no—
29D4×Dic163C2211yescharacteristic
30D4×Dic163C2211yescharacteristic
31D4×Dic163C2211yescharacteristic
32D4×Dic163C2211yescharacteristic
33#19104694C2211yescharacteristic
34#19104694C2211yescharacteristic
35C2×C22⋊Dic163—21no—
36C2×C22⋊Dic163—21no—
37#19104669C2211yescharacteristic, complemented
38#19104668C2211yescharacteristic, complemented
39#19104667C2211yescharacteristic, complemented
40#19104666C2211yescharacteristic, complemented
41Dic163⋊C8C2211yescharacteristic, complemented
42Dic163⋊C8C2211yescharacteristic, complemented
43#19104660C2211yescharacteristic, complemented
44#19104660C2211yescharacteristic, complemented
45#19104659C2211yescharacteristic, complemented
46#19104659C2211yescharacteristic, complemented
47#19104656C2211yescharacteristic
48#19104656C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50#19104645C2211yescharacteristic, complemented
51C8×Dic163C2211yescharacteristic, complemented
52C22×D4×C163C2211yescharacteristic
53C2×SD16×C163C2211yescharacteristic
54C2×SD16×C163C2211yescharacteristic
55C22⋊Q8×C163C2211yescharacteristic
56#102×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic
59C2×Dic326C2311yescharacteristic
60C22×Dic163—21no—
61C22×Dic163—21no—
62C22×Dic163—21no—
63C22×Dic163—21no—
64#14579687—22no—
65#14579687—22no—
66C2×C163⋊C8C2311yescharacteristic
67C2×C163⋊C8C2311yescharacteristic
68C4⋊Dic163—21no—
69C4⋊Dic163—21no—
70C4⋊Dic163D411yescharacteristic
71C4⋊Dic163D411yescharacteristic
72C4⋊Dic163C2311yescharacteristic
73C4⋊Dic163C2311yescharacteristic
74C4⋊Dic163C2311yescharacteristic
75Dic163⋊C4—21no—
76Dic163⋊C4—21no—
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79Dic163⋊C4—21no—
80C22⋊Dic163—21no—
81C22⋊Dic163—21no—
82C22⋊Dic163—21no—
83C22⋊Dic163—21no—
84C22⋊Dic163—21no—
85C22⋊Dic163—21no—
86C22⋊Dic163D411yescharacteristic
87C22⋊Dic163D411yescharacteristic
88C4×Dic163—21no—
89C4×Dic163—21no—
90C4×Dic163—21no—
91C4×Dic163C2311yescharacteristic
92C4×Dic163C2311yescharacteristic
93C24×C163—21no—
94C2×Q8×C163C2311yescharacteristic
95C2×D4×C163—22no—
96C2×D4×C163—22no—
97C2×D4×C163—21no—
98C2×D4×C163C2311yescharacteristic
99C2×D4×C163C2311yescharacteristic
100C22×C4×C163C2311yescharacteristic
101SD16×C163—22no—
102SD16×C163—22no—
103C2×C8×C163C2311yescharacteristic
104C2×C8×C163C2311yescharacteristic
105C4⋊C4×C163—21no—
106C4⋊C4×C163C2311yescharacteristic
107C22⋊C4×C163—21no—
108Dic326—21no—
109Dic326—21no—
110C2×Dic163—21no—
111C2×Dic163—21no—
112C2×Dic163—21no—
113C2×Dic163—21no—
114C2×Dic163—81no—
115C2×Dic163—81no—
116C2×Dic163—81no—
117C2×Dic163—81no—
118C2×Dic163C2×D411yescharacteristic
119C2×Dic163C2×D411yescharacteristic
120C2×Dic163C2×D411yescharacteristic
121C2×Dic163C2×D411yescharacteristic
122C163⋊C8—21no—
123C163⋊C8—21no—
124C23×C163—24no—
125C23×C163—21no—
126C23×C163—21no—
127C23×C163—81no—
128C23×C163—81no—
129C23×C163C2×D411yescharacteristic
130Q8×C163—21no—
131Q8×C163—21no—
132D4×C163—24no—
133D4×C163—21no—
134D4×C163—21no—
135D4×C163#4112yes—
136D4×C163#4112yes—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163C2×D411yescharacteristic
142C2×C4×C163C2411yescharacteristic
143C8×C163—21no—
144C8×C163—21no—
145C22×C163—21no—
146C22×C163—22no—
147C22×C163—22no—
148C22×C163—21no—
149C22×C163—21no—
150C22×C163—41no—
151C22×C163—41no—
152C22×C163—41no—
153C22×C163—41no—
154C22×C163—84no—
155C22×C163C22×D411yescharacteristic
156C4×C163—21no—
157C4×C163—41no—
158C4×C163—41no—
159C4×C163C2×#4111yescharacteristic
160C4×C163C2×#4111yescharacteristic
161Dic163—21no—
162Dic163—21no—
163Dic163—21no—
164Dic163—21no—
165Dic163—41no—
166Dic163—41no—
167Dic163—41no—
168Dic163—41no—
169C2×C163—41no—
170C2×C163—42no—
171C2×C163—42no—
172C2×C163#52511yescharacteristic
173C2×C163#57711yescharacteristic
174C2×C163#57711yescharacteristic
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#461—1631no—
185#450—1631no—
186#450D16311yescharacteristic
187#440—1631no—
188#440—1631no—
189C4×SD16—1631no—
190C4×SD16—1631no—
191#433—1631no—
192C22×D4C2×D16311yescharacteristic
193C2×SD16—1631no—
194C2×SD16—1631no—
195C2×SD16C2×D16311yescharacteristic
196C2×SD16C2×D16311yescharacteristic
197C4⋊Q8—1631no—
198#126—3261no—
199#126—3261no—
200#125—3261no—
201#125—3261no—
202#125—1631no—
203C22⋊Q8—1631no—
204C22⋊Q8C2×D16311yescharacteristic
205C4×Q8—1631no—
206C4×Q8—1631no—
207C4×D4—3261no—
208C4×D4—3261no—
209C4×D4—1631no—
210C4×D4—1631no—
211C4×D4—1631no—
212C4×D4—1631no—
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⋊C8—1631no—
223#103—1631no—
224#103—1631no—
225#103—1631no—
226#103—1631no—
227#102—1631no—
228#102—1631no—
229#102C2×D16311yescharacteristic
230#102C2×D16311yescharacteristic
231C22⋊C8—1631no—
232C22⋊C8C2×D16311yescharacteristic
233#97—1631no—
234C4×C8—1631no—
235C24—21no—
236C2×Q8—1631no—
237C2×Q8C22×D16311yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4C22×D16311yescharacteristic
242C2×D4C22×D16311yescharacteristic
243C22×C4—3261no—
244C22×C4—3261no—
245C22×C4—3261no—
246C22×C4—3261no—
247C22×C4C22×D16311yescharacteristic
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—3261no—
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—
283C42—1631no—
284C23—24no—
285C23—21no—
286C23—21no—
287C23—81no—
288C23—81no—
289C23D4×D16311yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—3261no—
293Q8—3261no—
294D4—24no—
295D4—21no—
296D4—21no—
297D4#1457969812yes—
298D4#1457969812yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—13041no—
304C2×C4—13041no—
305C2×C4—13041no—
306C2×C4—13041no—
307C2×C4—3261no—
308C2×C4—3261no—
309C2×C4—3261no—
310C2×C4—3261no—
311C2×C4—1631no—
312C2×C4—1631no—
313C2×C4—1631no—
314C2×C4—1631no—
315C2×C4D4×D16311yescharacteristic
316C2×C4C23×D16311yescharacteristic
317C8—21no—
318C8—21no—
319C8—3261no—
320C8—3261no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—21no—
325C22—21no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—84no—
331C22C2×D4×D16311yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—3261no—
336C4—3261no—
337C4—3261no—
338C4—3261no—
339C4—6521no—
340C4—6521no—
341C4—6521no—
342C4—6521no—
343C4C2×#1457969811yescharacteristic
344C4C2×#1457969811yescharacteristic
345C2—41no—
346C2—42no—
347C2—42no—
348C2#2378217111yescharacteristic
349C2#2378247111yescharacteristic
350C2#2378247511yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#450×C163C2
socleC22×C163C22×D4
prime radicalC22×D4×C163C22
hypercenter#450D163
hypocenterC163#3123
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #23782275C2semidirect
3. #23782007C2semidirect
4. #23782003C2semidirect
5. #23781967C2semidirect
6. #23781963C2semidirect
7. #23781868C2semidirect
8. #23781868C2semidirect
10. #23781766C2semidirect
11. #23781762C2semidirect
12. #23781756C2semidirect
13. SD16×Dic163C2semidirect
14. #23781723C2semidirect
18. #19104733C22semidirect
22. #19104726C22semidirect
25. Q8×Dic163C22semidirect
26. #19104700C22semidirect
37. #19104669C22semidirect
38. #19104668C22semidirect
39. #19104667C22semidirect
40. #19104666C22semidirect
41. Dic163⋊C8C22semidirect
42. Dic163⋊C8C22semidirect
43. #19104660C22semidirect
44. #19104660C22semidirect
45. #19104659C22semidirect
46. #19104659C22semidirect
50. #19104645C22semidirect
51. C8×Dic163C22semidirect
175. C163#3123semidirect

Maximal subgroups

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

subgroupindex
#237822752
#237821402
#237820072
#237820032
#237819672
#237819632
#237818682
#237818162
#237817662
#237817622
#237817562
SD16×Dic1632
#237817232
#450×C1632
#3123163