Finite Group Databasebeta

← #35224751

Subgroups — #35224751

Order 20864.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782275C211yesmaximal, characteristic, complemented
2#23782148C211yesmaximal, characteristic
3#23782001C211yesmaximal, characteristic, complemented
4#23781993C211yesmaximal, characteristic, complemented
5#23781974C211yesmaximal, characteristic, complemented
6#23781963C211yesmaximal, characteristic, complemented
7#23781873C211yesmaximal, characteristic, complemented
8#23781868C211yesmaximal, characteristic, complemented
9#23781827C211yesmaximal, characteristic
10#23781769C211yesmaximal, characteristic, complemented
11#23781766C211yesmaximal, characteristic, complemented
12Q16×Dic163C211yesmaximal, characteristic, complemented
13#23781756C211yesmaximal, characteristic, complemented
14#23781723C211yesmaximal, characteristic, complemented
15#452×C163C211yesmaximal, characteristic
16C2×#14579689C2211yescharacteristic
17C2×#14579687C2211yescharacteristic
18#19104733C2211yescharacteristic
19#19104728—21no—
20#19104728—21no—
21#19104726—21no—
22#19104726C2211yescharacteristic
23#19104725—21no—
24#19104714C2211yescharacteristic
25Q8×Dic163C2211yescharacteristic
26Q8×Dic163C2211yescharacteristic
27Q8×Dic163C2211yescharacteristic
28#19104700C2211yescharacteristic
29D4×Dic163—21no—
30D4×Dic163—21no—
31D4×Dic163C2211yescharacteristic
32D4×Dic163C2211yescharacteristic
33#19104694—21no—
34#19104694C2211yescharacteristic
35C2×C4⋊Dic163—21no—
36C2×C4⋊Dic163C2211yescharacteristic
37#19104669C2211yescharacteristic
38#19104669C2211yescharacteristic
39#19104666C2211yescharacteristic
40#19104666C2211yescharacteristic
41Dic163⋊C8C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104659C2211yescharacteristic
47#19104659C2211yescharacteristic
48#19104656C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50#19104645C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C2×#41×C163C2211yescharacteristic
53C2×Q16×C163C2211yescharacteristic
54C2×SD16×C163C2211yescharacteristic
55C22⋊Q8×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic
59C2×Dic326C2311yescharacteristic
60C22×Dic163—21no—
61C22×Dic163—21no—
62C22×Dic163—21no—
63#14579689—22no—
64#14579687—22no—
65C2×C163⋊C8C2311yescharacteristic
66C2×C163⋊C8C2311yescharacteristic
67C4⋊Dic163—21no—
68C4⋊Dic163—21no—
69C4⋊Dic163—21no—
70C4⋊Dic163—21no—
71C4⋊Dic163—21no—
72C4⋊Dic163D411yescharacteristic
73C4⋊Dic163D411yescharacteristic
74C4⋊Dic163C2311yescharacteristic
75C4⋊Dic163C2311yescharacteristic
76C4⋊Dic163C2311yescharacteristic
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82C22⋊Dic163—21no—
83C22⋊Dic163—21no—
84C22⋊Dic163—21no—
85C22⋊Dic163D411yescharacteristic
86C22⋊Dic163D411yescharacteristic
87C4×Dic163—21no—
88C4×Dic163—21no—
89C4×Dic163—21no—
90C4×Dic163—21no—
91C4×Dic163C2311yescharacteristic
92C4×Dic163C2311yescharacteristic
93#41×C163—22no—
94#41×C163—22no—
95C2×Q8×C163C2311yescharacteristic
96C2×Q8×C163C2311yescharacteristic
97C2×D4×C163—21no—
98C2×D4×C163C2311yescharacteristic
99C22×C4×C163—21no—
100C22×C4×C163C2311yescharacteristic
101Q16×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×Dic163C2×D411yescharacteristic
118C2×Dic163C2×D411yescharacteristic
119C2×Dic163C2×D411yescharacteristic
120C2×Dic163C2×D411yescharacteristic
121C163⋊C8—21no—
122C163⋊C8—21no—
123C23×C163—21no—
124C23×C163C2×D411yescharacteristic
125Q8×C163—21no—
126Q8×C163—21no—
127Q8×C163—21no—
128Q8×C163#4112yes—
129D4×C163—24no—
130D4×C163—21no—
131D4×C163#4112yes—
132C2×C4×C163—24no—
133C2×C4×C163—21no—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163C2×D411yescharacteristic
139C2×C4×C163C2411yescharacteristic
140C8×C163—21no—
141C8×C163—21no—
142C22×C163—21no—
143C22×C163—22no—
144C22×C163—21no—
145C22×C163—21no—
146C22×C163—41no—
147C22×C163—41no—
148C22×C163C22×D411yescharacteristic
149C4×C163—21no—
150C4×C163—22no—
151C4×C163—41no—
152C4×C163—41no—
153C4×C163C2×#4111yescharacteristic
154C4×C163C2×#4111yescharacteristic
155Dic163—21no—
156Dic163—21no—
157Dic163—21no—
158Dic163—21no—
159Dic163—41no—
160Dic163—41no—
161Dic163—41no—
162Dic163—41no—
163C2×C163—41no—
164C2×C163—42no—
165C2×C163#52511yescharacteristic
166C2×C163#57711yescharacteristic
167C2×C163#57811yescharacteristic
168C163#312011yescharacteristic, complemented, Sylow
169#3120—1631nomaximal, Sylow
170#541—1631no—
171#520—1631no—
172#484—1631no—
173#483—1631no—
174#479—1631no—
175#478—1631no—
176#462—1631no—
177#461—1631no—
178#452—1631no—
179#452D16311yescharacteristic
180#441—1631no—
181#440—1631no—
182C4×Q16—1631no—
183C4×SD16—1631no—
184#433—1631no—
185C2×#41C2×D16311yescharacteristic
186C2×Q16—1631no—
187C2×Q16C2×D16311yescharacteristic
188C2×SD16—1631no—
189C2×SD16C2×D16311yescharacteristic
190C4⋊Q8—1631no—
191#126—3261no—
192#126—3261no—
193#125—3261no—
194#125—3261no—
195#125—1631no—
196C22⋊Q8—1631no—
197C22⋊Q8C2×D16311yescharacteristic
198C4×Q8—1631no—
199C4×Q8—1631no—
200C4×Q8—1631no—
201C4×Q8—1631no—
202C4×D4—3261no—
203C4×D4—3261no—
204C4×D4—1631no—
205C4×D4—1631no—
206#117—3261no—
207#117—1631no—
208C2×C4⋊C4—3261no—
209C2×C4⋊C4—1631no—
210#107—1631no—
211#107—1631no—
212#106—1631no—
213#106—1631no—
214C4⋊C8—1631no—
215C4⋊C8—1631no—
216#103—1631no—
217#103—1631no—
218#103—1631no—
219#103—1631no—
220#103—1631no—
221#103C2×D16311yescharacteristic
222#102—1631no—
223#102C2×D16311yescharacteristic
224C22⋊C8—1631no—
225C22⋊C8C2×D16311yescharacteristic
226#97—1631no—
227C4×C8—1631no—
228#41—22no—
229#41—22no—
230C2×Q8—1631no—
231C2×Q8C22×D16311yescharacteristic
232C2×Q8C22×D16311yescharacteristic
233C2×D4—21no—
234C2×D4C22×D16311yescharacteristic
235C22×C4—21no—
236C22×C4—3261no—
237C22×C4—3261no—
238C22×C4—3261no—
239C22×C4C22×D16311yescharacteristic
240Q16—22no—
241Q16—3262no—
242SD16—22no—
243SD16—3262no—
244C2×C8—1631no—
245C2×C8—1631no—
246C2×C8C22×D16311yescharacteristic
247C2×C8C22×D16311yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—3261no—
250C4⋊C4—3261no—
251C4⋊C4—3261no—
252C4⋊C4—3261no—
253C4⋊C4—3261no—
254C4⋊C4—3261no—
255C4⋊C4—3261no—
256C4⋊C4—3261no—
257C4⋊C4—3261no—
258C4⋊C4—3261no—
259C4⋊C4—1631no—
260C4⋊C4—1631no—
261C4⋊C4—1631no—
262C4⋊C4—1631no—
263C4⋊C4—1631no—
264C4⋊C4C22×D16311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—3261no—
267C22⋊C4—3261no—
268C22⋊C4—3261no—
269C22⋊C4—1631no—
270C22⋊C4—1631no—
271C42—3261no—
272C42—3261no—
273C42—3261no—
274C42—3261no—
275C42—1631no—
276C42—1631no—
277C23—21no—
278C23D4×D16311yescharacteristic
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—3261no—
283Q8—3261no—
284Q8#1457969812yes—
285D4—24no—
286D4—21no—
287D4#1457969812yes—
288C2×C4—24no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—13041no—
295C2×C4—13041no—
296C2×C4—13041no—
297C2×C4—3261no—
298C2×C4—3261no—
299C2×C4—3261no—
300C2×C4—3261no—
301C2×C4—1631no—
302C2×C4—1631no—
303C2×C4—1631no—
304C2×C4—1631no—
305C2×C4D4×D16311yescharacteristic
306C2×C4C23×D16311yescharacteristic
307C8—21no—
308C8—21no—
309C8—3261no—
310C8—3261no—
311C22—21no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—41no—
316C22—41no—
317C22C2×D4×D16311yescharacteristic
318C4—21no—
319C4—22no—
320C4—41no—
321C4—41no—
322C4—3261no—
323C4—3261no—
324C4—3261no—
325C4—3261no—
326C4—6521no—
327C4—6521no—
328C4—6521no—
329C4—6521no—
330C4C2×#1457969811yescharacteristic
331C4C2×#1457969811yescharacteristic
332C2—41no—
333C2—42no—
334C2#2378217111yescharacteristic
335C2#2378247411yescharacteristic
336C2#2378247911yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#452×C163C2
socleC22×C163C22×D4
prime radicalC2×#41×C163C22
hypercenter#452D163
hypocenterC163#3120
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #23782275C2semidirect
3. #23782001C2semidirect
4. #23781993C2semidirect
5. #23781974C2semidirect
6. #23781963C2semidirect
7. #23781873C2semidirect
8. #23781868C2semidirect
10. #23781769C2semidirect
11. #23781766C2semidirect
12. Q16×Dic163C2semidirect
13. #23781756C2semidirect
14. #23781723C2semidirect
168. C163#3120semidirect

Maximal subgroups

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

subgroupindex
#237822752
#237821482
#237820012
#237819932
#237819742
#237819632
#237818732
#237818682
#237818272
#237817692
#237817662
Q16×Dic1632
#237817562
#237817232
#452×C1632
#3120163