Finite Group Databasebeta

← #35225138

Subgroups — #35225138

Order 20864.

364 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#23782219C211yesmaximal, characteristic, complemented
2#23782197C211yesmaximal, characteristic
3#23782072C211yesmaximal, characteristic, complemented
4#23782056C211yesmaximal, characteristic, complemented
5#23782032C211yesmaximal, characteristic, complemented
6#23782023C211yesmaximal, characteristic, complemented
7#23781924C211yesmaximal, characteristic, complemented
8#23781922C211yesmaximal, characteristic, complemented
9#23781910C211yesmaximal, characteristic
10#23781908C211yesmaximal, characteristic
11C4×Dic652C211yesmaximal, characteristic, complemented
12C4×#14579689C211yesmaximal, characteristic, complemented
13C4×#14579687C211yesmaximal, characteristic, complemented
14C4×#14579682C211yesmaximal, characteristic, complemented
15C4×SD16×C163C211yesmaximal, characteristic, complemented
16C2×Dic652C2211yescharacteristic
17C2×#14579689C2211yescharacteristic
18C2×#14579687C2211yescharacteristic
19C2×#14579682C2211yescharacteristic
20C2×#14579682C2211yescharacteristic
21#19104734C2211yescharacteristic
22#19104733C2211yescharacteristic
23#19104728—21no—
24#19104726C2211yescharacteristic
25#19104725—21no—
26#19104722C2211yescharacteristic
27#19104717—21no—
28#19104715C2211yescharacteristic
29#19104715C2211yescharacteristic
30#19104713—21no—
31#19104711C2211yescharacteristic
32#19104706C2211yescharacteristic
33C4×Dic326—21no—
34C4×Dic326C2211yescharacteristic
35C4×Dic326C2211yescharacteristic
36C4×C22⋊D163—21no—
37#19104694—21no—
38#19104692—21no—
39C42×D163C2211yescharacteristic
40#19104669C2211yescharacteristic
41#19104668C2211yescharacteristic
42#19104666C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104658C2211yescharacteristic
47#19104658C2211yescharacteristic
48#19104656C2211yescharacteristic
49#19104645C2211yescharacteristic
50#19104645C2211yescharacteristic
51C4×C163⋊C8C2211yescharacteristic
52C2×SD16×C163C2211yescharacteristic
53C4×Q8×C163C2211yescharacteristic
54C4×D4×C163C2211yescharacteristic
55#106×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58C4×C8×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326C2311yescharacteristic
61C2×Dic326C2311yescharacteristic
62C2×C22⋊D163—21no—
63C2×C4×D163D411yescharacteristic
64C2×C4×D163D411yescharacteristic
65C2×C4×D163C2311yescharacteristic
66Dic652—22no—
67#14579689—22no—
68#14579687—22no—
69#14579682—22no—
70#14579682—22no—
71C2×C163⋊C8C2311yescharacteristic
72C2×C163⋊C8C2311yescharacteristic
73C4⋊Dic163—21no—
74C4⋊Dic163—21no—
75C4⋊Dic163—21no—
76C4⋊Dic163C2311yescharacteristic
77C4⋊Dic163C2311yescharacteristic
78Dic163⋊C4—21no—
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85C22⋊Dic163—21no—
86C22⋊Dic163—21no—
87C22⋊Dic163—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×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×D163C2×D411yescharacteristic
109Dic326—21no—
110Dic326—21no—
111Dic326—21no—
112Dic326—21no—
113Dic326—42no—
114C22⋊D163—42no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163C2×D411yescharacteristic
120C2×Dic163C2×D411yescharacteristic
121C2×Dic163C2×D411yescharacteristic
122C4×D163—22no—
123C4×D163—21no—
124C4×D163—21no—
125C4×D163—41no—
126C163⋊C8—21no—
127C163⋊C8#4112yes—
128C23×C163—21no—
129Q8×C163—21no—
130Q8×C163—21no—
131D4×C163—21no—
132D4×C163—21no—
133C2×C4×C163—21no—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—42no—
137C2×C4×C163C2×D411yescharacteristic
138C2×C4×C163C2×D411yescharacteristic
139C2×C4×C163C2411yescharacteristic
140C8×C163—21no—
141C8×C163#4112yes—
142C22×C163—41no—
143C22×C163—41no—
144C22×C163—41no—
145C22×C163C22×D411yescharacteristic
146C2×D163—21no—
147C2×D163—21no—
148C2×D163—21no—
149C4×C163—21no—
150C4×C163—41no—
151C4×C163—41no—
152C4×C163—41no—
153C4×C163#13712yes—
154C4×C163C2×#4111yescharacteristic
155C4×C163C2×#4111yescharacteristic
156Dic163—21no—
157Dic163—21no—
158Dic163—21no—
159Dic163—41no—
160Dic163—41no—
161Dic163—41no—
162Dic163—41no—
163C2×C163—81no—
164C2×C163#53211yescharacteristic
165C2×C163C2×#13711yescharacteristic
166C2×C163#57511yescharacteristic
167D163—41no—
168C163#317211yescharacteristic, complemented, Sylow
169#3172—1631nomaximal, Sylow
170#533—1631no—
171#529—1631no—
172#501—1631no—
173#497—1631no—
174#491—1631no—
175#489—1631no—
176#470—1631no—
177#470—1631no—
178#468—1631no—
179#468—1631no—
180C4×Q16—1631no—
181C4×Q16—1631no—
182C4×SD16—1631no—
183C4×SD16D16311yescharacteristic, complemented
184C4×M16—1631no—
185C2×Q16—1631no—
186C2×Q16—1631no—
187C2×SD16—1631no—
188C2×SD16C2×D16311yescharacteristic
189C2×M16—1631no—
190C2×M16—1631no—
191C4⋊Q8—1631no—
192C4⋊Q8—1631no—
193#126—3261no—
194#125—3261no—
195#125—1631no—
196#124—1631no—
197#123—3261no—
198C22⋊Q8—3261no—
199C22⋊Q8—1631no—
200C22⋊Q8—1631no—
201C22⋊Q8—1631no—
202#121—1631no—
203C4×Q8—3261no—
204C4×Q8—1631no—
205C4×Q8—1631no—
206C4×Q8C2×D16311yescharacteristic
207C4×D4—3261no—
208C4×D4C2×D16311yescharacteristic
209#117—3261no—
210#117—3261no—
211C2×C42—1631no—
212#107—1631no—
213#107—1631no—
214#106—1631no—
215#106C2×D16311yescharacteristic
216#103—1631no—
217#103—1631no—
218#103—1631no—
219#103—1631no—
220#103—1631no—
221#103C2×D16311yescharacteristic
222#102—1631no—
223#102C2×D16311yescharacteristic
224#97—1631no—
225#97—1631no—
226C4×C8—1631no—
227C4×C8C2×D16311yescharacteristic
228C2×Q8—3261no—
229C2×Q8—1631no—
230C2×Q8—1631no—
231C2×Q8C22×D16311yescharacteristic
232C2×D4—3261no—
233C2×D4C22×D16311yescharacteristic
234C22×C4—21no—
235C22×C4—1631no—
236C22×C4—1631no—
237C22×C4—1631no—
238Q16—3262no—
239Q16—3262no—
240SD16—22no—
241SD16—3262no—
242M16—3262no—
243M16—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⋊C4C22×D16311yescharacteristic
262C4⋊C4C22×D16311yescharacteristic
263C22⋊C4—21no—
264C22⋊C4—3261no—
265C22⋊C4—3261no—
266C22⋊C4—3261no—
267C22⋊C4—3261no—
268C22⋊C4—3261no—
269C22⋊C4—3261no—
270C42—21no—
271C42—3261no—
272C42—3261no—
273C42—1631no—
274C42—1631no—
275C42—1631no—
276C42C22×D16311yescharacteristic
277C23—21no—
278C23—1631no—
279Q8—21no—
280Q8—21no—
281Q8—3261no—
282Q8—3261no—
283Q8—3261no—
284Q8—3261no—
285Q8—6522no—
286D4—21no—
287D4—21no—
288D4—6522no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—3261no—
294C2×C4—3261no—
295C2×C4—3262no—
296C2×C4—3261no—
297C2×C4—3261no—
298C2×C4—3261no—
299C2×C4—3261no—
300C2×C4—6521no—
301C2×C4—1631no—
302C2×C4—1631no—
303C2×C4—1631no—
304C2×C4D4×D16311yescharacteristic
305C2×C4D4×D16311yescharacteristic
306C2×C4C23×D16311yescharacteristic
307C8—21no—
308C8—3261no—
309C8—1632no—
310C8#1457969912yes—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—3261no—
315C22—3261no—
316C22—3261no—
317C22C2×D4×D16311yescharacteristic
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—3261no—
323C4—3261no—
324C4—3261no—
325C4—6521no—
326C4—6521no—
327C4—6521no—
328C4—6521no—
329C4#1910476412yes—
330C4#41×D16311yescharacteristic
331C4C2×#1457969911yescharacteristic
332C2—81no—
333C2—6521no—
334C2#2378221211yescharacteristic
335C2C2×#1910476411yescharacteristic
336C2#2378246211yescharacteristic
subgroupG/H
centerC2×C4D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC4×SD16×C163C2
socleC22×C163C22×D4
prime radical#19104706C22
hypercenterC4×SD16D163
hypocenterC163#3172
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #23782219C2semidirect
3. #23782072C2semidirect
4. #23782056C2semidirect
5. #23782032C2semidirect
6. #23782023C2semidirect
7. #23781924C2semidirect
8. #23781922C2semidirect
11. C4×Dic652C2semidirect
12. C4×#14579689C2semidirect
13. C4×#14579687C2semidirect
14. C4×#14579682C2semidirect
15. C4×SD16×C163C2semidirect
168. C163#3172semidirect
183. C4×SD16D163semidirect

Maximal subgroups

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

subgroupindex
#237822192
#237821972
#237820722
#237820562
#237820322
#237820232
#237819242
#237819222
#237819102
#237819082
C4×Dic6522
C4×#145796892
C4×#145796872
C4×#145796822
C4×SD16×C1632
#3172163