Finite Group Databasebeta

← #35223352

Subgroups — #35223352

Order 20864.

484 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104767C211yesmaximal, characteristic
2C4×#14579700C211yesmaximal, characteristic
3C4×Q8×D163C211yesmaximal, characteristic, complemented
4#23781770C211yesmaximal, characteristic, complemented
5#23781767C211yesmaximal, characteristic, complemented
6#23781763C211yesmaximal, characteristic, complemented
7Q16×Dic163C211yesmaximal, characteristic, complemented
8#23781757C211yesmaximal, characteristic, complemented
9#23781752C211yesmaximal, characteristic, complemented
10#23781750C211yesmaximal, characteristic, complemented
11#23781704C211yesmaximal, characteristic, complemented
12#23781672C211yesmaximal, characteristic, complemented
13#23781665C211yesmaximal, characteristic
14#23781617C211yesmaximal, characteristic, complemented
15#441×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×Q8×D163C2211yescharacteristic
18#19104767C418yescomplemented
19C2×#14579689C2211yescharacteristic
20C2×#14579688C2211yescharacteristic
21C2×#14579686C2211yescharacteristic
22C2×#14579682C2211yescharacteristic
23C2×#14579682C2211yescharacteristic
24Q8×Dic163C2211yescharacteristic
25Q8×Dic163C2211yescharacteristic
26C4×Dic326—21no—
27C4×Dic326C2211yescharacteristic
28#19104700—21no—
29#19104700C2211yescharacteristic
30C4×D652—21no—
31C4×D652C2211yescharacteristic
32#19104697—21no—
33#19104697C2211yescharacteristic
34#19104693—21no—
35#19104693C2211yescharacteristic
36C4⋊C4×D163—21no—
37C4⋊C4×D163C2211yescharacteristic
38C42×D163—21no—
39C42×D163—21no—
40C42×D163C2211yescharacteristic
41#19104669C2211yescharacteristic
42#19104668C2211yescharacteristic
43#19104666C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104659C2211yescharacteristic
47#19104658C2211yescharacteristic
48#19104657C2211yescharacteristic
49#19104655C2211yescharacteristic
50#19104646C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C8×Dic163C2211yescharacteristic
53C2×Q16×C163C2211yescharacteristic
54C4×Q8×C163C2211yescharacteristic
55C4×Q8×C163C2211yescharacteristic
56#106×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#103×C163C2211yescharacteristic
59#97×C163C2211yescharacteristic
60#14579700—22no—
61#14579700C2×C414yes—
62C2×Dic326—21no—
63C2×Dic326C2311yescharacteristic
64Q8×D163—22no—
65Q8×D163C2×C414yescomplemented
66C2×D652—21no—
67C2×D652C2311yescharacteristic
68C2×C4×D163—21no—
69C2×C4×D163—21no—
70C2×C4×D163—21no—
71C2×C4×D163—21no—
72C2×C4×D163—21no—
73C2×C4×D163D411yescharacteristic
74C2×C4×D163D411yescharacteristic
75C2×C4×D163C2311yescharacteristic
76#14579689C2×C414yescomplemented
77#14579688C2×C414yescomplemented
78#14579686C2×C414yescomplemented
79#14579682—22no—
80#14579682C2×C414yescomplemented
81C2×C163⋊C8C2311yescharacteristic
82C2×C163⋊C8C2311yescharacteristic
83C4⋊Dic163—21no—
84C4⋊Dic163—21no—
85C4⋊Dic163C2311yescharacteristic
86C4⋊Dic163C2311yescharacteristic
87Dic163⋊C4—21no—
88Dic163⋊C4—21no—
89Dic163⋊C4—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93C4×Dic163—21no—
94C4×Dic163—21no—
95C4×Dic163—21no—
96C4×Dic163—21no—
97C4×Dic163—21no—
98C4×Dic163D411yescharacteristic
99C4×Dic163D411yescharacteristic
100C4×Dic163C2311yescharacteristic
101C2×Q8×C163C2311yescharacteristic
102C2×Q8×C163C2311yescharacteristic
103Q16×C163C2×C414yescomplemented
104C2×C8×C163C2311yescharacteristic
105C2×C8×C163C2311yescharacteristic
106C4⋊C4×C163—21no—
107C4⋊C4×C163—21no—
108C4⋊C4×C163C2311yescharacteristic
109C4⋊C4×C163C2311yescharacteristic
110C42×C163—21no—
111C42×C163—21no—
112C42×C163C2311yescharacteristic
113C22×D163—21no—
114C22×D163C2×D411yescharacteristic
115Dic326—21no—
116Dic326—24no—
117Dic326C22×C412yes—
118D652—24no—
119D652—21no—
120D652C22×C412yes—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163C2×D411yescharacteristic
125C2×Dic163C2×D411yescharacteristic
126C2×Dic163C2×D411yescharacteristic
127C4×D163—24no—
128C4×D163—24no—
129C4×D163—22no—
130C4×D163—41no—
131C4×D163—42no—
132C4×D163—42no—
133C4×D163—42no—
134C4×D163C22×C412yes—
135C4×D163#4112yes—
136C163⋊C8—21no—
137C163⋊C8C22×C412yes—
138Q8×C163—21no—
139Q8×C163—21no—
140Q8×C163C22×C412yes—
141Q8×C163C22×C412yes—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163C2×D411yescharacteristic
147C2×C4×C163C2×D411yescharacteristic
148C2×C4×C163C2411yescharacteristic
149C8×C163—21no—
150C8×C163C22×C412yes—
151C22×C163C22×D411yescharacteristic
152C2×D163—21no—
153C2×D163—21no—
154C2×D163—22no—
155C2×D163—41no—
156C2×D163—41no—
157C2×D163C4×D412yes—
158C4×C163—22no—
159C4×C163—22no—
160C4×C163—21no—
161C4×C163—21no—
162C4×C163—41no—
163C4×C163—41no—
164C4×C163C23×C411yescharacteristic
165C4×C163C2×#4111yescharacteristic
166Dic163—22no—
167Dic163—21no—
168Dic163—41no—
169Dic163—41no—
170Dic163C4×D412yes—
171Dic163#13712yescomplemented
172C2×C163C2×C4×D411yescharacteristic
173C2×C163C2×#13711yescharacteristic
174C2×C163#57511yescharacteristic
175D163—22no—
176D163—42no—
177C163C4×#13711yescharacteristic, complemented, Sylow
178C4×#137—1631nomaximal, Sylow
179C2×#137—1631no—
180C4×#41—1631no—
181C2×C4×Q8—1631no—
182#441—1631no—
183#441D16311yescharacteristic, complemented
184#440—1631no—
185#440—1631no—
186C4×Q16—1631no—
187C4×Q16—1631no—
188C4×SD16—1631no—
189C4×SD16—1631no—
190#428—1631no—
191#419—1631no—
192#417—1631no—
193C4×M16—1631no—
194C2×#41—1631no—
195C22×Q8—1631no—
196#137—1638no—
197C2×Q16—1631no—
198C2×Q16C2×D16311yescharacteristic
199C2×SD16—1631no—
200C2×SD16—1631no—
201C2×M16—1631no—
202C2×M16—1631no—
203C4×Q8—3261no—
204C4×Q8—3261no—
205C4×Q8—1631no—
206C4×Q8—1631no—
207C4×Q8—1631no—
208C4×Q8—1631no—
209C4×Q8C2×D16311yescharacteristic
210C4×Q8C2×D16311yescharacteristic
211C4×D4—3261no—
212C4×D4—3261no—
213C4×D4—1631no—
214C4×D4—1631no—
215#117—3261no—
216#117—1631no—
217C2×C4⋊C4—3261no—
218C2×C4⋊C4—1631no—
219C2×C42—3261no—
220C2×C42—3261no—
221C2×C42—1631no—
222#107—1631no—
223#107—1631no—
224#106—1631no—
225#106C2×D16311yescharacteristic
226#103—1631no—
227#103—1631no—
228#103—1631no—
229#103—1631no—
230#103C2×D16311yescharacteristic
231#103C2×D16311yescharacteristic
232#102—1631no—
233#102—1631no—
234#97—1631no—
235#97C2×D16311yescharacteristic
236C4×C8—1631no—
237C4×C8—1631no—
238#41—3262no—
239#41—1634no—
240C2×Q8—3261no—
241C2×Q8—3262no—
242C2×Q8—1634no—
243C2×Q8—1631no—
244C2×Q8C22×D16311yescharacteristic
245C2×Q8C22×D16311yescharacteristic
246C2×D4—3261no—
247C2×D4—1631no—
248C22×C4—3261no—
249C22×C4—3261no—
250C22×C4—3261no—
251C22×C4—3261no—
252C22×C4—3261no—
253C22×C4—1631no—
254C22×C4—1631no—
255C22×C4—1631no—
256Q16—1634no—
257Q16C4×D16314yescomplemented
258SD16—1634no—
259SD16—1634no—
260M16—3262no—
261M16—1634no—
262C2×C8—1631no—
263C2×C8—1631no—
264C2×C8C22×D16311yescharacteristic
265C2×C8C22×D16311yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—3261no—
269C4⋊C4—3261no—
270C4⋊C4—3261no—
271C4⋊C4—3261no—
272C4⋊C4—3261no—
273C4⋊C4—1631no—
274C4⋊C4—1631no—
275C4⋊C4C22×D16311yescharacteristic
276C4⋊C4C22×D16311yescharacteristic
277C22⋊C4—3261no—
278C22⋊C4—3261no—
279C22⋊C4—3261no—
280C42—21no—
281C42—21no—
282C42—3261no—
283C42—3261no—
284C42—3261no—
285C42—3261no—
286C42—3261no—
287C42—1631no—
288C42—1631no—
289C42—1631no—
290C42C22×D16311yescharacteristic
291C23—3261no—
292C23—1631no—
293Q8—21no—
294Q8—21no—
295Q8—3264no—
296Q8—3261no—
297Q8—1632no—
298Q8C2×C4×D16312yes—
299Q8C2×C4×D16312yes—
300D4—3264no—
301D4—3261no—
302D4—1632no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—3264no—
308C2×C4—3264no—
309C2×C4—3262no—
310C2×C4—3261no—
311C2×C4—3261no—
312C2×C4—3261no—
313C2×C4—6521no—
314C2×C4—6522no—
315C2×C4—6522no—
316C2×C4—6522no—
317C2×C4—1631no—
318C2×C4—1631no—
319C2×C4—1631no—
320C2×C4—1632no—
321C2×C4—1632no—
322C2×C4D4×D16311yescharacteristic
323C2×C4D4×D16311yescharacteristic
324C2×C4C23×D16311yescharacteristic
325C8—21no—
326C8—3261no—
327C8—1632no—
328C8C2×C4×D16312yes—
329C22—3261no—
330C22—3261no—
331C22—3262no—
332C22—6521no—
333C22—6521no—
334C22—1632no—
335C22C2×D4×D16311yescharacteristic
336C4—21no—
337C4—21no—
338C4—22no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—3261no—
343C4—3262no—
344C4—6521no—
345C4—6521no—
346C4—1632no—
347C4—1632no—
348C4C22×C4×D16311yescharacteristic
349C4#41×D16311yescharacteristic
350C2—3262no—
351C2—6522no—
352C2C4×D4×D16311yescharacteristic
353C2#137×D16311yescharacteristic
354C2#2378246411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutatorC4×C163C23×C4
FrattiniC2×C4C23×D163
Fitting#441×C163C2
socleC22×C163C22×D4
prime radicalC2×#14579700C22
cosocleC2×C4×C163C24
hypercenter#441D163
hypocenterC163C4×#137
LCSG ▷ C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
3. C4×Q8×D163C2semidirect
4. #23781770C2semidirect
5. #23781767C2semidirect
6. #23781763C2semidirect
7. Q16×Dic163C2semidirect
8. #23781757C2semidirect
9. #23781752C2semidirect
10. #23781750C2semidirect
11. #23781704C2semidirect
12. #23781672C2semidirect
14. #23781617C2semidirect
15. #441×C163C2semidirect
18. #19104767C4semidirect
65. Q8×D163C2×C4semidirect
76. #14579689C2×C4semidirect
77. #14579688C2×C4semidirect
78. #14579686C2×C4semidirect
80. #14579682C2×C4semidirect
103. Q16×C163C2×C4semidirect
171. Dic163#137semidirect
177. C163C4×#137semidirect
183. #441D163semidirect
257. Q16C4×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047672
C4×#145797002
C4×Q8×D1632
#237817702
#237817672
#237817632
Q16×Dic1632
#237817572
#237817522
#237817502
#237817042
#237816722
#237816652
#237816172
#441×C1632
C4×#137163