Finite Group Databasebeta

← #35223378

Subgroups — #35223378

Order 20864.

470 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
1C2×#19104767C211yesmaximal, characteristic
2#23782147C211yesmaximal, characteristic
3#23782141C211yesmaximal, characteristic, complemented
4#23781769C211yesmaximal, characteristic, complemented
5#23781768C211yesmaximal, characteristic, complemented
6#23781767C211yesmaximal, characteristic, complemented
7Q16×Dic163C211yesmaximal, characteristic, complemented
8C4×#14579689C211yesmaximal, characteristic, complemented
9C4×#14579686C211yesmaximal, characteristic, complemented
10#23781752C211yesmaximal, characteristic, complemented
11#23781704C211yesmaximal, characteristic, complemented
12#23781660C211yesmaximal, characteristic
13#103×D163C211yesmaximal, characteristic, complemented
14#23781622C211yesmaximal, characteristic, complemented
15#441×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×Q8×D163C2211yescharacteristic
18#19104767C418yescomplemented
19C2×#14579689C2211yescharacteristic
20C2×#14579688C2211yescharacteristic
21C2×#14579686C2211yescharacteristic
22C2×#14579682C2211yescharacteristic
23C2×C8×D163C2211yescharacteristic
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
36#19104692—21no—
37#19104692—21no—
38#19104692C2211yescharacteristic
39C4⋊C4×D163—21no—
40C4⋊C4×D163C2211yescharacteristic
41#19104669C2211yescharacteristic
42#19104668C2211yescharacteristic
43#19104666C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104659C2211yescharacteristic
47#19104658C2211yescharacteristic
48#19104657C2211yescharacteristic
49#19104655C2211yescharacteristic
50#19104645C2211yescharacteristic
51C4×C163⋊C8C2211yescharacteristic
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×D163C2311yescharacteristic
74#14579689C2×C414yescomplemented
75#14579688C2×C414yescomplemented
76#14579686C2×C414yescomplemented
77#14579682C2×C414yescomplemented
78C2×C163⋊C8C2311yescharacteristic
79C2×C163⋊C8C2311yescharacteristic
80C8×D163—22no—
81C4⋊Dic163—21no—
82C4⋊Dic163—21no—
83C4⋊Dic163C2311yescharacteristic
84C4⋊Dic163C2311yescharacteristic
85Dic163⋊C4—21no—
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88Dic163⋊C4D411yescharacteristic
89Dic163⋊C4D411yescharacteristic
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4D411yescharacteristic
94(C2×D163)⋊C4D411yescharacteristic
95C4×Dic163—21no—
96C4×Dic163—21no—
97C4×Dic163—21no—
98C4×Dic163—21no—
99C4×Dic163—21no—
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—81no—
130C4×D163—81no—
131C4×D163—81no—
132C4×D163C22×C412yes—
133C4×D163#4112yes—
134C163⋊C8—21no—
135C163⋊C8C22×C412yes—
136Q8×C163—21no—
137Q8×C163—21no—
138Q8×C163C22×C412yes—
139Q8×C163C22×C412yes—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163C2×D411yescharacteristic
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2411yescharacteristic
147C8×C163—21no—
148C8×C163C22×C412yes—
149C22×C163C22×D411yescharacteristic
150C2×D163—21no—
151C2×D163—21no—
152C2×D163—22no—
153C2×D163—41no—
154C2×D163—41no—
155C2×D163C4×D412yes—
156C4×C163—22no—
157C4×C163—22no—
158C4×C163—21no—
159C4×C163—21no—
160C4×C163—41no—
161C4×C163—41no—
162C4×C163C23×C411yescharacteristic
163C4×C163C2×#4111yescharacteristic
164Dic163—22no—
165Dic163—21no—
166Dic163—21no—
167Dic163—41no—
168Dic163—41no—
169Dic163C4×D412yes—
170C2×C163C2×C4×D411yescharacteristic
171C2×C163#57711yescharacteristic
172C2×C163#57811yescharacteristic
173D163—22no—
174D163—42no—
175C163#294711yescharacteristic, complemented, Sylow
176#2947—1631nomaximal, Sylow
177C2×#137—1631no—
178#520—1631no—
179#519—1631no—
180#441—1631no—
181#441D16311yescharacteristic, complemented
182#440—1631no—
183#440—1631no—
184C4×Q16—1631no—
185C4×Q16—1631no—
186C4×SD16—1631no—
187C4×SD16—1631no—
188#428—1631no—
189#416—1631no—
190C2×#103—1631no—
191#405—1631no—
192C2×#41—1631no—
193C22×Q8—1631no—
194#137—1638no—
195C2×Q16—1631no—
196C2×Q16C2×D16311yescharacteristic
197C2×SD16—1631no—
198C2×SD16—1631no—
199C2×M16—1631no—
200C22×C8—1631no—
201C4×Q8—3261no—
202C4×Q8—3261no—
203C4×Q8—1631no—
204C4×Q8—1631no—
205C4×Q8—1631no—
206C4×Q8—1631no—
207C4×Q8C2×D16311yescharacteristic
208C4×Q8C2×D16311yescharacteristic
209C4×D4—3261no—
210C4×D4—3261no—
211C4×D4—1631no—
212C4×D4—1631no—
213#117—3261no—
214#117—3261no—
215#117—3261no—
216#117—1631no—
217#117—1631no—
218C2×C4⋊C4—3261no—
219C2×C4⋊C4—1631no—
220#107—1631no—
221#107—1631no—
222#106—1631no—
223#106C2×D16311yescharacteristic
224#103—1631no—
225#103—1631no—
226#103—1631no—
227#103—1631no—
228#103C2×D16311yescharacteristic
229#103C2×D16311yescharacteristic
230#102—1631no—
231#102—1631no—
232#97—1631no—
233#97C2×D16311yescharacteristic
234C4×C8—1631no—
235C4×C8—1631no—
236#41—3262no—
237#41—1634no—
238C2×Q8—3261no—
239C2×Q8—3262no—
240C2×Q8—1634no—
241C2×Q8—1631no—
242C2×Q8C22×D16311yescharacteristic
243C2×Q8C22×D16311yescharacteristic
244C2×D4—3261no—
245C2×D4—1631no—
246C22×C4—3261no—
247C22×C4—3261no—
248C22×C4—3261no—
249C22×C4—3261no—
250C22×C4—3261no—
251C22×C4—1631no—
252Q16—1634no—
253Q16C4×D16314yescomplemented
254SD16—1634no—
255SD16—1634no—
256M16—1634no—
257C2×C8—3262no—
258C2×C8—1631no—
259C2×C8—1631no—
260C2×C8C22×D16311yescharacteristic
261C2×C8C22×D16311yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—3261no—
265C4⋊C4—3261no—
266C4⋊C4—3261no—
267C4⋊C4—3261no—
268C4⋊C4—3261no—
269C4⋊C4—1631no—
270C4⋊C4—1631no—
271C4⋊C4—1631no—
272C4⋊C4—1631no—
273C4⋊C4C22×D16311yescharacteristic
274C4⋊C4C22×D16311yescharacteristic
275C22⋊C4—3261no—
276C22⋊C4—3261no—
277C22⋊C4—3261no—
278C22⋊C4—1631no—
279C22⋊C4—1631no—
280C42—21no—
281C42—21no—
282C42—3261no—
283C42—3261no—
284C42—3261no—
285C42—3261no—
286C42—3261no—
287C42—1631no—
288C42C22×D16311yescharacteristic
289C23—3261no—
290C23—1631no—
291Q8—21no—
292Q8—21no—
293Q8—3264no—
294Q8—3261no—
295Q8—1632no—
296Q8C2×C4×D16312yes—
297Q8C2×C4×D16312yes—
298D4—3264no—
299D4—3261no—
300D4—1632no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—13041no—
306C2×C4—13041no—
307C2×C4—13041no—
308C2×C4—3264no—
309C2×C4—3264no—
310C2×C4—3261no—
311C2×C4—3261no—
312C2×C4—3261no—
313C2×C4—1631no—
314C2×C4—1631no—
315C2×C4—1631no—
316C2×C4—1632no—
317C2×C4—1632no—
318C2×C4D4×D16311yescharacteristic
319C2×C4D4×D16311yescharacteristic
320C2×C4C23×D16311yescharacteristic
321C8—21no—
322C8—3261no—
323C8—1632no—
324C8C2×C4×D16312yes—
325C22—3261no—
326C22—3261no—
327C22—3262no—
328C22—6521no—
329C22—6521no—
330C22—1632no—
331C22C2×D4×D16311yescharacteristic
332C4—21no—
333C4—21no—
334C4—22no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—3261no—
339C4—3261no—
340C4—3262no—
341C4—6521no—
342C4—6521no—
343C4—1632no—
344C4C22×C4×D16311yescharacteristic
345C4#41×D16311yescharacteristic
346C2—3262no—
347C2—6522no—
348C2C4×D4×D16311yescharacteristic
349C2#2378247411yescharacteristic
350C2#2378248111yescharacteristic
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
hypocenterC163#2947
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. #23782141C2semidirect
4. #23781769C2semidirect
5. #23781768C2semidirect
6. #23781767C2semidirect
7. Q16×Dic163C2semidirect
8. C4×#14579689C2semidirect
9. C4×#14579686C2semidirect
10. #23781752C2semidirect
11. #23781704C2semidirect
13. #103×D163C2semidirect
14. #23781622C2semidirect
15. #441×C163C2semidirect
18. #19104767C4semidirect
65. Q8×D163C2×C4semidirect
74. #14579689C2×C4semidirect
75. #14579688C2×C4semidirect
76. #14579686C2×C4semidirect
77. #14579682C2×C4semidirect
103. Q16×C163C2×C4semidirect
175. C163#2947semidirect
181. #441D163semidirect
253. Q16C4×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047672
#237821472
#237821412
#237817692
#237817682
#237817672
Q16×Dic1632
C4×#145796892
C4×#145796862
#237817522
#237817042
#237816602
#103×D1632
#237816222
#441×C1632
#2947163