Finite Group Databasebeta

← #35223414

Subgroups — #35223414

Order 20864.

484 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104756C211yesmaximal, characteristic
2#23782147C211yesmaximal, characteristic, complemented
3#23782144C211yesmaximal, characteristic, complemented
4#23781775C211yesmaximal, characteristic, complemented
5#23781774C211yesmaximal, characteristic, complemented
6#23781770C211yesmaximal, characteristic, complemented
7#23781769C211yesmaximal, characteristic, complemented
8C4×#14579688C211yesmaximal, characteristic, complemented
9SD16×Dic163C211yesmaximal, characteristic, complemented
10#23781752C211yesmaximal, characteristic, complemented
11#106×D163C211yesmaximal, characteristic, complemented
12#23781665C211yesmaximal, characteristic, complemented
13#23781663C211yesmaximal, characteristic, complemented
14#23781619C211yesmaximal, characteristic, complemented
15C4×SD16×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579698C2211yescharacteristic
18#19104756C418yescomplemented
19C2×#14579689C2211yescharacteristic
20C2×#14579688C2211yescharacteristic
21C2×#14579684C2211yescharacteristic
22C2×#14579682C2211yescharacteristic
23C2×C8×D163C2211yescharacteristic
24Q8×Dic163C2211yescharacteristic
25C4×Dic326C2211yescharacteristic
26#19104700C2211yescharacteristic
27C4×D652—21no—
28C4×D652C2211yescharacteristic
29D4×Dic163C2211yescharacteristic
30#19104697—21no—
31#19104697C2211yescharacteristic
32C4×C22⋊D163—21no—
33(C22⋊D163)⋊C4—21no—
34#19104692—21no—
35#19104692C2211yescharacteristic
36#19104691—21no—
37C2×Dic163⋊C4—21no—
38C4⋊C4×D163—21no—
39C4⋊C4×D163C2211yescharacteristic
40C4⋊C4×D163C2211yescharacteristic
41#19104667C2211yescharacteristic
42#19104666C2211yescharacteristic
43#19104666C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104658C2211yescharacteristic
47#19104657C2211yescharacteristic
48#19104656C2211yescharacteristic
49#19104655C2211yescharacteristic
50#19104646C2211yescharacteristic
51#19104645C2211yescharacteristic
52C8×Dic163C2211yescharacteristic
53C2×SD16×C163C2211yescharacteristic
54C4×Q8×C163C2211yescharacteristic
55C4×D4×C163C2211yescharacteristic
56#106×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59C4×C8×C163C2211yescharacteristic
60#14579700—22no—
61#14579700C2×C414yescomplemented
62#14579698—22no—
63#14579698C2×C414yescomplemented
64C2×Dic326C2311yescharacteristic
65C2×D652—21no—
66C2×D652C2311yescharacteristic
67C2×C22⋊D163—21no—
68C22×Dic163—21no—
69C22×Dic163—21no—
70C2×C4×D163—21no—
71C2×C4×D163—21no—
72C2×C4×D163—21no—
73C2×C4×D163—21no—
74C2×C4×D163—21no—
75C2×C4×D163C2311yescharacteristic
76#14579689C2×C414yescomplemented
77#14579688C2×C414yescomplemented
78#14579684C2×C414yescomplemented
79#14579682—22no—
80C2×C163⋊C8C2311yescharacteristic
81C2×C163⋊C8C2311yescharacteristic
82C8×D163C2×C414yescomplemented
83C4⋊Dic163—21no—
84C4⋊Dic163C2311yescharacteristic
85C4⋊Dic163C2311yescharacteristic
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88Dic163⋊C4—21no—
89Dic163⋊C4—21no—
90Dic163⋊C4D411yescharacteristic
91Dic163⋊C4D411yescharacteristic
92C22⋊Dic163—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4D411yescharacteristic
96(C2×D163)⋊C4D411yescharacteristic
97C4×Dic163—21no—
98C4×Dic163—21no—
99C4×Dic163—21no—
100C4×Dic163C2311yescharacteristic
101C2×Q8×C163C2311yescharacteristic
102C2×D4×C163C2311yescharacteristic
103C22×C4×C163—21no—
104SD16×C163C2×C414yescomplemented
105C2×C8×C163C2311yescharacteristic
106C2×C8×C163C2311yescharacteristic
107C4⋊C4×C163—21no—
108C4⋊C4×C163C2311yescharacteristic
109C4⋊C4×C163C2311yescharacteristic
110C22⋊C4×C163—21no—
111C42×C163—21no—
112C42×C163C2311yescharacteristic
113C22×D163—21no—
114C22×D163C2×D411yescharacteristic
115Dic326—21no—
116Dic326C22×C412yes—
117D652—24no—
118D652—21no—
119D652C22×C412yes—
120C22⋊D163—24no—
121C2×Dic163—21no—
122C2×Dic163—24no—
123C2×Dic163—21no—
124C2×Dic163—21no—
125C2×Dic163—81no—
126C2×Dic163C2×D411yescharacteristic
127C2×Dic163C2×D411yescharacteristic
128C2×Dic163C2×D411yescharacteristic
129C4×D163—24no—
130C4×D163—81no—
131C4×D163—81no—
132C4×D163—81no—
133C4×D163—81no—
134C4×D163C22×C412yes—
135C4×D163#4112yes—
136C163⋊C8—21no—
137C163⋊C8C22×C412yes—
138C23×C163—21no—
139Q8×C163—21no—
140Q8×C163C22×C412yes—
141D4×C163—21no—
142D4×C163C22×C412yes—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163—81no—
147C2×C4×C163C2×D411yescharacteristic
148C2×C4×C163C2×D411yescharacteristic
149C2×C4×C163C2411yescharacteristic
150C8×C163—21no—
151C8×C163C22×C412yes—
152C22×C163—22no—
153C22×C163—41no—
154C22×C163—41no—
155C22×C163C22×D411yescharacteristic
156C2×D163—21no—
157C2×D163—21no—
158C2×D163—22no—
159C2×D163—41no—
160C2×D163—41no—
161C2×D163C4×D412yes—
162C4×C163—22no—
163C4×C163—21no—
164C4×C163—21no—
165C4×C163—41no—
166C4×C163—41no—
167C4×C163C23×C411yescharacteristic
168C4×C163C2×#4111yescharacteristic
169Dic163—22no—
170Dic163—21no—
171Dic163—21no—
172Dic163—41no—
173Dic163—41no—
174Dic163C4×D412yes—
175C2×C163—42no—
176C2×C163C2×C4×D411yescharacteristic
177C2×C163#57711yescharacteristic
178C2×C163#57711yescharacteristic
179D163—22no—
180D163—42no—
181C163#295211yescharacteristic, complemented, Sylow
182#2952—1631nomaximal, Sylow
183C2×#135—1631no—
184#520—1631no—
185#520—1631no—
186#442—1631no—
187#442—1631no—
188#441—1631no—
189#441—1631no—
190C4×SD16—1631no—
191C4×SD16—1631no—
192C4×SD16—1631no—
193C4×SD16D16311yescharacteristic, complemented
194C2×#106—1631no—
195#417—1631no—
196#417—1631no—
197#405—1631no—
198C2×#41—1631no—
199C2×#41—1631no—
200#135—1638no—
201C2×Q16—1631no—
202C2×SD16—1631no—
203C2×SD16C2×D16311yescharacteristic
204C2×D8—1631no—
205C2×M16—1631no—
206C22×C8—1631no—
207C4×Q8—1631no—
208C4×Q8—1631no—
209C4×Q8—1631no—
210C4×Q8C2×D16311yescharacteristic
211C4×D4—3261no—
212C4×D4—3261no—
213C4×D4—3261no—
214C4×D4—3261no—
215C4×D4—1631no—
216C4×D4—1631no—
217C4×D4—1631no—
218C4×D4C2×D16311yescharacteristic
219#117—3261no—
220#117—3261no—
221#117—1631no—
222C2×C4⋊C4—3261no—
223C2×C4⋊C4—3261no—
224C2×C4⋊C4—1631no—
225C2×C4⋊C4—1631no—
226#106—1631no—
227#106—1631no—
228#106—1631no—
229#106C2×D16311yescharacteristic
230#103—1631no—
231#103—1631no—
232#103—1631no—
233#103C2×D16311yescharacteristic
234#102—1631no—
235#102—1631no—
236#102—1631no—
237#102C2×D16311yescharacteristic
238#97—1631no—
239#97—1631no—
240C4×C8—1631no—
241C4×C8C2×D16311yescharacteristic
242#41—3262no—
243#41—3262no—
244#41—1634no—
245#41—1634no—
246C2×Q8—1631no—
247C2×Q8C22×D16311yescharacteristic
248C2×D4—3261no—
249C2×D4—3261no—
250C2×D4—1631no—
251C2×D4C22×D16311yescharacteristic
252C22×C4—21no—
253C22×C4—3261no—
254C22×C4—3261no—
255C22×C4—3261no—
256C22×C4—3261no—
257C22×C4—3261no—
258C22×C4—3261no—
259C22×C4—3261no—
260C22×C4—1631no—
261Q16—1634no—
262SD16—1634no—
263SD16C4×D16314yescomplemented
264D8—1634no—
265M16—3262no—
266C2×C8—1631no—
267C2×C8—1631no—
268C2×C8—1634no—
269C2×C8C22×D16311yescharacteristic
270C2×C8C22×D16311yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—3261no—
273C4⋊C4—3261no—
274C4⋊C4—3261no—
275C4⋊C4—3261no—
276C4⋊C4—3261no—
277C4⋊C4—1631no—
278C4⋊C4—1631no—
279C4⋊C4—1631no—
280C4⋊C4—1631no—
281C4⋊C4C22×D16311yescharacteristic
282C4⋊C4C22×D16311yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—3261no—
285C22⋊C4—3261no—
286C22⋊C4—3261no—
287C22⋊C4—1631no—
288C22⋊C4—1631no—
289C42—21no—
290C42—3261no—
291C42—3261no—
292C42—3261no—
293C42—1631no—
294C42C22×D16311yescharacteristic
295C23—21no—
296C23—3261no—
297C23—1631no—
298Q8—21no—
299Q8—3261no—
300Q8—1632no—
301Q8C2×C4×D16312yes—
302D4—21no—
303D4—3264no—
304D4—3264no—
305D4—3261no—
306D4—1632no—
307D4C2×C4×D16312yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—13041no—
313C2×C4—13041no—
314C2×C4—13041no—
315C2×C4—13041no—
316C2×C4—13041no—
317C2×C4—3264no—
318C2×C4—3261no—
319C2×C4—3264no—
320C2×C4—3261no—
321C2×C4—3261no—
322C2×C4—1631no—
323C2×C4—1631no—
324C2×C4—1631no—
325C2×C4—1632no—
326C2×C4—1632no—
327C2×C4D4×D16311yescharacteristic
328C2×C4D4×D16311yescharacteristic
329C2×C4C23×D16311yescharacteristic
330C8—21no—
331C8—3261no—
332C8—1632no—
333C8C2×C4×D16312yes—
334C22—22no—
335C22—41no—
336C22—41no—
337C22—3261no—
338C22—3261no—
339C22—3262no—
340C22—6521no—
341C22—6521no—
342C22—1632no—
343C22C2×D4×D16311yescharacteristic
344C4—21no—
345C4—21no—
346C4—22no—
347C4—41no—
348C4—41no—
349C4—3261no—
350C4—3261no—
351C4—3262no—
352C4—6521no—
353C4—6521no—
354C4—1632no—
355C4C22×C4×D16311yescharacteristic
356C4#41×D16311yescharacteristic
357C2—42no—
358C2—3262no—
359C2—6522no—
360C2C4×D4×D16311yescharacteristic
361C2#2378247111yescharacteristic
362C2#2378247211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutatorC4×C163C23×C4
FrattiniC2×C4C23×D163
FittingC4×SD16×C163C2
socleC22×C163C22×D4
prime radicalC2×#19104756C2
cosocleC2×C4×C163C24
hypercenterC4×SD16D163
hypocenterC163#2952
LCSG ▷ C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #23782147C2semidirect
3. #23782144C2semidirect
4. #23781775C2semidirect
5. #23781774C2semidirect
6. #23781770C2semidirect
7. #23781769C2semidirect
8. C4×#14579688C2semidirect
9. SD16×Dic163C2semidirect
10. #23781752C2semidirect
11. #106×D163C2semidirect
12. #23781665C2semidirect
13. #23781663C2semidirect
14. #23781619C2semidirect
15. C4×SD16×C163C2semidirect
18. #19104756C4semidirect
61. #14579700C2×C4semidirect
63. #14579698C2×C4semidirect
76. #14579689C2×C4semidirect
77. #14579688C2×C4semidirect
78. #14579684C2×C4semidirect
82. C8×D163C2×C4semidirect
104. SD16×C163C2×C4semidirect
181. C163#2952semidirect
193. C4×SD16D163semidirect
263. SD16C4×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047562
#237821472
#237821442
#237817752
#237817742
#237817702
#237817692
C4×#145796882
SD16×Dic1632
#237817522
#106×D1632
#237816652
#237816632
#237816192
C4×SD16×C1632
#2952163