Finite Group Databasebeta

← #35223845

Subgroups — #35223845

Order 20864.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104780C211yesmaximal, characteristic, complemented
2C2×#19104761C211yesmaximal, characteristic, complemented
3C2×#19104757C211yesmaximal, characteristic, complemented
4#23782246C211yesmaximal, characteristic, complemented
5#23781828C212yesmaximal, complemented
6#23781817C212yesmaximal, complemented
7#23781811C212yesmaximal, complemented
8#23781800C212yesmaximal, complemented
9#23781664C211yesmaximal, characteristic, complemented
10#23781631C211yesmaximal, characteristic, complemented
11C2×#102×C163C211yesmaximal, characteristic, complemented
12#19104780—24no—
13#19104780—24no—
14C2×#14579699C2211yescharacteristic
15C2×#14579699C2211yescharacteristic
16C2×#14579699C2211yescharacteristic, complemented
17C2×#14579698—21no—
18C2×#14579698C2212yes—
19C22×C22⋊D163—22no—
20C2×D4×D163—21no—
21C2×D4×D163C2212yes—
22#19104761—24no—
23#19104757—24no—
24C2×Dic652C2211yescharacteristic, complemented
25C2×#14579688C2211yescharacteristic, complemented
26C2×#14579688C2211yescharacteristic, complemented
27C2×#14579687C2212yescomplemented
28C2×D1304C2211yescharacteristic, complemented
29C2×#14579684C2212yescomplemented
30C2×#14579683C2211yescharacteristic, complemented
31#19104715—21no—
32#19104715C2212yescomplemented
33#19104710—21no—
34#19104710C2212yescomplemented
35#19104709—22no—
36#19104658C2212yescomplemented
37#19104655C2212yescomplemented
38(C2×D163)⋊C8C2212yescomplemented
39(C2×D163)⋊C8C2212yescomplemented
40C22×D4×C163C2211yescharacteristic
41C22×C8×C163C2211yescharacteristic, complemented
42C2×C4⋊C4×C163C2211yescharacteristic, complemented
43#102×C163C2212yescomplemented
44#102×C163C2212yescomplemented
45C23×D163—22no—
46#14579699—22no—
47#14579699—22no—
48#14579699—22no—
49#14579699—22no—
50#14579699—22no—
51#14579699D414yescomplemented
52#14579698—28no—
53#14579698—24no—
54#14579698—24no—
55C2×Dic326D411yescharacteristic
56C2×Dic326C2311yescharacteristic
57C2×Dic326C2311yescharacteristic
58C2×D652—22no—
59C2×D652D411yescharacteristic
60C2×D652C2311yescharacteristic
61C2×D652C2311yescharacteristic
62C2×C22⋊D163—216no—
63C2×C22⋊D163—22no—
64C2×C22⋊D163—22no—
65C2×C22⋊D163—21no—
66C2×C22⋊D163—21no—
67C2×C22⋊D163D412yes—
68D4×D163—28no—
69D4×D163—24no—
70D4×D163—24no—
71C22×Dic163—22no—
72C2×C4×D163—21no—
73C2×C4×D163—21no—
74C2×C4×D163C2312yes—
75Dic652—22no—
76#14579688—22no—
77#14579688—22no—
78#14579687—24no—
79D1304—22no—
80#14579684—24no—
81#14579683—22no—
82C2×C163⋊C8C2312yes—
83C4⋊Dic163—22no—
84(C2×D163)⋊C4—22no—
85(C2×D163)⋊C4—22no—
86C24×C163—21no—
87C2×D4×C163—21no—
88C2×D4×C163—22no—
89C2×D4×C163D414yescomplemented
90C2×D4×C163C2312yes—
91C22×C4×C163—21no—
92C22×C4×C163C2311yescharacteristic
93C2×C8×C163—22no—
94C2×C8×C163C2311yescharacteristic
95C2×C8×C163C2311yescharacteristic
96C4⋊C4×C163—21no—
97C4⋊C4×C163C2312yes—
98C22×D163—21no—
99C22×D163—28no—
100C22×D163—21no—
101C22×D163—82no—
102C22×D163—82no—
103C22×D163C2×D412yes—
104Dic326—22no—
105Dic326—21no—
106Dic326—21no—
107Dic326—21no—
108Dic326C2×D412yes—
109D652—22no—
110D652—21no—
111D652—21no—
112D652—21no—
113D652—82no—
114D652C2×D412yes—
115C22⋊D163—28no—
116C22⋊D163—28no—
117C22⋊D163—24no—
118C22⋊D163—24no—
119C22⋊D163—42no—
120C2×Dic163—28no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163C2×D412yes—
124C4×D163—24no—
125C4×D163—22no—
126C4×D163—22no—
127C4×D163—42no—
128C163⋊C8—22no—
129C23×C163—22no—
130C23×C163—24no—
131C23×C163—81no—
132C23×C163—81no—
133C23×C163C2×D411yescharacteristic
134D4×C163—24no—
135D4×C163—22no—
136D4×C163C2×D414yes—
137C2×C4×C163—22no—
138C2×C4×C163—81no—
139C2×C4×C163C2×D411yescharacteristic
140C2×C4×C163C2×D412yes—
141C2×C4×C163C2×D412yes—
142C2×C4×C163C2411yescharacteristic
143C8×C163—21no—
144C8×C163—21no—
145C22×C163—21no—
146C22×C163—21no—
147C22×C163—24no—
148C22×C163—42no—
149C22×C163—42no—
150C22×C163—84no—
151C22×C163#12012yes—
152C22×C163C22×D411yescharacteristic
153C2×D163—22no—
154C2×D163—22no—
155C2×D163—22no—
156C2×D163—22no—
157C2×D163—41no—
158C2×D163—41no—
159C2×D163—41no—
160C2×D163—41no—
161C2×D163—41no—
162C2×D163—88no—
163C4×C163—42no—
164C4×C163#12012yes—
165C4×C163C22×D411yescharacteristic
166C4×C163C22×D411yescharacteristic
167Dic163—22no—
168Dic163—22no—
169Dic163—41no—
170C2×C163—22no—
171C2×C163—44no—
172C2×C163C2×#12011yescharacteristic
173C2×C163#57611yescharacteristic
174C2×C163#57711yescharacteristic
175D163—42no—
176D163—42no—
177D163—81no—
178C163#301211yescharacteristic, complemented, Sylow
179#3012—1631nomaximal, Sylow
180C2×H(5,𝔽2)—1631no—
181C2×AGL(1,ℤ/8)—1631no—
182C2×#135—1631no—
183#537—1631no—
184#452—1632no—
185#450—1632no—
186#449—1632no—
187#447—1632no—
188#417—1631no—
189C2×#102D16311yescharacteristic, complemented
190#408—1631no—
191H(5,𝔽2)—3264no—
192H(5,𝔽2)—3264no—
193C2×#41—3261no—
194C2×#41—1632no—
195C2×#41—1631no—
196C2×#41—1631no—
197C2×#41—1631no—
198C22×D4—3261no—
199C22×D4—3262no—
200C22×D4—1632no—
201C22×D4C2×D16311yescharacteristic
202AGL(1,ℤ/8)—3264no—
203#135—3264no—
204C2×Q16—1631no—
205C2×SD16—1632no—
206C2×SD16—1631no—
207C2×SD16—1631no—
208C2×D8—1632no—
209C2×D8—1631no—
210C2×M16—1631no—
211C22×C8C2×D16311yescharacteristic, complemented
212C22⋊Q8—3261no—
213C22⋊Q8—1632no—
214#121—3261no—
215#121—3262no—
216#121—1632no—
217C2×C4⋊C4C2×D16311yescharacteristic, complemented
218#103—1632no—
219#102—1632no—
220#102C2×D16312yescomplemented
221#102C2×D16312yescomplemented
222C22⋊C8—1632no—
223C22⋊C8—1632no—
224C24—21no—
225C24—3262no—
226#41—3268no—
227#41—3264no—
228#41—3264no—
229#41—3262no—
230#41—3262no—
231#41—3262no—
232#41—3262no—
233#41—3262no—
234#41—1634no—
235C2×Q8—1631no—
236C2×Q8—1631no—
237C2×Q8—1631no—
238C2×D4—21no—
239C2×D4—22no—
240C2×D4—3262no—
241C2×D4—3268no—
242C2×D4—3264no—
243C2×D4—32616no—
244C2×D4—3262no—
245C2×D4—3261no—
246C2×D4—3264no—
247C2×D4—3261no—
248C2×D4—3262no—
249C2×D4—1632no—
250C2×D4—1631no—
251C2×D4—1631no—
252C2×D4—1631no—
253C2×D4D65214yescomplemented
254C2×D4C22×D16312yes—
255C22×C4—21no—
256C22×C4—3262no—
257C22×C4—3261no—
258C22×C4—3261no—
259C22×C4—1632no—
260C22×C4C22×D16311yescharacteristic
261Q16—3262no—
262SD16—3262no—
263SD16—3262no—
264SD16—3264no—
265D8—3262no—
266D8—3264no—
267M16—3262no—
268C2×C8—22no—
269C2×C8—1632no—
270C2×C8C22×D16311yescharacteristic
271C2×C8C22×D16311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—3262no—
274C4⋊C4C22×D16312yes—
275C22⋊C4—3262no—
276C22⋊C4—3262no—
277C23—22no—
278C23—24no—
279C23—81no—
280C23—81no—
281C23—13042no—
282C23—13042no—
283C23—3268no—
284C23—3261no—
285C23—3261no—
286C23—1632no—
287C23D4×D16311yescharacteristic
288Q8—3261no—
289Q8—3262no—
290Q8—3261no—
291Q8—3261no—
292Q8—1632no—
293D4—22no—
294D4—24no—
295D4—13042no—
296D4—3268no—
297D4—3268no—
298D4—3264no—
299D4—3264no—
300D4—3261no—
301D4—3261no—
302D4—3261no—
303D4—3262no—
304D4—6522no—
305D4—1632no—
306D4C2×D65214yes—
307C2×C4—22no—
308C2×C4—81no—
309C2×C4—3268no—
310C2×C4—3261no—
311C2×C4—3262no—
312C2×C4—3262no—
313C2×C4—3264no—
314C2×C4—3261no—
315C2×C4—6522no—
316C2×C4—1632no—
317C2×C4D4×D16312yes—
318C2×C4D4×D16311yescharacteristic
319C2×C4C2×D65212yes—
320C2×C4C23×D16311yescharacteristic
321C8—21no—
322C8—21no—
323C8—3262no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—84no—
330C22—13048no—
331C22—3262no—
332C22—3262no—
333C22—3262no—
334C22—3262no—
335C22—6521no—
336C22—6521no—
337C22—6521no—
338C22—6521no—
339C22—6521no—
340C22#1910470512yes—
341C22C2×D4×D16311yescharacteristic
342C4—42no—
343C4—3262no—
344C4—3262no—
345C4—6521no—
346C4#1910470512yes—
347C4C2×D4×D16311yescharacteristic
348C4C22×D65211yescharacteristic
349C2—22no—
350C2—44no—
351C2—13041no—
352C2—6522no—
353C2—6522no—
354C2C2×#1910470511yescharacteristic
355C2#2378246611yescharacteristic
356C2#2378247111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×#102×C163C2
socleC22×C163C22×D4
hypercenterC2×#102D163
hypocenterC163#3012
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#19104780C2semidirect
2. C2×#19104761C2semidirect
3. C2×#19104757C2semidirect
4. #23782246C2semidirect
5. #23781828C2semidirect
6. #23781817C2semidirect
7. #23781811C2semidirect
8. #23781800C2semidirect
9. #23781664C2semidirect
10. #23781631C2semidirect
11. C2×#102×C163C2semidirect
16. C2×#14579699C22semidirect
24. C2×Dic652C22semidirect
25. C2×#14579688C22semidirect
26. C2×#14579688C22semidirect
27. C2×#14579687C22semidirect
28. C2×D1304C22semidirect
29. C2×#14579684C22semidirect
30. C2×#14579683C22semidirect
32. #19104715C22semidirect
34. #19104710C22semidirect
36. #19104658C22semidirect
37. #19104655C22semidirect
38. (C2×D163)⋊C8C22semidirect
39. (C2×D163)⋊C8C22semidirect
41. C22×C8×C163C22semidirect
42. C2×C4⋊C4×C163C22semidirect
43. #102×C163C22semidirect
44. #102×C163C22semidirect
51. #14579699D4semidirect
89. C2×D4×C163D4semidirect
178. C163#3012semidirect
189. C2×#102D163semidirect
211. C22×C8C2×D163semidirect
217. C2×C4⋊C4C2×D163semidirect
220. #102C2×D163semidirect
221. #102C2×D163semidirect
253. C2×D4D652semidirect

Maximal subgroups

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

subgroupindex
C2×#191047802
C2×#191047612
C2×#191047572
#237822462
#237818282
#237818172
#237818112
#237818002
#237816642
#237816312
C2×#102×C1632
#3012163