Finite Group Databasebeta

← #35223839

Subgroups — #35223839

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×#19104755C211yesmaximal, characteristic, complemented
3#23782243C211yesmaximal, characteristic, complemented
4#23781824C212yesmaximal, complemented
5#23781814C212yesmaximal, complemented
6#23781807C212yesmaximal, complemented
7#23781797C212yesmaximal, complemented
8#23781667C211yesmaximal, characteristic, complemented
9C2×#19104656C211yesmaximal, characteristic, complemented
10#23781633C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C163C211yesmaximal, characteristic, complemented
12#19104780—24no—
13#19104780—24no—
14C2×#14579699C2211yescharacteristic
15C2×#14579699C2211yescharacteristic
16C2×#14579698—21no—
17C2×#14579698C2212yes—
18C22×C22⋊D163—22no—
19C2×D4×D163—21no—
20C2×D4×D163C2212yes—
21#19104755—24no—
22C2×#14579689C2211yescharacteristic, complemented
23C2×#14579687C2211yescharacteristic, complemented
24C2×#14579686C2211yescharacteristic, complemented
25C2×#14579684C2211yescharacteristic, complemented
26C22×C163⋊C8C2211yescharacteristic, complemented
27#19104714—21no—
28#19104711C2212yescomplemented
29#19104709—21no—
30#19104707—22no—
31#19104706C2212yescomplemented
32C2×C4⋊Dic163C2211yescharacteristic, complemented
33#19104660C2212yescomplemented
34#19104657C2212yescomplemented
35#19104656C2212yescomplemented
36#19104656C2212yescomplemented
37(C2×D163)⋊C8C2212yescomplemented
38(C2×D163)⋊C8C2212yescomplemented
39C2×#41×C163C2211yescharacteristic, complemented
40C22×D4×C163C2211yescharacteristic
41AGL(1,ℤ/8)×C163—24no—
42C2×SD16×C163C2212yescomplemented
43C2×D8×C163C2212yescomplemented
44C2×M16×C163C2211yescharacteristic, complemented
45C23×D163—22no—
46#14579699—22no—
47#14579699—22no—
48#14579699—22no—
49#14579699D414yescomplemented
50#14579698—28no—
51#14579698—24no—
52#14579698—24no—
53C2×Dic326D411yescharacteristic
54C2×Dic326C2311yescharacteristic
55C2×D652D411yescharacteristic
56C2×D652C2311yescharacteristic
57C2×C22⋊D163—216no—
58C2×C22⋊D163—22no—
59C2×C22⋊D163—22no—
60C2×C22⋊D163—22no—
61C2×C22⋊D163—21no—
62C2×C22⋊D163D412yes—
63D4×D163—28no—
64D4×D163—24no—
65D4×D163—24no—
66C22×Dic163—22no—
67C22×Dic163—21no—
68C2×C4×D163—21no—
69C2×C4×D163C2312yes—
70#14579689—22no—
71#14579687—22no—
72#14579686—22no—
73#14579684—22no—
74C2×C163⋊C8—22no—
75C2×C163⋊C8C2311yescharacteristic
76C2×C163⋊C8C2311yescharacteristic
77C4⋊Dic163—21no—
78C4⋊Dic163C2312yes—
79Dic163⋊C4—22no—
80C22⋊Dic163—22no—
81(C2×D163)⋊C4—22no—
82C24×C163—21no—
83#41×C163—22no—
84#41×C163—22no—
85C2×Q8×C163C2311yescharacteristic
86C2×D4×C163—21no—
87C2×D4×C163—21no—
88C2×D4×C163—22no—
89C2×D4×C163D414yescomplemented
90C2×D4×C163C2311yescharacteristic
91C2×D4×C163C2312yes—
92C22×C4×C163—21no—
93C22×C4×C163C2311yescharacteristic
94SD16×C163—24no—
95D8×C163—24no—
96M16×C163—22no—
97C2×C8×C163C2312yes—
98C22×D163—21no—
99C22×D163—28no—
100C22×D163—82no—
101C22×D163—82no—
102C22×D163C2×D412yes—
103Dic326—22no—
104Dic326—21no—
105Dic326C2×D412yes—
106D652—22no—
107D652—21no—
108D652C2×D412yes—
109C22⋊D163—28no—
110C22⋊D163—28no—
111C22⋊D163—24no—
112C22⋊D163—24no—
113C22⋊D163—82no—
114C2×Dic163—28no—
115C2×Dic163—21no—
116C2×Dic163—22no—
117C2×Dic163—81no—
118C2×Dic163C2×D412yes—
119C4×D163—24no—
120C4×D163—22no—
121C4×D163—22no—
122C163⋊C8—21no—
123C163⋊C8—21no—
124C23×C163—22no—
125C23×C163—24no—
126C23×C163—21no—
127C23×C163—81no—
128C23×C163—81no—
129C23×C163C2×D411yescharacteristic
130Q8×C163—21no—
131Q8×C163—21no—
132D4×C163—24no—
133D4×C163—22no—
134D4×C163—21no—
135D4×C163—21no—
136D4×C163—42no—
137D4×C163C2×D414yes—
138C2×C4×C163—21no—
139C2×C4×C163—42no—
140C2×C4×C163C2×D411yescharacteristic
141C2×C4×C163C2×D412yes—
142C2×C4×C163C2×D412yes—
143C2×C4×C163C2411yescharacteristic
144C8×C163—22no—
145C22×C163—21no—
146C22×C163—21no—
147C22×C163—24no—
148C22×C163—42no—
149C22×C163—42no—
150C22×C163—41no—
151C22×C163—41no—
152C22×C163—41no—
153C22×C163—84no—
154C22×C163#12012yes—
155C22×C163C22×D411yescharacteristic
156C2×D163—22no—
157C2×D163—22no—
158C2×D163—22no—
159C2×D163—22no—
160C2×D163—41no—
161C2×D163—41no—
162C2×D163—88no—
163C4×C163—41no—
164C4×C163#12012yes—
165C4×C163C22×D411yescharacteristic
166C4×C163C22×D411yescharacteristic
167Dic163—22no—
168Dic163—22no—
169Dic163—42no—
170C2×C163—22no—
171C2×C163—44no—
172C2×C163—81no—
173C2×C163C2×#12011yescharacteristic
174C2×C163#57611yescharacteristic
175C2×C163#57711yescharacteristic
176D163—42no—
177D163—42no—
178C163#301211yescharacteristic, complemented, Sylow
179#3012—1631nomaximal, Sylow
180C2×H(5,𝔽2)—1631no—
181C2×AGL(1,ℤ/8)D16311yescharacteristic, complemented
182C2×#135—1631no—
183#537—1631no—
184#452—1632no—
185#450—1632no—
186#449—1632no—
187#447—1632no—
188#417—1631no—
189C2×#102—1631no—
190#408—1631no—
191H(5,𝔽2)—3264no—
192H(5,𝔽2)—3264no—
193C2×#41—3261no—
194C2×#41—1632no—
195C2×#41—1631no—
196C2×#41—1631no—
197C2×#41C2×D16311yescharacteristic, complemented
198C22×D4—3261no—
199C22×D4—3262no—
200C22×D4—1632no—
201C22×D4C2×D16311yescharacteristic
202AGL(1,ℤ/8)—24no—
203#135—3264no—
204C2×Q16—1631no—
205C2×SD16—1631no—
206C2×SD16—1631no—
207C2×SD16C2×D16312yescomplemented
208C2×D8—1631no—
209C2×D8C2×D16312yescomplemented
210C2×M16C2×D16311yescharacteristic, complemented
211C22×C8—1631no—
212C22⋊Q8—3261no—
213C22⋊Q8—1632no—
214#121—3261no—
215#121—3262no—
216#121—1632no—
217C2×C4⋊C4—1631no—
218#103—1632no—
219#102—1632no—
220#102—1632no—
221#102—1632no—
222C22⋊C8—1632no—
223C22⋊C8—1632no—
224C24—21no—
225C24—3262no—
226#41—22no—
227#41—22no—
228#41—3268no—
229#41—3264no—
230#41—3264no—
231#41—3262no—
232#41—3262no—
233#41—3262no—
234#41—1634no—
235C2×Q8—1631no—
236C2×Q8—1631no—
237C2×Q8C22×D16311yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—3268no—
242C2×D4—3264no—
243C2×D4—3262no—
244C2×D4—32616no—
245C2×D4—3262no—
246C2×D4—3261no—
247C2×D4—3264no—
248C2×D4—3262no—
249C2×D4—1632no—
250C2×D4—1631no—
251C2×D4—1631no—
252C2×D4C22⋊D16314yescomplemented
253C2×D4C22×D16311yescharacteristic
254C2×D4C22×D16312yes—
255C22×C4—21no—
256C22×C4—3261no—
257C22×C4—3262no—
258C22×C4—3261no—
259C22×C4—1632no—
260C22×C4C22×D16311yescharacteristic
261Q16—3262no—
262SD16—24no—
263SD16—3262no—
264SD16—3262no—
265D8—24no—
266D8—3262no—
267M16—22no—
268C2×C8—3262no—
269C2×C8—1631no—
270C2×C8—1631no—
271C2×C8C22×D16312yes—
272C4⋊C4—3262no—
273C4⋊C4—3261no—
274C4⋊C4—1632no—
275C22⋊C4—3262no—
276C22⋊C4—3262no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23—13042no—
283C23—13042no—
284C23—3268no—
285C23—3261no—
286C23—1632no—
287C23D4×D16311yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—3261no—
291Q8—3262no—
292Q8—1632no—
293D4—22no—
294D4—21no—
295D4—21no—
296D4—24no—
297D4—42no—
298D4—13042no—
299D4—3268no—
300D4—3268no—
301D4—3264no—
302D4—3264no—
303D4—3261no—
304D4—3262no—
305D4—1632no—
306D4C2×C22⋊D16314yes—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—13041no—
310C2×C4—3268no—
311C2×C4—3261no—
312C2×C4—3262no—
313C2×C4—3262no—
314C2×C4—3264no—
315C2×C4—3262no—
316C2×C4—1632no—
317C2×C4D4×D16312yes—
318C2×C4D4×D16311yescharacteristic
319C2×C4C2×C22⋊D16312yes—
320C2×C4C23×D16311yescharacteristic
321C8—22no—
322C8—3261no—
323C8—3261no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—84no—
333C22—13048no—
334C22—3262no—
335C22—3262no—
336C22—3262no—
337C22—3262no—
338C22—6521no—
339C22—6521no—
340C22#1910470312yes—
341C22C2×D4×D16311yescharacteristic
342C4—41no—
343C4—3262no—
344C4—3262no—
345C4—6522no—
346C4#1910470312yes—
347C4C2×D4×D16311yescharacteristic
348C4C22×C22⋊D16311yescharacteristic
349C2—22no—
350C2—44no—
351C2—81no—
352C2—6522no—
353C2—6522no—
354C2C2×#1910470311yescharacteristic
355C2#2378246911yescharacteristic
356C2#2378247511yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×AGL(1,ℤ/8)×C163C2
socleC22×C163C22×D4
hypercenterC2×AGL(1,ℤ/8)D163
hypocenterC163#3012
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19104780C2semidirect
2. C2×#19104755C2semidirect
3. #23782243C2semidirect
4. #23781824C2semidirect
5. #23781814C2semidirect
6. #23781807C2semidirect
7. #23781797C2semidirect
8. #23781667C2semidirect
9. C2×#19104656C2semidirect
10. #23781633C2semidirect
11. C2×AGL(1,ℤ/8)×C163C2semidirect
22. C2×#14579689C22semidirect
23. C2×#14579687C22semidirect
24. C2×#14579686C22semidirect
25. C2×#14579684C22semidirect
26. C22×C163⋊C8C22semidirect
28. #19104711C22semidirect
31. #19104706C22semidirect
32. C2×C4⋊Dic163C22semidirect
33. #19104660C22semidirect
34. #19104657C22semidirect
35. #19104656C22semidirect
36. #19104656C22semidirect
37. (C2×D163)⋊C8C22semidirect
38. (C2×D163)⋊C8C22semidirect
39. C2×#41×C163C22semidirect
42. C2×SD16×C163C22semidirect
43. C2×D8×C163C22semidirect
44. C2×M16×C163C22semidirect
49. #14579699D4semidirect
89. C2×D4×C163D4semidirect
178. C163#3012semidirect
181. C2×AGL(1,ℤ/8)D163semidirect
197. C2×#41C2×D163semidirect
207. C2×SD16C2×D163semidirect
209. C2×D8C2×D163semidirect
210. C2×M16C2×D163semidirect
252. C2×D4C22⋊D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047802
C2×#191047552
#237822432
#237818242
#237818142
#237818072
#237817972
#237816672
C2×#191046562
#237816332
C2×AGL(1,ℤ/8)×C1632
#3012163