Finite Group Databasebeta

← #35224716

Subgroups — #35224716

Order 20864.

406 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
1#23782280C211yesmaximal, characteristic, complemented
2#23782137C211yesmaximal, characteristic, complemented
3#23782008C211yesmaximal, characteristic, complemented
4#23781988C211yesmaximal, characteristic, complemented
5#23781953C211yesmaximal, characteristic, complemented
6#23781944C211yesmaximal, characteristic, complemented
7#23781879C211yesmaximal, characteristic, complemented
8#23781878C211yesmaximal, characteristic, complemented
9#23781817C211yesmaximal, characteristic, complemented
10#23781797C211yesmaximal, characteristic
11#23781775C211yesmaximal, characteristic, complemented
12#23781765C211yesmaximal, characteristic, complemented
13C4×#14579688C211yesmaximal, characteristic, complemented
14#23781724C211yesmaximal, characteristic, complemented
15C4×D8×C163C211yesmaximal, characteristic, complemented
16C2×D4×D163C2211yescharacteristic
17C2×#14579688C2211yescharacteristic
18C2×#14579687C2211yescharacteristic
19C2×#14579684C2211yescharacteristic
20#19104734C2211yescharacteristic, complemented
21#19104728—21no—
22#19104722C2211yescharacteristic, complemented
23#19104721—21no—
24#19104718—21no—
25#19104715C2211yescharacteristic
26#19104714—21no—
27#19104706C2211yescharacteristic
28C4×Dic326C2211yescharacteristic, complemented
29C4×D652C2211yescharacteristic
30D4×Dic163C2211yescharacteristic
31#19104697C2211yescharacteristic
32C4×C22⋊D163—21no—
33(C22⋊D163)⋊C4—21no—
34#19104693C2211yescharacteristic
35#19104692C2211yescharacteristic
36C2×(C2×D163)⋊C4—21no—
37C22⋊C4×D163—21no—
38#19104668C2211yescharacteristic, complemented
39#19104667C2211yescharacteristic, complemented
40#19104666C2211yescharacteristic, complemented
41Dic163⋊C8C2211yescharacteristic, complemented
42Dic163⋊C8C2211yescharacteristic, complemented
43#19104660C2211yescharacteristic, complemented
44#19104658C2211yescharacteristic, complemented
45#19104657C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104656C2211yescharacteristic, complemented
48#19104655C2211yescharacteristic
49(C2×D163)⋊C8C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51#19104646C2211yescharacteristic, complemented
52C2×D8×C163C2211yescharacteristic
53C4×D4×C163C2211yescharacteristic
54C4×D4×C163C2211yescharacteristic, complemented
55#107×C163C2211yescharacteristic, complemented
56#102×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic, complemented
58C4×C8×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60C2×Dic326—21no—
61C2×Dic326C2311yescharacteristic
62C2×D652C2311yescharacteristic
63C2×C22⋊D163—21no—
64C2×C22⋊D163—21no—
65D4×D163—22no—
66D4×D163—22no—
67C22×Dic163—21no—
68C2×C4×D163—21no—
69C2×C4×D163—21no—
70C2×C4×D163C2311yescharacteristic
71#14579688—22no—
72#14579687—22no—
73#14579684—22no—
74C2×C163⋊C8C2311yescharacteristic
75C2×C163⋊C8C2311yescharacteristic
76C4⋊Dic163—21no—
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4D411yescharacteristic
82Dic163⋊C4D411yescharacteristic
83C22⋊Dic163—21no—
84C22⋊Dic163—21no—
85C22⋊Dic163—21no—
86C22⋊Dic163—21no—
87(C2×D163)⋊C4—21no—
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4D411yescharacteristic
92(C2×D163)⋊C4D411yescharacteristic
93C4×Dic163—21no—
94C4×Dic163—21no—
95C4×Dic163C2311yescharacteristic
96C2×D4×C163C2311yescharacteristic
97C2×D4×C163C2311yescharacteristic
98C22×C4×C163—21no—
99C22×C4×C163—21no—
100D8×C163—22no—
101C2×C8×C163C2311yescharacteristic
102C2×C8×C163C2311yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C4⋊C4×C163C2311yescharacteristic
105C22⋊C4×C163—21no—
106C22⋊C4×C163—21no—
107C42×C163C2311yescharacteristic
108C22×D163—21no—
109C22×D163—24no—
110C22×D163—81no—
111C22×D163—81no—
112C22×D163C2×D411yescharacteristic
113Dic326—21no—
114Dic326—21no—
115Dic326—81no—
116D652—21no—
117D652#4112yes—
118C22⋊D163—24no—
119C22⋊D163—81no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—81no—
124C2×Dic163C2×D411yescharacteristic
125C2×Dic163C2×D411yescharacteristic
126C2×Dic163C2×D411yescharacteristic
127C4×D163—21no—
128C4×D163—21no—
129C4×D163—81no—
130C4×D163—81no—
131C163⋊C8—21no—
132C163⋊C8—21no—
133C23×C163—21no—
134C23×C163—21no—
135D4×C163—21no—
136D4×C163—21no—
137D4×C163—21no—
138D4×C163#4112yes—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—81no—
142C2×C4×C163—81no—
143C2×C4×C163C2×D411yescharacteristic
144C2×C4×C163C2×D411yescharacteristic
145C2×C4×C163C2411yescharacteristic
146C8×C163—21no—
147C8×C163—21no—
148C22×C163—22no—
149C22×C163—41no—
150C22×C163—41no—
151C22×C163—41no—
152C22×C163—41no—
153C22×C163—41no—
154C22×C163C22×D411yescharacteristic
155C2×D163—21no—
156C2×D163—22no—
157C2×D163—21no—
158C2×D163—21no—
159C2×D163—41no—
160C2×D163—41no—
161C2×D163—84no—
162C4×C163—21no—
163C4×C163—21no—
164C4×C163—41no—
165C4×C163—41no—
166C4×C163C2×#4111yescharacteristic
167C4×C163C2×#4111yescharacteristic
168Dic163—21no—
169Dic163—21no—
170Dic163—21no—
171Dic163—41no—
172Dic163—41no—
173Dic163—41no—
174C2×C163—42no—
175C2×C163—81no—
176C2×C163#52511yescharacteristic
177C2×C163#57611yescharacteristic
178C2×C163#57711yescharacteristic
179D163—41no—
180D163—42no—
181C163#311811yescharacteristic, complemented, Sylow
182#3118—1631nomaximal, Sylow
183#542—1631no—
184#518—1631no—
185#485—1631no—
186#482—1631no—
187#476—1631no—
188#474—1631no—
189#463—1631no—
190#463—1631no—
191#450—1631no—
192#447—1631no—
193#442—1631no—
194#440—1631no—
195C4×SD16—1631no—
196C4×D8D16311yescharacteristic, complemented
197#433—1631no—
198C22×D4—1631no—
199C2×SD16—1631no—
200C2×SD16—1631no—
201C2×D8—1631no—
202C2×D8C2×D16311yescharacteristic
203C4⋊Q8—1631no—
204#126—3261no—
205#124—3261no—
206#124—1631no—
207#123—3261no—
208C22⋊Q8—3261no—
209C22⋊Q8—1631no—
210#121—1631no—
211C4×Q8—1631no—
212C4×D4—3261no—
213C4×D4—3261no—
214C4×D4—1631no—
215C4×D4—1631no—
216C4×D4—1631no—
217C4×D4C2×D16311yescharacteristic
218C4×D4C2×D16311yescharacteristic, complemented
219#117—1631no—
220#117—1631no—
221C2×C22⋊C4—3261no—
222C2×C22⋊C4—3261no—
223#107—1631no—
224#107C2×D16311yescharacteristic, complemented
225#106—1631no—
226#106—1631no—
227C4⋊C8—1631no—
228C4⋊C8—1631no—
229#103—1631no—
230#103—1631no—
231#102—1631no—
232#102—1631no—
233#102—1631no—
234#102—1631no—
235#102C2×D16311yescharacteristic
236#102C2×D16311yescharacteristic, complemented
237C22⋊C8—1631no—
238C22⋊C8—1631no—
239#97—1631no—
240C4×C8C2×D16311yescharacteristic, complemented
241C24—3261no—
242C2×Q8—3261no—
243C2×Q8—1631no—
244C2×D4—3262no—
245C2×D4—3261no—
246C2×D4—3262no—
247C2×D4—3261no—
248C2×D4—1631no—
249C2×D4C22×D16311yescharacteristic
250C2×D4C22×D16311yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—3261no—
254C22×C4—3261no—
255C22×C4—3261no—
256C22×C4—1631no—
257SD16—3262no—
258SD16—3262no—
259D8—22no—
260D8—3262no—
261C2×C8—1631no—
262C2×C8—1631no—
263C2×C8C22×D16311yescharacteristic
264C2×C8C22×D16311yescharacteristic
265C4⋊C4—3261no—
266C4⋊C4—3261no—
267C4⋊C4—3261no—
268C4⋊C4—1631no—
269C4⋊C4—1631no—
270C4⋊C4—1631no—
271C4⋊C4—1631no—
272C4⋊C4C22×D16311yescharacteristic
273C4⋊C4C22×D16311yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—3261no—
277C22⋊C4—3261no—
278C22⋊C4—3261no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—3261no—
284C22⋊C4—1631no—
285C22⋊C4—1631no—
286C42—3261no—
287C42—3261no—
288C42—1631no—
289C42C22×D16311yescharacteristic
290C23—21no—
291C23—21no—
292C23—13041no—
293C23—13041no—
294C23—3264no—
295C23—3261no—
296C23—1631no—
297Q8—13041no—
298Q8—3261no—
299Q8—3261no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—13041no—
304D4—3264no—
305D4—3261no—
306D4—1632no—
307D4#1457969912yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—13041no—
313C2×C4—13041no—
314C2×C4—13041no—
315C2×C4—3261no—
316C2×C4—3261no—
317C2×C4—3261no—
318C2×C4—3261no—
319C2×C4—3261no—
320C2×C4—1631no—
321C2×C4—1631no—
322C2×C4—1631no—
323C2×C4D4×D16311yescharacteristic
324C2×C4D4×D16311yescharacteristic
325C2×C4C23×D16311yescharacteristic
326C8—21no—
327C8—21no—
328C8—3261no—
329C8—3261no—
330C22—22no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—13044no—
337C22—3261no—
338C22—3262no—
339C22—3261no—
340C22—3261no—
341C22—6521no—
342C22—6521no—
343C22C2×D4×D16311yescharacteristic
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—3261no—
349C4—3261no—
350C4—3261no—
351C4—6521no—
352C4—6521no—
353C4—6521no—
354C4#41×D16311yescharacteristic
355C4C2×#1457969911yescharacteristic
356C2—42no—
357C2—81no—
358C2—6521no—
359C2—6522no—
360C2#2378217411yescharacteristic
361C2#2378246611yescharacteristic
362C2#2378247211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC4×D8×C163C2
socleC22×C163C22×D4
prime radical#23781797C2
hypercenterC4×D8D163
hypocenterC163#3118
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #23782280C2semidirect
2. #23782137C2semidirect
3. #23782008C2semidirect
4. #23781988C2semidirect
5. #23781953C2semidirect
6. #23781944C2semidirect
7. #23781879C2semidirect
8. #23781878C2semidirect
9. #23781817C2semidirect
11. #23781775C2semidirect
12. #23781765C2semidirect
13. C4×#14579688C2semidirect
14. #23781724C2semidirect
15. C4×D8×C163C2semidirect
20. #19104734C22semidirect
22. #19104722C22semidirect
28. C4×Dic326C22semidirect
38. #19104668C22semidirect
39. #19104667C22semidirect
40. #19104666C22semidirect
41. Dic163⋊C8C22semidirect
42. Dic163⋊C8C22semidirect
43. #19104660C22semidirect
44. #19104658C22semidirect
47. #19104656C22semidirect
51. #19104646C22semidirect
54. C4×D4×C163C22semidirect
55. #107×C163C22semidirect
57. #102×C163C22semidirect
58. C4×C8×C163C22semidirect
181. C163#3118semidirect
196. C4×D8D163semidirect
218. C4×D4C2×D163semidirect
224. #107C2×D163semidirect
236. #102C2×D163semidirect
240. C4×C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237822802
#237821372
#237820082
#237819882
#237819532
#237819442
#237818792
#237818782
#237818172
#237817972
#237817752
#237817652
C4×#145796882
#237817242
C4×D8×C1632
#3118163