Finite Group Databasebeta

← #35226614

Subgroups — #35226614

Order 20864.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782376C211yesmaximal, characteristic, complemented
2#23782310C211yesmaximal, characteristic
3#23782027C211yesmaximal, characteristic, complemented
4#23782002C211yesmaximal, characteristic, complemented
5#23781980C211yesmaximal, characteristic, complemented
6#23781912C211yesmaximal, characteristic, complemented
7#23781907C211yesmaximal, characteristic, complemented
8#23781888C211yesmaximal, characteristic, complemented
9#23781828C211yesmaximal, characteristic, complemented
10#23781817C211yesmaximal, characteristic, complemented
11#23781765C211yesmaximal, characteristic, complemented
12#23781735C211yesmaximal, characteristic, complemented
13#23781663C211yesmaximal, characteristic, complemented
14#102×D163C211yesmaximal, characteristic, complemented
15#474×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C2×D4×D163C2211yescharacteristic
18C2×Dic652C2211yescharacteristic, complemented
19C2×#14579688C2211yescharacteristic
20C2×#14579687C2211yescharacteristic
21C2×#14579682C2211yescharacteristic, complemented
22C2×C8×D163C2211yescharacteristic
23#19104726—21no—
24#19104723C2211yescharacteristic
25#19104719—21no—
26#19104715C2211yescharacteristic
27#19104715C2211yescharacteristic
28#19104715C2211yescharacteristic, complemented
29#19104711—21no—
30#19104711—21no—
31#19104710C2211yescharacteristic
32#19104709—21no—
33#19104705—21no—
34C4×Dic326C2211yescharacteristic, complemented
35C4×D652C2211yescharacteristic
36#19104697—21no—
37C4⋊C4×D163C2211yescharacteristic
38C4⋊C4×D163C2211yescharacteristic, complemented
39C2×(C2×D163)⋊C4—21no—
40#19104668C2211yescharacteristic, complemented
41#19104660C2211yescharacteristic, complemented
42#19104658C2211yescharacteristic, complemented
43#19104658C2211yescharacteristic, complemented
44#19104658C2211yescharacteristic, complemented
45#19104657C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104655C2211yescharacteristic
48#19104655C2211yescharacteristic
49(C2×D163)⋊C8C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51#19104646C2211yescharacteristic, complemented
52C4⋊Q8×C163C2211yescharacteristic, complemented
53C4×D4×C163C2211yescharacteristic
54#107×C163C2211yescharacteristic, complemented
55#107×C163C2211yescharacteristic, complemented
56C4⋊C8×C163C2211yescharacteristic, complemented
57#102×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59C23×D163—21no—
60#14579698—22no—
61#14579698—41no—
62C2×Dic326C2311yescharacteristic
63C2×Dic326C2311yescharacteristic
64C2×D652—21no—
65C2×D652D411yescharacteristic
66C2×D652C2311yescharacteristic
67C2×C22⋊D163—21no—
68C2×C22⋊D163—21no—
69D4×D163—22no—
70D4×D163—41no—
71C22×Dic163—21no—
72C2×C4×D163—21no—
73C2×C4×D163—21no—
74C2×C4×D163C2311yescharacteristic
75C2×C4×D163C2311yescharacteristic
76Dic652—41no—
77#14579688—41no—
78#14579687—22no—
79#14579682—41no—
80C2×C163⋊C8C2311yescharacteristic
81C2×C163⋊C8C2311yescharacteristic
82C8×D163—41no—
83C4⋊Dic163—21no—
84C4⋊Dic163—21no—
85C4⋊Dic163—21no—
86C4⋊Dic163D411yescharacteristic
87C4⋊Dic163C2311yescharacteristic
88Dic163⋊C4—21no—
89Dic163⋊C4—21no—
90Dic163⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4—21no—
96(C2×D163)⋊C4—21no—
97(C2×D163)⋊C4D411yescharacteristic
98(C2×D163)⋊C4D411yescharacteristic
99C4×Dic163—21no—
100C2×Q8×C163—21no—
101C2×D4×C163C2311yescharacteristic
102C22×C4×C163—21no—
103C2×C8×C163C2311yescharacteristic
104C2×C8×C163C2311yescharacteristic
105C4⋊C4×C163—21no—
106C4⋊C4×C163C2311yescharacteristic
107C4⋊C4×C163C2311yescharacteristic
108C4⋊C4×C163C2311yescharacteristic
109C22⋊C4×C163—21no—
110C42×C163C2311yescharacteristic
111C22×D163—21no—
112C22×D163—42no—
113C22×D163—42no—
114C22×D163—81no—
115C22×D163C2×D411yescharacteristic
116C22×D163C2×D411yescharacteristic
117Dic326—21no—
118Dic326—21no—
119Dic326—21no—
120Dic326—21no—
121D652—22no—
122D652—21no—
123D652—21no—
124D652—81no—
125C22⋊D163—42no—
126C22⋊D163—42no—
127C2×Dic163—21no—
128C2×Dic163—21no—
129C2×Dic163—21no—
130C2×Dic163—42no—
131C2×Dic163C2×D411yescharacteristic
132C2×Dic163C2×D411yescharacteristic
133C4×D163—21no—
134C4×D163—21no—
135C4×D163—21no—
136C4×D163—21no—
137C4×D163—81no—
138C4×D163—81no—
139C163⋊C8—21no—
140C163⋊C8—21no—
141C23×C163—21no—
142Q8×C163—81no—
143D4×C163—21no—
144D4×C163#4112yes—
145C2×C4×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—81no—
149C2×C4×C163C2×D411yescharacteristic
150C2×C4×C163C2×D411yescharacteristic
151C2×C4×C163C2411yescharacteristic
152C8×C163—21no—
153C8×C163—21no—
154C22×C163—22no—
155C22×C163—41no—
156C22×C163—41no—
157C22×C163C22×D411yescharacteristic
158C2×D163—21no—
159C2×D163—21no—
160C2×D163—21no—
161C2×D163—21no—
162C2×D163—21no—
163C2×D163—41no—
164C2×D163—41no—
165C2×D163—41no—
166C2×D163—84no—
167C2×D163#13712yes—
168C4×C163—21no—
169C4×C163—21no—
170C4×C163—41no—
171C4×C163—41no—
172C4×C163—41no—
173C4×C163C2×#4111yescharacteristic
174C4×C163H(5,𝔽2)11yescharacteristic
175Dic163—21no—
176Dic163—21no—
177Dic163—41no—
178Dic163—41no—
179Dic163—41no—
180C2×C163—42no—
181C2×C163#54611yescharacteristic
182C2×C163C2×#13711yescharacteristic
183C2×C163#57611yescharacteristic
184D163—22no—
185D163—41no—
186D163—81no—
187C163#331411yescharacteristic, complemented, Sylow
188#3314—1631nomaximal, Sylow
189#554—1631no—
190#546—1631no—
191#489—1631no—
192#484—1631no—
193#480—1631no—
194#474D16311yescharacteristic, complemented
195#468—1631no—
196#467—1631no—
197#464—1631no—
198#452—1631no—
199#450—1631no—
200#440—1631no—
201#435—1631no—
202#417—1631no—
203C2×#102—1631no—
204C2×#41—1631no—
205C22×D4—1631no—
206C2×Q16—1631no—
207C2×SD16—1631no—
208C2×SD16—1631no—
209C2×M16—1631no—
210C22×C8—1631no—
211C4⋊Q8C2×D16311yescharacteristic, complemented
212#125—3261no—
213#124—1631no—
214#123—3261no—
215C22⋊Q8—3261no—
216C22⋊Q8—3261no—
217C22⋊Q8—1631no—
218C22⋊Q8—1631no—
219C22⋊Q8—1631no—
220#121—3261no—
221#121—1631no—
222#120—3261no—
223C4×Q8—1631no—
224C4×D4—3261no—
225C4×D4—1631no—
226C4×D4C2×D16311yescharacteristic
227C2×C4⋊C4—1631no—
228C2×C4⋊C4—1631no—
229C2×C22⋊C4—3261no—
230#107—1631no—
231#107C2×D16311yescharacteristic, complemented
232#107C2×D16311yescharacteristic, complemented
233C4⋊C8C2×D16311yescharacteristic, complemented
234#103—1631no—
235#103—1631no—
236#103—1631no—
237#103—1631no—
238#102—1631no—
239#102—1631no—
240#102—1631no—
241#102—1631no—
242#102C2×D16311yescharacteristic
243#102C2×D16311yescharacteristic
244C22⋊C8—1631no—
245C22⋊C8—1631no—
246#97—1631no—
247C24—3261no—
248#41—3262no—
249#41—6521no—
250C2×Q8—21no—
251C2×Q8—1631no—
252C2×Q8—1631no—
253C2×D4—3261no—
254C2×D4—3262no—
255C2×D4—3261no—
256C2×D4—3261no—
257C2×D4—6521no—
258C2×D4—1631no—
259C2×D4—1631no—
260C2×D4C22×D16311yescharacteristic
261C22×C4—21no—
262C22×C4—3261no—
263C22×C4—3261no—
264C22×C4—3261no—
265C22×C4—1631no—
266C22×C4—1631no—
267Q16—6521no—
268SD16—3262no—
269SD16—6521no—
270M16—6521no—
271C2×C8—6521no—
272C2×C8—1631no—
273C2×C8—1631no—
274C2×C8C22×D16311yescharacteristic
275C2×C8C22×D16311yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—3261no—
278C4⋊C4—3261no—
279C4⋊C4—3261no—
280C4⋊C4—3261no—
281C4⋊C4—3261no—
282C4⋊C4—3261no—
283C4⋊C4—1631no—
284C4⋊C4—1631no—
285C4⋊C4C22×D16311yescharacteristic
286C4⋊C4C22×D16311yescharacteristic
287C4⋊C4C22×D16311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—3261no—
290C22⋊C4—3261no—
291C22⋊C4—3261no—
292C22⋊C4—3261no—
293C22⋊C4—3261no—
294C22⋊C4—3261no—
295C22⋊C4—1631no—
296C22⋊C4—1631no—
297C42—3261no—
298C42C22×D16311yescharacteristic
299C23—21no—
300C23—13041no—
301C23—3261no—
302C23—6522no—
303C23—6522no—
304C23—1631no—
305C23—1631no—
306Q8—81no—
307Q8—3261no—
308Q8—3261no—
309Q8—3261no—
310Q8—3261no—
311D4—21no—
312D4—13041no—
313D4—3262no—
314D4—3261no—
315D4—3261no—
316D4—6522no—
317D4—6522no—
318D4#1457970012yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—81no—
323C2×C4—13041no—
324C2×C4—13041no—
325C2×C4—3261no—
326C2×C4—3261no—
327C2×C4—3261no—
328C2×C4—3261no—
329C2×C4—3261no—
330C2×C4—3261no—
331C2×C4—3261no—
332C2×C4—6522no—
333C2×C4—1631no—
334C2×C4—1631no—
335C2×C4D4×D16311yescharacteristic
336C2×C4D4×D16311yescharacteristic
337C2×C4C23×D16311yescharacteristic
338C8—21no—
339C8—21no—
340C8—3261no—
341C8—3261no—
342C22—22no—
343C22—41no—
344C22—41no—
345C22—13044no—
346C22—3261no—
347C22—3261no—
348C22—3261no—
349C22—3261no—
350C22—3261no—
351C22—6521no—
352C22—6521no—
353C22—6521no—
354C22—1632no—
355C22C2×D4×D16311yescharacteristic
356C4—21no—
357C4—21no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—3261no—
362C4—3261no—
363C4—6521no—
364C4—6521no—
365C4—6521no—
366C4C2×#1457970011yescharacteristic
367C4#1910478111yescharacteristic
368C2—42no—
369C2—13041no—
370C2—3262no—
371C2—6521no—
372C2#2378230611yescharacteristic
373C2#137×D16311yescharacteristic
374C2#2378246611yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#474×C163C2
socleC22×C163C22×D4
prime radical#23782310C2
hypercenter#474D163
hypocenterC163#3314
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #23782376C2semidirect
3. #23782027C2semidirect
4. #23782002C2semidirect
5. #23781980C2semidirect
6. #23781912C2semidirect
7. #23781907C2semidirect
8. #23781888C2semidirect
9. #23781828C2semidirect
10. #23781817C2semidirect
11. #23781765C2semidirect
12. #23781735C2semidirect
13. #23781663C2semidirect
14. #102×D163C2semidirect
15. #474×C163C2semidirect
18. C2×Dic652C22semidirect
21. C2×#14579682C22semidirect
28. #19104715C22semidirect
34. C4×Dic326C22semidirect
38. C4⋊C4×D163C22semidirect
40. #19104668C22semidirect
41. #19104660C22semidirect
42. #19104658C22semidirect
43. #19104658C22semidirect
44. #19104658C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. #19104646C22semidirect
52. C4⋊Q8×C163C22semidirect
54. #107×C163C22semidirect
55. #107×C163C22semidirect
56. C4⋊C8×C163C22semidirect
187. C163#3314semidirect
194. #474D163semidirect
211. C4⋊Q8C2×D163semidirect
231. #107C2×D163semidirect
232. #107C2×D163semidirect
233. C4⋊C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823762
#237823102
#237820272
#237820022
#237819802
#237819122
#237819072
#237818882
#237818282
#237818172
#237817652
#237817352
#237816632
#102×D1632
#474×C1632
#3314163