Finite Group Databasebeta

← #35226619

Subgroups — #35226619

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#23782372C211yesmaximal, characteristic, complemented
2#23782311C211yesmaximal, characteristic
3#23782025C211yesmaximal, characteristic, complemented
4#23782001C211yesmaximal, characteristic, complemented
5#23781979C211yesmaximal, characteristic, complemented
6#23781946C211yesmaximal, characteristic, complemented
7#23781911C211yesmaximal, characteristic, complemented
8#23781906C211yesmaximal, characteristic, complemented
9#23781888C211yesmaximal, characteristic, complemented
10#23781829C211yesmaximal, characteristic, complemented
11#23781818C211yesmaximal, characteristic, complemented
12#23781767C211yesmaximal, characteristic, complemented
13#23781667C211yesmaximal, characteristic, complemented
14C2×#19104657C211yesmaximal, characteristic, complemented
15#435×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C22×D652C2211yescharacteristic
18C2×Dic652C2211yescharacteristic, complemented
19C2×#14579688C2211yescharacteristic
20C2×#14579688C2211yescharacteristic
21#19104735C2211yescharacteristic, complemented
22#19104726—21no—
23#19104723C2211yescharacteristic
24#19104720—21no—
25#19104714C2211yescharacteristic
26#19104714C2211yescharacteristic
27#19104714C2211yescharacteristic, complemented
28#19104712—21no—
29#19104712—21no—
30#19104709C2211yescharacteristic
31#19104708—21no—
32#19104703—21no—
33C4×Dic326C2211yescharacteristic, complemented
34C4×D652C2211yescharacteristic
35D4×Dic163—21no—
36C2×C4⋊Dic163C2211yescharacteristic
37C2×C4⋊Dic163C2211yescharacteristic, complemented
38C22⋊C4×D163—21no—
39#19104669C2211yescharacteristic, complemented
40#19104669C2211yescharacteristic, complemented
41#19104669C2211yescharacteristic, complemented
42#19104660C2211yescharacteristic, complemented
43#19104660C2211yescharacteristic, complemented
44#19104660C2211yescharacteristic, complemented
45#19104660C2211yescharacteristic, complemented
46#19104657C2211yescharacteristic
47#19104657C2211yescharacteristic
48#19104657C2211yescharacteristic
49#19104657C2211yescharacteristic
50#19104657C2211yescharacteristic
51#19104657C2211yescharacteristic
52C2×M16×C163C2211yescharacteristic, complemented
53C22×C8×C163C2211yescharacteristic
54C4×D4×C163C2211yescharacteristic
55C4⋊C8×C163C2211yescharacteristic, complemented
56C22⋊C8×C163C2211yescharacteristic
57C22⋊C8×C163C2211yescharacteristic, complemented
58#97×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60#14579699—22no—
61#14579699—41no—
62C2×Dic326—21no—
63C2×Dic326C2311yescharacteristic
64C2×Dic326C2311yescharacteristic
65C2×D652—22no—
66C2×D652—21no—
67C2×D652—41no—
68C2×D652C2311yescharacteristic
69C2×D652C2311yescharacteristic
70C2×C22⋊D163—21no—
71C2×C22⋊D163—21no—
72C22×Dic163—21no—
73C22×Dic163—21no—
74C2×C4×D163—21no—
75C2×C4×D163—21no—
76Dic652—41no—
77#14579688—22no—
78#14579688—41no—
79C4⋊Dic163—21no—
80C4⋊Dic163—21no—
81C4⋊Dic163—21no—
82C4⋊Dic163C2311yescharacteristic
83C4⋊Dic163C2311yescharacteristic
84C4⋊Dic163C2311yescharacteristic
85C4⋊Dic163C2311yescharacteristic
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88Dic163⋊C4—21no—
89Dic163⋊C4—21no—
90C22⋊Dic163—21no—
91C22⋊Dic163—21no—
92C22⋊Dic163—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4—21no—
96(C2×D163)⋊C4—21no—
97C4×Dic163—21no—
98C2×D4×C163D411yescharacteristic
99C22×C4×C163C2311yescharacteristic
100C22×C4×C163C2311yescharacteristic
101M16×C163—41no—
102C2×C8×C163—41no—
103C2×C8×C163C2311yescharacteristic
104C2×C8×C163C2311yescharacteristic
105C2×C8×C163C2311yescharacteristic
106C2×C8×C163C2311yescharacteristic
107C4⋊C4×C163D411yescharacteristic
108C22⋊C4×C163D411yescharacteristic
109C22⋊C4×C163D411yescharacteristic
110C42×C163C2311yescharacteristic
111C22×D163—21no—
112C22×D163—21no—
113C22×D163—42no—
114C22×D163—42no—
115C22×D163—81no—
116Dic326—21no—
117Dic326—21no—
118Dic326—21no—
119Dic326—21no—
120Dic326—81no—
121D652—21no—
122D652—21no—
123D652—21no—
124D652—42no—
125D652#4112yes—
126C22⋊D163—42no—
127C22⋊D163—81no—
128C2×Dic163—21no—
129C2×Dic163—21no—
130C2×Dic163—21no—
131C2×Dic163—21no—
132C2×Dic163—21no—
133C2×Dic163—21no—
134C2×Dic163—81no—
135C2×Dic163—81no—
136C4×D163—42no—
137C4×D163—81no—
138C23×C163C2×D411yescharacteristic
139C23×C163C2×D411yescharacteristic
140D4×C163—22no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2×D411yescharacteristic
147C2×C4×C163C2×D411yescharacteristic
148C2×C4×C163C2×D411yescharacteristic
149C2×C4×C163C2411yescharacteristic
150C8×C163—21no—
151C8×C163—21no—
152C8×C163—21no—
153C8×C163—21no—
154C22×C163—21no—
155C22×C163—21no—
156C22×C163—21no—
157C22×C163—21no—
158C22×C163—21no—
159C22×C163#13712yes—
160C22×C163C22×D411yescharacteristic
161C2×D163—22no—
162C2×D163—41no—
163C2×D163—41no—
164C2×D163—41no—
165C2×D163—41no—
166C2×D163—41no—
167C2×D163—84no—
168C4×C163—21no—
169C4×C163—21no—
170C4×C163—21no—
171C4×C163—21no—
172C4×C163C2×#4111yescharacteristic
173C4×C163H(5,𝔽2)11yescharacteristic
174Dic163—41no—
175Dic163—41no—
176Dic163—41no—
177Dic163—41no—
178Dic163—41no—
179Dic163—41no—
180C2×C163—22no—
181C2×C163—41no—
182C2×C163#54611yescharacteristic
183C2×C163C2×#13711yescharacteristic
184C2×C163#57611yescharacteristic
185D163—42no—
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#474—1631no—
195#468—1631no—
196#467—1631no—
197#464—1631no—
198#452—1631no—
199#450—1631no—
200#440—1631no—
201#435D16311yescharacteristic, complemented
202#417—1631no—
203C2×#102—1631no—
204C2×#41—1631no—
205C22×D4—1631no—
206C2×Q16—1631no—
207C2×SD16—1631no—
208C2×SD16—1631no—
209C2×M16C2×D16311yescharacteristic, complemented
210C22×C8C2×D16311yescharacteristic
211C4⋊Q8—1631no—
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#107—1631no—
232#107—1631no—
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#102—1631no—
243#102—1631no—
244C22⋊C8C2×D16311yescharacteristic
245C22⋊C8C2×D16311yescharacteristic, complemented
246#97C2×D16311yescharacteristic, complemented
247C24—3261no—
248#41—3262no—
249#41—6521no—
250C2×Q8—3261no—
251C2×Q8—1631no—
252C2×Q8—1631no—
253C2×D4—3262no—
254C2×D4—3261no—
255C2×D4—3261no—
256C2×D4—3261no—
257C2×D4—6521no—
258C2×D4—1631no—
259C2×D4—1631no—
260C2×D4D65211yescharacteristic
261C22×C4—3261no—
262C22×C4—3261no—
263C22×C4—3261no—
264C22×C4—3261no—
265C22×C4C22×D16311yescharacteristic
266C22×C4C22×D16311yescharacteristic
267Q16—6521no—
268SD16—3262no—
269SD16—6521no—
270M16—41no—
271C2×C8—41no—
272C2×C8C22×D16311yescharacteristic
273C2×C8C22×D16311yescharacteristic
274C2×C8C22×D16311yescharacteristic
275C2×C8C22×D16311yescharacteristic
276C4⋊C4—3261no—
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⋊C4—1631no—
286C4⋊C4—1631no—
287C4⋊C4D65211yescharacteristic
288C22⋊C4—3261no—
289C22⋊C4—3261no—
290C22⋊C4—3261no—
291C22⋊C4—3261no—
292C22⋊C4—3261no—
293C22⋊C4—3261no—
294C22⋊C4—3261no—
295C22⋊C4D65211yescharacteristic
296C22⋊C4D65211yescharacteristic
297C42—3261no—
298C42C22×D16311yescharacteristic
299C23—13041no—
300C23—3261no—
301C23—3261no—
302C23—6522no—
303C23—6522no—
304C23C2×D65211yescharacteristic
305C23C2×D65211yescharacteristic
306Q8—13041no—
307Q8—3261no—
308Q8—3261no—
309Q8—3261no—
310Q8—3261no—
311D4—22no—
312D4—13041no—
313D4—3261no—
314D4—3261no—
315D4—3261no—
316D4—6522no—
317D4—6522no—
318D4—1632no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—13041no—
324C2×C4—13041no—
325C2×C4—13041no—
326C2×C4—3261no—
327C2×C4—3261no—
328C2×C4—3261no—
329C2×C4—3261no—
330C2×C4—3261no—
331C2×C4—3261no—
332C2×C4—6522no—
333C2×C4C2×D65211yescharacteristic
334C2×C4C2×D65211yescharacteristic
335C2×C4C2×D65211yescharacteristic
336C2×C4C2×D65211yescharacteristic
337C2×C4C23×D16311yescharacteristic
338C8—21no—
339C8—21no—
340C8—21no—
341C8—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—13044no—
348C22—3262no—
349C22—6521no—
350C22—6521no—
351C22—6521no—
352C22—6521no—
353C22—6521no—
354C22#1910476812yes—
355C22C22×D65211yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—6521no—
361C4—6521no—
362C4—6521no—
363C4—6521no—
364C4—6521no—
365C4—6521no—
366C4#41×D16311yescharacteristic
367C4#1910478011yescharacteristic
368C2—22no—
369C2—41no—
370C2—13041no—
371C2—6522no—
372C2#2378231011yescharacteristic
373C2C2×#1910476811yescharacteristic
374C2#2378247011yescharacteristic
subgroupG/H
centerC22C22×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#435×C163C2
socleC22×C163C22×D4
prime radical#23782311C2
hypercenter#435D163
hypocenterC163#3314
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #23782372C2semidirect
3. #23782025C2semidirect
4. #23782001C2semidirect
5. #23781979C2semidirect
6. #23781946C2semidirect
7. #23781911C2semidirect
8. #23781906C2semidirect
9. #23781888C2semidirect
10. #23781829C2semidirect
11. #23781818C2semidirect
12. #23781767C2semidirect
13. #23781667C2semidirect
14. C2×#19104657C2semidirect
15. #435×C163C2semidirect
18. C2×Dic652C22semidirect
21. #19104735C22semidirect
27. #19104714C22semidirect
33. C4×Dic326C22semidirect
37. C2×C4⋊Dic163C22semidirect
39. #19104669C22semidirect
40. #19104669C22semidirect
41. #19104669C22semidirect
42. #19104660C22semidirect
43. #19104660C22semidirect
44. #19104660C22semidirect
45. #19104660C22semidirect
52. C2×M16×C163C22semidirect
55. C4⋊C8×C163C22semidirect
57. C22⋊C8×C163C22semidirect
58. #97×C163C22semidirect
187. C163#3314semidirect
201. #435D163semidirect
209. C2×M16C2×D163semidirect
233. C4⋊C8C2×D163semidirect
245. C22⋊C8C2×D163semidirect
246. #97C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823722
#237823112
#237820252
#237820012
#237819792
#237819462
#237819112
#237819062
#237818882
#237818292
#237818182
#237817672
#237816672
C2×#191046572
#435×C1632
#3314163