Finite Group Databasebeta

← #35226344

Subgroups — #35226344

Order 20864.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782315C211yesmaximal, characteristic, complemented
2#23782299C211yesmaximal, characteristic, complemented
3#23782010C211yesmaximal, characteristic, complemented
4#23781980C211yesmaximal, characteristic, complemented
5#23781976C211yesmaximal, characteristic, complemented
6#23781944C211yesmaximal, characteristic, complemented
7#23781900C211yesmaximal, characteristic, complemented
8#23781858C211yesmaximal, characteristic, complemented
9#23781798C211yesmaximal, characteristic, complemented
10#23781797C211yesmaximal, characteristic, complemented
11#23781745C211yesmaximal, characteristic, complemented
12C8×C22⋊D163C211yesmaximal, characteristic, complemented
13C2×#19104657C211yesmaximal, characteristic, complemented
14#102×D163C211yesmaximal, characteristic, complemented
15#466×C163C211yesmaximal, characteristic, complemented
16C2×#14579698—21no—
17C22×D652C2211yescharacteristic
18C2×D4×D163—21no—
19C2×D4×D163C2211yescharacteristic
20C2×#14579684C2211yescharacteristic
21C2×#14579684C2211yescharacteristic, complemented
22C2×C8×D163C2211yescharacteristic, complemented
23#19104734C2211yescharacteristic, complemented
24#19104731C2211yescharacteristic
25#19104717—21no—
26#19104716—21no—
27#19104712—21no—
28#19104710C2211yescharacteristic
29#19104708—21no—
30#19104706C2211yescharacteristic, complemented
31#19104705—21no—
32#19104703—21no—
33D4×Dic163C2211yescharacteristic, complemented
34#19104697C2211yescharacteristic
35C4×C22⋊D163C2211yescharacteristic
36C2×C4⋊Dic163C2211yescharacteristic, complemented
37C4⋊C4×D163C2211yescharacteristic, complemented
38#19104668C2211yescharacteristic, complemented
39#19104668C2211yescharacteristic, complemented
40Dic163⋊C8C2211yescharacteristic, complemented
41#19104657C2211yescharacteristic
42#19104657C2211yescharacteristic
43#19104657C2211yescharacteristic, complemented
44#19104657C2211yescharacteristic, complemented
45#19104656C2211yescharacteristic, complemented
46#19104656C2211yescharacteristic, complemented
47#19104655C2211yescharacteristic
48#19104655C2211yescharacteristic, complemented
49C22⋊(C163⋊C8)C2211yescharacteristic, complemented
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51C8×Dic163C2211yescharacteristic, complemented
52C2×D8×C163C2211yescharacteristic, complemented
53C22×C8×C163C2211yescharacteristic, complemented
54#121×C163C2211yescharacteristic
55#121×C163C2211yescharacteristic, complemented
56#107×C163C2211yescharacteristic, complemented
57#102×C163C2211yescharacteristic, complemented
58#102×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60C23×D163—21no—
61#14579698—44no—
62C2×Dic326—21no—
63C2×D652—22no—
64C2×D652—21no—
65C2×D652—21no—
66C2×D652—41no—
67C2×D652C2311yescharacteristic
68C2×D652C2311yescharacteristic
69C2×C22⋊D163—21no—
70C2×C22⋊D163—21no—
71C2×C22⋊D163—21no—
72C2×C22⋊D163—21no—
73C2×C22⋊D163D411yescharacteristic
74D4×D163—22no—
75D4×D163—41no—
76D4×D163—44no—
77C22×Dic163—21no—
78C22×Dic163—21no—
79C2×C4×D163—21no—
80C2×C4×D163—21no—
81C2×C4×D163C2311yescharacteristic
82#14579684—22no—
83#14579684—41no—
84C2×C163⋊C8C2311yescharacteristic
85C2×C163⋊C8C2311yescharacteristic
86C8×D163—41no—
87C4⋊Dic163—21no—
88C4⋊Dic163C2311yescharacteristic
89C4⋊Dic163C2311yescharacteristic
90Dic163⋊C4—21no—
91Dic163⋊C4—21no—
92Dic163⋊C4D411yescharacteristic
93C22⋊Dic163—21no—
94C22⋊Dic163—21no—
95C22⋊Dic163D411yescharacteristic
96(C2×D163)⋊C4—21no—
97(C2×D163)⋊C4—21no—
98(C2×D163)⋊C4D411yescharacteristic
99C4×Dic163C2311yescharacteristic
100C2×D4×C163—21no—
101C2×D4×C163—21no—
102C2×D4×C163C2311yescharacteristic
103C2×D4×C163C2311yescharacteristic
104C22×C4×C163C2311yescharacteristic
105D8×C163—41no—
106C2×C8×C163—41no—
107C2×C8×C163C2311yescharacteristic
108C2×C8×C163C2311yescharacteristic
109C4⋊C4×C163C2311yescharacteristic
110C4⋊C4×C163C2311yescharacteristic
111C22⋊C4×C163—21no—
112C22⋊C4×C163—21no—
113C22×D163—21no—
114C22×D163—21no—
115C22×D163—42no—
116C22×D163—42no—
117C22×D163—42no—
118C22×D163—42no—
119C22×D163—81no—
120C22×D163—81no—
121C22×D163C2×D411yescharacteristic
122Dic326—42no—
123D652—21no—
124D652—21no—
125D652—21no—
126D652—42no—
127D652—42no—
128D652#4112yescomplemented
129C22⋊D163—42no—
130C22⋊D163—42no—
131C22⋊D163—42no—
132C22⋊D163—81no—
133C22⋊D163D814yescomplemented
134C2×Dic163—21no—
135C2×Dic163—21no—
136C2×Dic163—42no—
137C2×Dic163—42no—
138C2×Dic163C2×D411yescharacteristic
139C2×Dic163C2×D411yescharacteristic
140C2×Dic163C2×D411yescharacteristic
141C4×D163—21no—
142C4×D163—21no—
143C4×D163—42no—
144C4×D163—42no—
145C163⋊C8—21no—
146C163⋊C8—21no—
147C23×C163—21no—
148C23×C163—21no—
149C23×C163C2×D411yescharacteristic
150D4×C163—21no—
151D4×C163—21no—
152D4×C163—21no—
153D4×C163—21no—
154D4×C163—42no—
155D4×C163—42no—
156C2×C4×C163—21no—
157C2×C4×C163—21no—
158C2×C4×C163—21no—
159C2×C4×C163—21no—
160C2×C4×C163C2×D411yescharacteristic
161C2×C4×C163C2411yescharacteristic
162C8×C163—21no—
163C8×C163—21no—
164C22×C163—21no—
165C22×C163—21no—
166C22×C163—41no—
167C22×C163—41no—
168C22×C163—41no—
169C22×C163—41no—
170C22×C163—41no—
171C22×C163—41no—
172C22×C163C2×D812yes—
173C22×C163C22×D411yescharacteristic
174C2×D163—21no—
175C2×D163—21no—
176C2×D163—22no—
177C2×D163—41no—
178C2×D163—41no—
179C2×D163—41no—
180C2×D163—41no—
181C2×D163—41no—
182C2×D163—84no—
183C2×D163—84no—
184C2×D163C2×D812yes—
185C4×C163—21no—
186C4×C163—41no—
187C4×C163—41no—
188C4×C163C2×#4111yescharacteristic
189C4×C163H(5,𝔽2)11yescharacteristic
190Dic163—21no—
191Dic163—21no—
192Dic163—41no—
193Dic163—41no—
194Dic163C2×D812yes—
195C2×C163—22no—
196C2×C163—81no—
197C2×C163—81no—
198C2×C163#54611yescharacteristic
199C2×C163C22×D811yescharacteristic
200C2×C163#57711yescharacteristic
201D163—22no—
202D163—42no—
203D163—81no—
204C163#329411yescharacteristic, complemented, Sylow
205#3294—1631nomaximal, Sylow
206#547—1631no—
207D42—1631no—
208#486—1631no—
209#480—1631no—
210#480—1631no—
211#474—1631no—
212#466—1631no—
213#466D16311yescharacteristic, complemented
214#459—1631no—
215#447—1631no—
216#447—1631no—
217C4×D8—1631no—
218C8×D4—1631no—
219C2×#102—1631no—
220C2×#102—1631no—
221C2×#41—3261no—
222C22×D4—3261no—
223C22×D4—1631no—
224C22×D4—1631no—
225C2×D8—1631no—
226C2×D8—1631no—
227C2×D8C2×D16311yescharacteristic, complemented
228C22×C8—1631no—
229C22×C8C2×D16311yescharacteristic, complemented
230C4⋊Q8—1631no—
231#127—1631no—
232#123—3261no—
233#123—3261no—
234C22⋊Q8—3261no—
235#121—3261no—
236#121—1631no—
237#121—1631no—
238#121C2×D16311yescharacteristic
239#121C2×D16311yescharacteristic, complemented
240#120—3261no—
241#120—3261no—
242C4×D4—1631no—
243C4×D4—1631no—
244C4×D4—1631no—
245C2×C4⋊C4—1631no—
246C2×C4⋊C4—1631no—
247#107—1631no—
248#107—1631no—
249#107C2×D16311yescharacteristic, complemented
250C4⋊C8—1631no—
251#102—1631no—
252#102—1631no—
253#102—1631no—
254#102—1631no—
255#102—1631no—
256#102—1631no—
257#102—1631no—
258#102—1631no—
259#102C2×D16311yescharacteristic, complemented
260#102C2×D16311yescharacteristic, complemented
261C22⋊C8—1631no—
262C22⋊C8—1631no—
263C4×C8—1631no—
264C24—3261no—
265C24—3261no—
266#41—6524no—
267C2×Q8—3261no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—3262no—
271C2×D4—3261no—
272C2×D4—3262no—
273C2×D4—3261no—
274C2×D4—3261no—
275C2×D4—3261no—
276C2×D4—3261no—
277C2×D4—3261no—
278C2×D4—6521no—
279C2×D4—6521no—
280C2×D4—6524no—
281C2×D4—1631no—
282C2×D4—1631no—
283C2×D4—1631no—
284C2×D4C22×D16311yescharacteristic
285C2×D4C22×D16311yescharacteristic
286C22×C4—3261no—
287C22×C4—3261no—
288C22×C4—3261no—
289C22×C4—3261no—
290C22×C4—1631no—
291C22×C4C22×D16311yescharacteristic
292D8—41no—
293D8—3262no—
294D8—6521no—
295C2×C8—41no—
296C2×C8—6521no—
297C2×C8—1631no—
298C2×C8—1631no—
299C2×C8C22×D16311yescharacteristic
300C2×C8C22×D16311yescharacteristic
301C4⋊C4—3261no—
302C4⋊C4—3261no—
303C4⋊C4—3261no—
304C4⋊C4—1631no—
305C4⋊C4—1631no—
306C4⋊C4—1631no—
307C4⋊C4C22×D16311yescharacteristic
308C4⋊C4C22×D16311yescharacteristic
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—3261no—
312C22⋊C4—3261no—
313C22⋊C4—3261no—
314C22⋊C4—3261no—
315C22⋊C4—1631no—
316C22⋊C4—1631no—
317C42—1631no—
318C23—21no—
319C23—21no—
320C23—13041no—
321C23—13041no—
322C23—3261no—
323C23—3261no—
324C23—6522no—
325C23—6522no—
326C23—6522no—
327C23—6522no—
328C23—1631no—
329C23D4×D16311yescharacteristic
330Q8—6522no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—21no—
335D4—42no—
336D4—42no—
337D4—13041no—
338D4—3261no—
339D4—3261no—
340D4—3261no—
341D4—6522no—
342D4—6522no—
343D4—6522no—
344D4—6522no—
345D4—6522no—
346D4—1634no—
347D4—1632no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—3261no—
353C2×C4—3261no—
354C2×C4—3261no—
355C2×C4—3261no—
356C2×C4—6522no—
357C2×C4—6522no—
358C2×C4—6522no—
359C2×C4—6522no—
360C2×C4—1631no—
361C2×C4—1631no—
362C2×C4—1631no—
363C2×C4D4×D16311yescharacteristic
364C2×C4C23×D16311yescharacteristic
365C8—21no—
366C8—21no—
367C8—3261no—
368C8—3261no—
369C22—21no—
370C22—21no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—13044no—
378C22—13044no—
379C22—3261no—
380C22—3261no—
381C22—3262no—
382C22—6521no—
383C22—6521no—
384C22—6521no—
385C22—6521no—
386C22—6521no—
387C22—1632no—
388C22D8×D16312yes—
389C22C2×D4×D16311yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—3261no—
394C4—3261no—
395C4—6521no—
396C4—6521no—
397C4—1632no—
398C4#41×D16311yescharacteristic
399C4#1910478011yescharacteristic
400C2—22no—
401C2—81no—
402C2—81no—
403C2—13041no—
404C2—3262no—
405C2—6522no—
406C2#2378230411yescharacteristic
407C2C2×D8×D16311yescharacteristic
408C2#2378247211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#466×C163C2
socleC22×C163C22×D4
hypercenter#466D163
hypocenterC163#3294
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #23782315C2semidirect
2. #23782299C2semidirect
3. #23782010C2semidirect
4. #23781980C2semidirect
5. #23781976C2semidirect
6. #23781944C2semidirect
7. #23781900C2semidirect
8. #23781858C2semidirect
9. #23781798C2semidirect
10. #23781797C2semidirect
11. #23781745C2semidirect
12. C8×C22⋊D163C2semidirect
13. C2×#19104657C2semidirect
14. #102×D163C2semidirect
15. #466×C163C2semidirect
21. C2×#14579684C22semidirect
22. C2×C8×D163C22semidirect
23. #19104734C22semidirect
30. #19104706C22semidirect
33. D4×Dic163C22semidirect
36. C2×C4⋊Dic163C22semidirect
37. C4⋊C4×D163C22semidirect
38. #19104668C22semidirect
39. #19104668C22semidirect
40. Dic163⋊C8C22semidirect
43. #19104657C22semidirect
44. #19104657C22semidirect
45. #19104656C22semidirect
46. #19104656C22semidirect
48. #19104655C22semidirect
49. C22⋊(C163⋊C8)C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. C8×Dic163C22semidirect
52. C2×D8×C163C22semidirect
53. C22×C8×C163C22semidirect
55. #121×C163C22semidirect
56. #107×C163C22semidirect
57. #102×C163C22semidirect
58. #102×C163C22semidirect
128. D652#41semidirect
133. C22⋊D163D8semidirect
204. C163#3294semidirect
213. #466D163semidirect
227. C2×D8C2×D163semidirect
229. C22×C8C2×D163semidirect
239. #121C2×D163semidirect
249. #107C2×D163semidirect
259. #102C2×D163semidirect
260. #102C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823152
#237822992
#237820102
#237819802
#237819762
#237819442
#237819002
#237818582
#237817982
#237817972
#237817452
C8×C22⋊D1632
C2×#191046572
#102×D1632
#466×C1632
#3294163