Finite Group Databasebeta

← #35226342

Subgroups — #35226342

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
2D4×C22⋊D163C211yesmaximal, characteristic, complemented
3#23782010C211yesmaximal, characteristic, complemented
4#23781979C211yesmaximal, characteristic, complemented
5#23781977C211yesmaximal, characteristic, complemented
6#23781943C211yesmaximal, characteristic, complemented
7#23781899C211yesmaximal, characteristic, complemented
8#23781898C211yesmaximal, characteristic, complemented
9#23781858C211yesmaximal, characteristic, complemented
10#23781797C211yesmaximal, characteristic, complemented
11D8×Dic163C211yesmaximal, characteristic, complemented
12(C22⋊D163)⋊C8C211yesmaximal, characteristic, complemented
13C2×#19104656C211yesmaximal, characteristic, complemented
14#102×D163C211yesmaximal, characteristic, complemented
15#447×C163C211yesmaximal, characteristic, complemented
16C2×#14579698—21no—
17C22×C22⋊D163—21no—
18C2×D4×D163C2211yescharacteristic
19C2×#14579684C2211yescharacteristic, complemented
20C22×C163⋊C8C2211yescharacteristic, complemented
21C2×C8×D163C2211yescharacteristic, complemented
22#19104734C2211yescharacteristic, complemented
23#19104731C2211yescharacteristic
24#19104717—21no—
25#19104716—21no—
26#19104712—21no—
27#19104709C2211yescharacteristic
28#19104707—21no—
29#19104706C2211yescharacteristic
30#19104706C2211yescharacteristic, complemented
31#19104704—21no—
32#19104703—21no—
33D4×Dic163C2211yescharacteristic
34D4×Dic163C2211yescharacteristic, complemented
35C4×C22⋊D163C2211yescharacteristic
36C2×C4⋊Dic163C2211yescharacteristic, complemented
37C4⋊C4×D163C2211yescharacteristic, complemented
38#19104669C2211yescharacteristic, complemented
39#19104669C2211yescharacteristic, complemented
40#19104668C2211yescharacteristic, complemented
41Dic163⋊C8C2211yescharacteristic, complemented
42#19104657C2211yescharacteristic, complemented
43#19104657C2211yescharacteristic, complemented
44#19104656C2211yescharacteristic
45#19104656C2211yescharacteristic
46#19104656C2211yescharacteristic, complemented
47#19104656C2211yescharacteristic, complemented
48#19104656C2211yescharacteristic, complemented
49#19104655C2211yescharacteristic, complemented
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51C8×Dic163C2211yescharacteristic, complemented
52C22×D4×C163C2211yescharacteristic
53C2×D8×C163C2211yescharacteristic
54C2×D8×C163C2211yescharacteristic, complemented
55#121×C163C2211yescharacteristic, complemented
56#102×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic, complemented
58C22⋊C8×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60#14579698—44no—
61C2×Dic326—21no—
62C2×D652C2311yescharacteristic
63C2×C22⋊D163—21no—
64C2×C22⋊D163—21no—
65C2×C22⋊D163—21no—
66C2×C22⋊D163—21no—
67C2×C22⋊D163—21no—
68C2×C22⋊D163—21no—
69C2×C22⋊D163—44no—
70C2×C22⋊D163D411yescharacteristic
71D4×D163—22no—
72D4×D163—41no—
73C22×Dic163—21no—
74C22×Dic163—21no—
75C22×Dic163—21no—
76C2×C4×D163—21no—
77C2×C4×D163C2311yescharacteristic
78#14579684—41no—
79C2×C163⋊C8—41no—
80C2×C163⋊C8C2311yescharacteristic
81C2×C163⋊C8C2311yescharacteristic
82C8×D163—41no—
83C4⋊Dic163—21no—
84C4⋊Dic163C2311yescharacteristic
85C4⋊Dic163C2311yescharacteristic
86C4⋊Dic163C2311yescharacteristic
87Dic163⋊C4—21no—
88Dic163⋊C4—21no—
89Dic163⋊C4D411yescharacteristic
90C22⋊Dic163—21no—
91C22⋊Dic163—21no—
92C22⋊Dic163—21no—
93C22⋊Dic163—21no—
94C22⋊Dic163D411yescharacteristic
95(C2×D163)⋊C4—21no—
96(C2×D163)⋊C4D411yescharacteristic
97C4×Dic163C2311yescharacteristic
98C24×C163—21no—
99C2×D4×C163—22no—
100C2×D4×C163—21no—
101C2×D4×C163—21no—
102C2×D4×C163—41no—
103C2×D4×C163C2311yescharacteristic
104C2×D4×C163C2311yescharacteristic
105C2×D4×C163C2311yescharacteristic
106C22×C4×C163C2311yescharacteristic
107D8×C163—22no—
108D8×C163—41no—
109C2×C8×C163C2311yescharacteristic
110C2×C8×C163C2311yescharacteristic
111C4⋊C4×C163C2311yescharacteristic
112C22⋊C4×C163—21no—
113C22×D163—21no—
114C22×D163—42no—
115C22×D163—42no—
116C22×D163—81no—
117C22×D163C2×D411yescharacteristic
118Dic326—42no—
119D652—21no—
120D652—21no—
121C22⋊D163—42no—
122C22⋊D163—42no—
123C22⋊D163—42no—
124C22⋊D163—42no—
125C22⋊D163—42no—
126C22⋊D163—81no—
127C22⋊D163D814yescomplemented
128C2×Dic163—21no—
129C2×Dic163—21no—
130C2×Dic163—21no—
131C2×Dic163—42no—
132C2×Dic163—42no—
133C2×Dic163—42no—
134C2×Dic163C2×D411yescharacteristic
135C2×Dic163C2×D411yescharacteristic
136C2×Dic163C2×D411yescharacteristic
137C4×D163—21no—
138C4×D163—21no—
139C4×D163—42no—
140C163⋊C8—21no—
141C163⋊C8—21no—
142C23×C163—21no—
143C23×C163—21no—
144C23×C163—21no—
145C23×C163—42no—
146C23×C163—42no—
147C23×C163—81no—
148C23×C163C2×D411yescharacteristic
149D4×C163—21no—
150D4×C163—21no—
151D4×C163—21no—
152D4×C163—21no—
153D4×C163—21no—
154D4×C163—42no—
155D4×C163—42no—
156D4×C163#4112yescomplemented
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—22no—
167C22×C163—41no—
168C22×C163—41no—
169C22×C163—41no—
170C22×C163—41no—
171C22×C163—41no—
172C22×C163—41no—
173C22×C163—41no—
174C22×C163—41no—
175C22×C163—84no—
176C22×C163C2×D812yes—
177C22×C163C22×D411yescharacteristic
178C2×D163—21no—
179C2×D163—21no—
180C2×D163—41no—
181C2×D163—41no—
182C2×D163—41no—
183C2×D163—84no—
184C2×D163C2×D812yes—
185C4×C163—21no—
186C4×C163—41no—
187C4×C163C2×#4111yescharacteristic
188C4×C163H(5,𝔽2)11yescharacteristic
189Dic163—21no—
190Dic163—21no—
191Dic163—41no—
192Dic163—41no—
193Dic163—41no—
194Dic163C2×D812yes—
195C2×C163—22no—
196C2×C163—42no—
197C2×C163—81no—
198C2×C163—81no—
199C2×C163#54611yescharacteristic
200C2×C163C22×D811yescharacteristic
201C2×C163#57711yescharacteristic
202D163—22no—
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#466—1631no—
214#459—1631no—
215#447—1631no—
216#447D16311yescharacteristic, complemented
217C4×D8—1631no—
218C8×D4—1631no—
219C2×#102—1631no—
220C2×#102—1631no—
221C2×#41—3261no—
222C22×D4—3261no—
223C22×D4—1631no—
224C22×D4C2×D16311yescharacteristic
225C2×D8—1631no—
226C2×D8C2×D16311yescharacteristic
227C2×D8C2×D16311yescharacteristic, complemented
228C22×C8—1631no—
229C22×C8—1631no—
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#121—1631no—
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#107—1631no—
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
260#102C2×D16311yescharacteristic, complemented
261C22⋊C8—1631no—
262C22⋊C8C2×D16311yescharacteristic, complemented
263C4×C8—1631no—
264C24—21no—
265C24—3261no—
266#41—6524no—
267C2×Q8—3261no—
268C2×D4—22no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—3262no—
273C2×D4—3261no—
274C2×D4—3261no—
275C2×D4—3261no—
276C2×D4—3261no—
277C2×D4—3261no—
278C2×D4—3261no—
279C2×D4—6521no—
280C2×D4—6524no—
281C2×D4—1631no—
282C2×D4—1631no—
283C2×D4C22×D16311yescharacteristic
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—22no—
293D8—41no—
294D8—6521no—
295C2×C8—6521no—
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⋊C4—1631no—
308C4⋊C4C22×D16311yescharacteristic
309C22⋊C4—21no—
310C22⋊C4—3261no—
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—21no—
321C23—42no—
322C23—42no—
323C23—81no—
324C23—13041no—
325C23—3261no—
326C23—6522no—
327C23—6522no—
328C23—1631no—
329C23D4×D16311yescharacteristic
330Q8—6522no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—42no—
338D4—13041no—
339D4—3261no—
340D4—3261no—
341D4—6522no—
342D4—6522no—
343D4—6522no—
344D4—6522no—
345D4—6522no—
346D4—1634no—
347D4#1457969812yescomplemented
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—3261no—
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—22no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—84no—
381C22—13044no—
382C22—3261no—
383C22—3261no—
384C22—6521no—
385C22—6521no—
386C22—6521no—
387C22—1632no—
388C22D8×D16312yes—
389C22C2×D4×D16311yescharacteristic
390C4—21no—
391C4—41no—
392C4—3261no—
393C4—3261no—
394C4—6521no—
395C4—6521no—
396C4—6521no—
397C4—1632no—
398C4C2×#1457969811yescharacteristic
399C4#1910478011yescharacteristic
400C2—22no—
401C2—42no—
402C2—81no—
403C2—81no—
404C2—13041no—
405C2—3262no—
406C2#2378230211yescharacteristic
407C2C2×D8×D16311yescharacteristic
408C2#2378247511yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#447×C163C2
socleC22×C163C22×D4
hypercenter#447D163
hypocenterC163#3294
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #23782315C2semidirect
2. D4×C22⋊D163C2semidirect
3. #23782010C2semidirect
4. #23781979C2semidirect
5. #23781977C2semidirect
6. #23781943C2semidirect
7. #23781899C2semidirect
8. #23781898C2semidirect
9. #23781858C2semidirect
10. #23781797C2semidirect
11. D8×Dic163C2semidirect
12. (C22⋊D163)⋊C8C2semidirect
13. C2×#19104656C2semidirect
14. #102×D163C2semidirect
15. #447×C163C2semidirect
19. C2×#14579684C22semidirect
20. C22×C163⋊C8C22semidirect
21. C2×C8×D163C22semidirect
22. #19104734C22semidirect
30. #19104706C22semidirect
34. D4×Dic163C22semidirect
36. C2×C4⋊Dic163C22semidirect
37. C4⋊C4×D163C22semidirect
38. #19104669C22semidirect
39. #19104669C22semidirect
40. #19104668C22semidirect
41. Dic163⋊C8C22semidirect
42. #19104657C22semidirect
43. #19104657C22semidirect
46. #19104656C22semidirect
47. #19104656C22semidirect
48. #19104656C22semidirect
49. #19104655C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. C8×Dic163C22semidirect
54. C2×D8×C163C22semidirect
55. #121×C163C22semidirect
57. #102×C163C22semidirect
58. C22⋊C8×C163C22semidirect
127. C22⋊D163D8semidirect
156. D4×C163#41semidirect
204. C163#3294semidirect
216. #447D163semidirect
227. C2×D8C2×D163semidirect
239. #121C2×D163semidirect
260. #102C2×D163semidirect
262. C22⋊C8C2×D163semidirect
347. D4#14579698semidirect

Maximal subgroups

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

subgroupindex
#237823152
D4×C22⋊D1632
#237820102
#237819792
#237819772
#237819432
#237818992
#237818982
#237818582
#237817972
D8×Dic1632
(C22⋊D163)⋊C82
C2×#191046562
#102×D1632
#447×C1632
#3294163