Finite Group Databasebeta

← #35226247

Subgroups — #35226247

Order 20864.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104754C211yesmaximal, characteristic, complemented
2C22×#14579687C211yesmaximal, characteristic, complemented
3#23782321C211yesmaximal, characteristic, complemented
4#23782309C211yesmaximal, characteristic, complemented
5#23782048C211yesmaximal, characteristic, complemented
6#23781893C211yesmaximal, characteristic, complemented
7#23781892C211yesmaximal, characteristic, complemented
8#23781886C211yesmaximal, characteristic, complemented
9#23781878C211yesmaximal, characteristic, complemented
10#23781828C211yesmaximal, characteristic, complemented
11#23781823C211yesmaximal, characteristic, complemented
12#23781807C211yesmaximal, characteristic, complemented
13#23781756C211yesmaximal, characteristic, complemented
14(C22⋊D163)⋊C8C211yesmaximal, characteristic, complemented
15#447×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C2×#14579698C2211yescharacteristic, complemented
18C22×Dic326C2211yescharacteristic
19#19104754—24no—
20C2×Dic652C2211yescharacteristic, complemented
21C2×Dic652C2211yescharacteristic, complemented
22C2×#14579687—24no—
23C2×#14579687C2211yescharacteristic
24C2×#14579687C2211yescharacteristic
25C2×#14579687C2211yescharacteristic
26C2×#14579687C2211yescharacteristic
27C2×#14579687C2211yescharacteristic, complemented
28C22×C163⋊C8C2211yescharacteristic
29C2×C8×D163C2211yescharacteristic, complemented
30#19104722C2211yescharacteristic
31#19104722C2211yescharacteristic, complemented
32#19104721—21no—
33#19104716—21no—
34#19104715C2211yescharacteristic, complemented
35#19104714C2211yescharacteristic
36#19104712—21no—
37#19104707—21no—
38#19104707—21no—
39#19104706C2211yescharacteristic
40#19104704—21no—
41#19104700C2211yescharacteristic, complemented
42D4×Dic163—21no—
43D4×Dic163C2211yescharacteristic
44C4×C22⋊D163C2211yescharacteristic
45C2×C22⋊Dic163—21no—
46#19104666C2211yescharacteristic, complemented
47Dic163⋊C8C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104660C2211yescharacteristic, complemented
50#19104658C2211yescharacteristic, complemented
51#19104656C2211yescharacteristic, complemented
52(C2×D163)⋊C8C2211yescharacteristic, complemented
53C8×Dic163C2211yescharacteristic, complemented
54C22×D4×C163C2211yescharacteristic
55C2×D8×C163C2211yescharacteristic
56C2×D8×C163C2211yescharacteristic
57#121×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59#102×C163C2211yescharacteristic
60C22⋊C8×C163C2211yescharacteristic
61#14579698—22no—
62#14579698—22no—
63#14579698—41no—
64#14579698—41no—
65C2×Dic326—22no—
66C2×Dic326—21no—
67C2×Dic326—41no—
68C2×Dic326C2311yescharacteristic
69C2×Dic326C2311yescharacteristic
70C2×Dic326C2311yescharacteristic
71C2×C22⋊D163—21no—
72C2×C22⋊D163—21no—
73C2×C22⋊D163—21no—
74C2×C22⋊D163D411yescharacteristic
75C22×Dic163—21no—
76C22×Dic163—21no—
77C22×Dic163—21no—
78C22×Dic163—21no—
79C2×C4×D163C2311yescharacteristic
80Dic652—22no—
81Dic652—41no—
82#14579687—22no—
83#14579687—22no—
84#14579687—22no—
85#14579687—22no—
86#14579687D414yescomplemented
87C2×C163⋊C8—22no—
88C2×C163⋊C8C2311yescharacteristic
89C2×C163⋊C8C2311yescharacteristic
90C8×D163—22no—
91C4⋊Dic163—21no—
92C4⋊Dic163C2311yescharacteristic
93Dic163⋊C4—21no—
94Dic163⋊C4—21no—
95Dic163⋊C4D411yescharacteristic
96C22⋊Dic163—21no—
97C22⋊Dic163—21no—
98C22⋊Dic163—21no—
99C22⋊Dic163—21no—
100C22⋊Dic163—21no—
101C22⋊Dic163—21no—
102C22⋊Dic163—21no—
103C22⋊Dic163D411yescharacteristic
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4D411yescharacteristic
106C4×Dic163—21no—
107C4×Dic163C2311yescharacteristic
108C24×C163—21no—
109C2×D4×C163—22no—
110C2×D4×C163—21no—
111C2×D4×C163—21no—
112C2×D4×C163—41no—
113C2×D4×C163C2311yescharacteristic
114C2×D4×C163C2311yescharacteristic
115C2×D4×C163C2311yescharacteristic
116C22×C4×C163C2311yescharacteristic
117D8×C163—22no—
118D8×C163—41no—
119C2×C8×C163C2311yescharacteristic
120C2×C8×C163C2311yescharacteristic
121C4⋊C4×C163C2311yescharacteristic
122C22⋊C4×C163—21no—
123C22×D163C2×D411yescharacteristic
124Dic326—21no—
125Dic326—21no—
126Dic326—21no—
127Dic326—21no—
128Dic326—21no—
129Dic326—42no—
130Dic326C2×D412yes—
131C22⋊D163—22no—
132C22⋊D163—42no—
133C22⋊D163—42no—
134C22⋊D163—81no—
135C2×Dic163—21no—
136C2×Dic163—21no—
137C2×Dic163—21no—
138C2×Dic163—21no—
139C2×Dic163—42no—
140C2×Dic163—42no—
141C2×Dic163—42no—
142C2×Dic163—81no—
143C2×Dic163C2×D411yescharacteristic
144C2×Dic163C2×D411yescharacteristic
145C2×Dic163C2×D411yescharacteristic
146C4×D163—21no—
147C4×D163—21no—
148C163⋊C8—21no—
149C163⋊C8C2×D412yes—
150C23×C163—21no—
151C23×C163—21no—
152C23×C163—21no—
153C23×C163—42no—
154C23×C163—42no—
155C23×C163—81no—
156C23×C163C2×D411yescharacteristic
157D4×C163—21no—
158D4×C163—21no—
159D4×C163—21no—
160D4×C163—21no—
161D4×C163—21no—
162D4×C163—42no—
163D4×C163—42no—
164D4×C163C2×D412yes—
165C2×C4×C163—21no—
166C2×C4×C163—21no—
167C2×C4×C163—21no—
168C2×C4×C163C2×D411yescharacteristic
169C2×C4×C163C2411yescharacteristic
170C8×C163—21no—
171C8×C163—21no—
172C22×C163—21no—
173C22×C163—21no—
174C22×C163—22no—
175C22×C163—41no—
176C22×C163—41no—
177C22×C163—41no—
178C22×C163—41no—
179C22×C163—41no—
180C22×C163—41no—
181C22×C163—41no—
182C22×C163—41no—
183C22×C163—84no—
184C22×C163#13512yescomplemented
185C22×C163C22×D411yescharacteristic
186C2×D163—21no—
187C2×D163—21no—
188C2×D163—21no—
189C4×C163—21no—
190C4×C163—41no—
191C4×C163C22×D411yescharacteristic
192C4×C163H(5,𝔽2)11yescharacteristic
193Dic163—21no—
194Dic163—22no—
195Dic163—21no—
196Dic163—21no—
197Dic163—41no—
198Dic163—41no—
199Dic163—41no—
200C2×C163—22no—
201C2×C163—42no—
202C2×C163—81no—
203C2×C163—81no—
204C2×C163D4211yescharacteristic
205C2×C163C2×#13511yescharacteristic
206C2×C163#57711yescharacteristic
207D163—41no—
208C163#328211yescharacteristic, complemented, Sylow
209#3282—1631nomaximal, Sylow
210C2×#135—1631no—
211C22×SD16—1631no—
212#548—1631no—
213#546—1631no—
214#495—1631no—
215#465—1631no—
216#465—1631no—
217#464—1631no—
218#463—1631no—
219#452—1631no—
220#451—1631no—
221#449—1631no—
222#447D16311yescharacteristic, complemented
223C4×SD16—1631no—
224C8×D4—1631no—
225C2×#41—1631no—
226C2×#41—1631no—
227C22×Q8—1631no—
228C22×D4C2×D16311yescharacteristic
229#135—3264no—
230C2×Q16—1631no—
231C2×Q16—1631no—
232C2×SD16—3264no—
233C2×SD16—1631no—
234C2×SD16—1631no—
235C2×SD16—1631no—
236C2×SD16—1631no—
237C2×SD16—1631no—
238C2×D8C2×D16311yescharacteristic
239C2×D8C2×D16311yescharacteristic
240C22×C8—1631no—
241C22×C8—1631no—
242#124—3261no—
243#124—1631no—
244#124—1631no—
245#123—3261no—
246C22⋊Q8—3261no—
247C22⋊Q8—1631no—
248C22⋊Q8—1631no—
249#121—3261no—
250#121—3261no—
251#121—1631no—
252#121C2×D16311yescharacteristic
253#120—3261no—
254C4×Q8—1631no—
255C4×D4—3261no—
256C4×D4—1631no—
257C4×D4—1631no—
258C2×C22⋊C4—3261no—
259#106—1631no—
260C4⋊C8—1631no—
261#103—1631no—
262#103—1631no—
263#103—1631no—
264#102—1631no—
265#102C2×D16311yescharacteristic
266#102C2×D16311yescharacteristic
267C22⋊C8—1631no—
268C22⋊C8C2×D16311yescharacteristic
269C4×C8—1631no—
270C24—21no—
271#41—3262no—
272#41—3262no—
273#41—6521no—
274#41—6521no—
275C2×Q8—3262no—
276C2×Q8—3261no—
277C2×Q8—6521no—
278C2×Q8—1631no—
279C2×Q8—1631no—
280C2×Q8—1631no—
281C2×D4—22no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—3261no—
286C2×D4—3261no—
287C2×D4—3261no—
288C2×D4—1631no—
289C2×D4C22×D16311yescharacteristic
290C2×D4C22×D16311yescharacteristic
291C2×D4C22×D16311yescharacteristic
292C22×C4—3261no—
293C22×C4—3261no—
294C22×C4—3261no—
295C22×C4—3261no—
296C22×C4—1631no—
297C22×C4C22×D16311yescharacteristic
298Q16—3262no—
299Q16—6521no—
300SD16—3262no—
301SD16—3262no—
302SD16—3262no—
303SD16—3262no—
304SD16—1634no—
305D8—22no—
306D8—41no—
307C2×C8—3262no—
308C2×C8—3262no—
309C2×C8—1631no—
310C2×C8—1631no—
311C2×C8C22×D16311yescharacteristic
312C2×C8C22×D16311yescharacteristic
313C4⋊C4—3261no—
314C4⋊C4—3261no—
315C4⋊C4—3261no—
316C4⋊C4—1631no—
317C4⋊C4—1631no—
318C4⋊C4C22×D16311yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—3261no—
321C22⋊C4—3261no—
322C22⋊C4—3261no—
323C22⋊C4—3261no—
324C22⋊C4—3261no—
325C22⋊C4—3261no—
326C22⋊C4—3261no—
327C22⋊C4—3261no—
328C22⋊C4—1631no—
329C22⋊C4—1631no—
330C42—3261no—
331C42—1631no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—1631no—
339C23D4×D16311yescharacteristic
340Q8—3261no—
341Q8—3261no—
342Q8—3261no—
343Q8—3261no—
344Q8—3261no—
345Q8—6522no—
346Q8—1632no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—21no—
352D4—42no—
353D4—42no—
354D4—13041no—
355D4—3262no—
356D4—6522no—
357D4—6522no—
358D4D4×D16312yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—13041no—
363C2×C4—3261no—
364C2×C4—3261no—
365C2×C4—3261no—
366C2×C4—3261no—
367C2×C4—3261no—
368C2×C4—3261no—
369C2×C4—6522no—
370C2×C4—6522no—
371C2×C4—6522no—
372C2×C4—1631no—
373C2×C4—1631no—
374C2×C4—1631no—
375C2×C4D4×D16311yescharacteristic
376C2×C4C23×D16311yescharacteristic
377C8—21no—
378C8—21no—
379C8—3261no—
380C8—1632no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—84no—
393C22—3261no—
394C22—3261no—
395C22—3261no—
396C22#1910475412yescomplemented
397C22C2×D4×D16311yescharacteristic
398C4—21no—
399C4—41no—
400C4—3261no—
401C4—3261no—
402C4—3261no—
403C4—3262no—
404C4—6521no—
405C4—6521no—
406C4—6521no—
407C4C2×D4×D16311yescharacteristic
408C4#1910478011yescharacteristic
409C2—22no—
410C2—42no—
411C2—81no—
412C2—81no—
413C2—6521no—
414C2#2378229711yescharacteristic
415C2C2×#1910475411yescharacteristic
416C2#2378247511yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#447×C163C2
socleC22×C163C22×D4
hypercenter#447D163
hypocenterC163#3282
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. C2×#19104754C2semidirect
2. C22×#14579687C2semidirect
3. #23782321C2semidirect
4. #23782309C2semidirect
5. #23782048C2semidirect
6. #23781893C2semidirect
7. #23781892C2semidirect
8. #23781886C2semidirect
9. #23781878C2semidirect
10. #23781828C2semidirect
11. #23781823C2semidirect
12. #23781807C2semidirect
13. #23781756C2semidirect
14. (C22⋊D163)⋊C8C2semidirect
15. #447×C163C2semidirect
17. C2×#14579698C22semidirect
20. C2×Dic652C22semidirect
21. C2×Dic652C22semidirect
27. C2×#14579687C22semidirect
29. C2×C8×D163C22semidirect
31. #19104722C22semidirect
34. #19104715C22semidirect
41. #19104700C22semidirect
46. #19104666C22semidirect
47. Dic163⋊C8C22semidirect
48. #19104660C22semidirect
49. #19104660C22semidirect
50. #19104658C22semidirect
51. #19104656C22semidirect
52. (C2×D163)⋊C8C22semidirect
53. C8×Dic163C22semidirect
86. #14579687D4semidirect
184. C22×C163#135semidirect
208. C163#3282semidirect
222. #447D163semidirect
396. C22#19104754semidirect

Maximal subgroups

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

subgroupindex
C2×#191047542
C22×#145796872
#237823212
#237823092
#237820482
#237818932
#237818922
#237818862
#237818782
#237818282
#237818232
#237818072
#237817562
(C22⋊D163)⋊C82
#447×C1632
#3282163