Finite Group Databasebeta

← #35226054

Subgroups — #35226054

Order 20864.

500 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×#19104759C211yesmaximal, characteristic, complemented
2C2×#19104755C211yesmaximal, characteristic, complemented
3#23782314C211yesmaximal, characteristic, complemented
4#23782312C211yesmaximal, characteristic, complemented
5#23782056C211yesmaximal, characteristic, complemented
6#23781922C211yesmaximal, characteristic, complemented
7#23781892C211yesmaximal, characteristic, complemented
8#23781878C211yesmaximal, characteristic, complemented
9#23781869C211yesmaximal, characteristic, complemented
10#23781829C211yesmaximal, characteristic, complemented
11#23781828C211yesmaximal, characteristic, complemented
12#23781814C211yesmaximal, characteristic, complemented
13#23781774C211yesmaximal, characteristic, complemented
14#23781733C211yesmaximal, characteristic, complemented
15#449×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C2×#14579698C2211yescharacteristic
18C2×D4×D163C2211yescharacteristic
19#19104759—24no—
20#19104755—24no—
21C2×Dic652C2211yescharacteristic, complemented
22C2×#14579689C2211yescharacteristic, complemented
23C2×#14579688C2211yescharacteristic
24C2×#14579687C2211yescharacteristic
25C2×#14579687C2211yescharacteristic, complemented
26C2×#14579686C2211yescharacteristic
27C2×#14579684C2211yescharacteristic
28C2×#14579682C2211yescharacteristic
29C22×C163⋊C8C2211yescharacteristic, complemented
30#19104733C2211yescharacteristic, complemented
31#19104722C2211yescharacteristic
32#19104718—21no—
33#19104717—21no—
34#19104716—21no—
35#19104715C2211yescharacteristic
36#19104714C2211yescharacteristic, complemented
37#19104711C2211yescharacteristic
38#19104707—21no—
39#19104706—21no—
40#19104703—21no—
41D4×Dic163C2211yescharacteristic, complemented
42#19104697C2211yescharacteristic
43C4×C22⋊D163C2211yescharacteristic
44C2×Dic163⋊C4—21no—
45C22⋊C4×D163—21no—
46#19104666C2211yescharacteristic, complemented
47Dic163⋊C8C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104658C2211yescharacteristic, complemented
50#19104657C2211yescharacteristic
51#19104656C2211yescharacteristic
52(C2×D163)⋊C8C2211yescharacteristic
53#19104645C2211yescharacteristic, complemented
54C2×#41×C163C2211yescharacteristic, complemented
55C2×SD16×C163C2211yescharacteristic
56C2×D8×C163C2211yescharacteristic
57#121×C163C2211yescharacteristic
58#103×C163C2211yescharacteristic, complemented
59#102×C163C2211yescharacteristic, complemented
60C22⋊C8×C163C2211yescharacteristic, complemented
61C23×D163—21no—
62#14579699—22no—
63#14579699—41no—
64#14579698—22no—
65#14579698—41no—
66C2×Dic326C2311yescharacteristic
67C2×Dic326C2311yescharacteristic
68C2×D652C2311yescharacteristic
69C2×C22⋊D163—21no—
70C2×C22⋊D163—21no—
71C2×C22⋊D163—21no—
72C2×C22⋊D163D411yescharacteristic
73D4×D163—22no—
74D4×D163—41no—
75C22×Dic163—21no—
76C22×Dic163—21no—
77C2×C4×D163—21no—
78C2×C4×D163—21no—
79C2×C4×D163C2311yescharacteristic
80Dic652—41no—
81#14579689—22no—
82#14579688—22no—
83#14579687—22no—
84#14579687—22no—
85#14579686—22no—
86#14579684D414yescomplemented
87#14579682—22no—
88C2×C163⋊C8—22no—
89C2×C163⋊C8C2311yescharacteristic
90C2×C163⋊C8C2311yescharacteristic
91C4⋊Dic163—21no—
92C4⋊Dic163C2311yescharacteristic
93Dic163⋊C4—21no—
94Dic163⋊C4—21no—
95Dic163⋊C4—21no—
96Dic163⋊C4—21no—
97Dic163⋊C4D411yescharacteristic
98C22⋊Dic163—21no—
99C22⋊Dic163—21no—
100C22⋊Dic163—21no—
101C22⋊Dic163—21no—
102C22⋊Dic163D411yescharacteristic
103(C2×D163)⋊C4—21no—
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4—21no—
106(C2×D163)⋊C4D411yescharacteristic
107C4×Dic163C2311yescharacteristic
108#41×C163—22no—
109#41×C163—41no—
110C2×Q8×C163C2311yescharacteristic
111C2×D4×C163—21no—
112C2×D4×C163—21no—
113C2×D4×C163C2311yescharacteristic
114C2×D4×C163C2311yescharacteristic
115C22×C4×C163—21no—
116C22×C4×C163C2311yescharacteristic
117SD16×C163—41no—
118D8×C163—22no—
119C2×C8×C163C2311yescharacteristic
120C2×C8×C163C2311yescharacteristic
121C4⋊C4×C163C2311yescharacteristic
122C22⋊C4×C163—21no—
123C22×D163—21no—
124C22×D163—42no—
125C22×D163—42no—
126C22×D163—81no—
127C22×D163C2×D411yescharacteristic
128Dic326—21no—
129Dic326—21no—
130Dic326—21no—
131Dic326—21no—
132D652—21no—
133D652C2×D412yes—
134C22⋊D163—41no—
135C22⋊D163—42no—
136C22⋊D163—42no—
137C22⋊D163—42no—
138C2×Dic163—21no—
139C2×Dic163—21no—
140C2×Dic163—21no—
141C2×Dic163—42no—
142C2×Dic163—42no—
143C2×Dic163C2×D411yescharacteristic
144C2×Dic163C2×D411yescharacteristic
145C2×Dic163C2×D411yescharacteristic
146C4×D163—21no—
147C4×D163—21no—
148C4×D163—42no—
149C4×D163—81no—
150C163⋊C8—21no—
151C163⋊C8C2×D412yes—
152C23×C163—21no—
153C23×C163—21no—
154C23×C163C2×D411yescharacteristic
155Q8×C163—21no—
156Q8×C163—21no—
157D4×C163—21no—
158D4×C163—21no—
159D4×C163—21no—
160D4×C163—42no—
161D4×C163—81no—
162D4×C163C2×D412yes—
163C2×C4×C163—21no—
164C2×C4×C163—21no—
165C2×C4×C163—21no—
166C2×C4×C163—21no—
167C2×C4×C163—42no—
168C2×C4×C163C2×D411yescharacteristic
169C2×C4×C163C2411yescharacteristic
170C8×C163—21no—
171C8×C163—21no—
172C22×C163—21no—
173C22×C163—22no—
174C22×C163—21no—
175C22×C163—21no—
176C22×C163—41no—
177C22×C163—41no—
178C22×C163—41no—
179C22×C163—41no—
180C22×C163—41no—
181C22×C163C22×D411yescharacteristic
182C2×D163—21no—
183C2×D163—22no—
184C2×D163—21no—
185C2×D163—21no—
186C2×D163—41no—
187C2×D163—41no—
188C2×D163—84no—
189C4×C163—21no—
190C4×C163—41no—
191C4×C163—41no—
192C4×C163C22×D411yescharacteristic
193C4×C163H(5,𝔽2)11yescharacteristic
194Dic163—21no—
195Dic163—21no—
196Dic163—21no—
197Dic163—41no—
198Dic163—41no—
199Dic163—41no—
200C2×C163—41no—
201C2×C163—42no—
202C2×C163—81no—
203C2×C163D4211yescharacteristic
204C2×C163#57511yescharacteristic
205C2×C163#57711yescharacteristic
206D163—41no—
207D163—42no—
208C163#326711yescharacteristic, complemented, Sylow
209#3267—1631nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1631no—
211C2×#135—1631no—
212#547—1631no—
213#546—1631no—
214#497—1631no—
215#470—1631no—
216#465—1631no—
217#463—1631no—
218#461—1631no—
219#452—1631no—
220#452—1631no—
221#450—1631no—
222#449D16311yescharacteristic, complemented
223#442—1631no—
224#435—1631no—
225C2×#41—1631no—
226C2×#41—1631no—
227C2×#41C2×D16311yescharacteristic, complemented
228C22×D4—1631no—
229AGL(1,ℤ/8)—3264no—
230#135—3264no—
231C2×Q16—1631no—
232C2×Q16—1631no—
233C2×SD16—1631no—
234C2×SD16—1631no—
235C2×SD16—1631no—
236C2×SD16—1631no—
237C2×SD16C2×D16311yescharacteristic
238C2×D8—1631no—
239C2×D8C2×D16311yescharacteristic
240C2×M16—1631no—
241C22×C8—1631no—
242C4⋊Q8—1631no—
243#124—1631no—
244#123—3261no—
245#123—3261no—
246#123—3261no—
247C22⋊Q8—1631no—
248C22⋊Q8—1631no—
249C22⋊Q8—1631no—
250#121—3261no—
251#121—3261no—
252#121C2×D16311yescharacteristic
253#120—3261no—
254C4×D4—1631no—
255C4×D4—1631no—
256C4×D4—1631no—
257C2×C4⋊C4—3261no—
258C2×C22⋊C4—3261no—
259#106—1631no—
260C4⋊C8—1631no—
261#103—1631no—
262#103—1631no—
263#103C2×D16311yescharacteristic, complemented
264#102—1631no—
265#102—1631no—
266#102C2×D16311yescharacteristic, complemented
267C22⋊C8—1631no—
268C22⋊C8C2×D16311yescharacteristic, complemented
269#97—1631no—
270C24—3261no—
271#41—22no—
272#41—41no—
273#41—3262no—
274#41—3262no—
275#41—6521no—
276#41—6521no—
277C2×Q8—1631no—
278C2×Q8—1631no—
279C2×Q8C22×D16311yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—3262no—
283C2×D4—3261no—
284C2×D4—3261no—
285C2×D4—3261no—
286C2×D4—6521no—
287C2×D4—1631no—
288C2×D4—1631no—
289C2×D4C22×D16311yescharacteristic
290C2×D4C22×D16311yescharacteristic
291C22×C4—21no—
292C22×C4—3261no—
293C22×C4—3261no—
294C22×C4—3261no—
295C22×C4—3261no—
296C22×C4—1631no—
297C22×C4C22×D16311yescharacteristic
298Q16—3262no—
299Q16—6521no—
300SD16—41no—
301SD16—3262no—
302SD16—3262no—
303SD16—3262no—
304SD16—3262no—
305D8—22no—
306D8—1634no—
307M16—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—3261no—
317C4⋊C4—3261no—
318C4⋊C4—1631no—
319C4⋊C4—1631no—
320C4⋊C4C22×D16311yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—3261no—
323C22⋊C4—3261no—
324C22⋊C4—3261no—
325C22⋊C4—3261no—
326C22⋊C4—3261no—
327C22⋊C4—3261no—
328C22⋊C4—3261no—
329C22⋊C4—1631no—
330C22⋊C4—1631no—
331C42—1631no—
332C23—21no—
333C23—21no—
334C23—13041no—
335C23—3261no—
336C23—6522no—
337C23—6522no—
338C23—1631no—
339C23D4×D16311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—3261no—
343Q8—3261no—
344Q8—3261no—
345Q8—3261no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—81no—
351D4—3261no—
352D4—6521no—
353D4—6522no—
354D4—6522no—
355D4—6522no—
356D4—1632no—
357D4D4×D16312yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—13041no—
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—21no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—13044no—
391C22—3261no—
392C22—3261no—
393C22—3261no—
394C22—3262no—
395C22—6521no—
396C22—6521no—
397C22C2×D4×D16311yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—3261no—
402C4—3261no—
403C4—3261no—
404C4—6521no—
405C4—6521no—
406C4—6521no—
407C4C2×D4×D16311yescharacteristic
408C4#1910478011yescharacteristic
409C2—41no—
410C2—42no—
411C2—81no—
412C2—6521no—
413C2—6522no—
414C2#2378229711yescharacteristic
415C2#2378246211yescharacteristic
416C2#2378247511yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#449×C163C2
socleC22×C163C22×D4
hypercenter#449D163
hypocenterC163#3267
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#19104759C2semidirect
2. C2×#19104755C2semidirect
3. #23782314C2semidirect
4. #23782312C2semidirect
5. #23782056C2semidirect
6. #23781922C2semidirect
7. #23781892C2semidirect
8. #23781878C2semidirect
9. #23781869C2semidirect
10. #23781829C2semidirect
11. #23781828C2semidirect
12. #23781814C2semidirect
13. #23781774C2semidirect
14. #23781733C2semidirect
15. #449×C163C2semidirect
21. C2×Dic652C22semidirect
22. C2×#14579689C22semidirect
25. C2×#14579687C22semidirect
29. C22×C163⋊C8C22semidirect
30. #19104733C22semidirect
36. #19104714C22semidirect
41. D4×Dic163C22semidirect
46. #19104666C22semidirect
47. Dic163⋊C8C22semidirect
48. #19104660C22semidirect
49. #19104658C22semidirect
53. #19104645C22semidirect
54. C2×#41×C163C22semidirect
58. #103×C163C22semidirect
59. #102×C163C22semidirect
60. C22⋊C8×C163C22semidirect
86. #14579684D4semidirect
208. C163#3267semidirect
222. #449D163semidirect
227. C2×#41C2×D163semidirect
263. #103C2×D163semidirect
266. #102C2×D163semidirect
268. C22⋊C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047592
C2×#191047552
#237823142
#237823122
#237820562
#237819222
#237818922
#237818782
#237818692
#237818292
#237818282
#237818142
#237817742
#237817332
#449×C1632
#3267163