Finite Group Databasebeta

← #35226069

Subgroups — #35226069

Order 20864.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104762C211yesmaximal, characteristic, complemented
2C2×#19104755C211yesmaximal, characteristic, complemented
3#23782334C211yesmaximal, characteristic, complemented
4#23782312C211yesmaximal, characteristic, complemented
5#23782053C211yesmaximal, characteristic, complemented
6#23781910C211yesmaximal, characteristic, complemented
7#23781899C211yesmaximal, characteristic, complemented
8#23781885C211yesmaximal, characteristic, complemented
9#23781859C211yesmaximal, characteristic, complemented
10#23781830C211yesmaximal, characteristic, complemented
11#23781812C211yesmaximal, characteristic, complemented
12#23781797C211yesmaximal, characteristic, complemented
13#23781768C211yesmaximal, characteristic, complemented
14#23781733C211yesmaximal, characteristic, complemented
15#449×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579699C2211yescharacteristic
18C2×D4×D163C2211yescharacteristic
19#19104762—24no—
20#19104755—24no—
21C2×#14579689C2211yescharacteristic, complemented
22C2×#14579688C2211yescharacteristic, complemented
23C2×#14579687C2211yescharacteristic, complemented
24C2×#14579686C2211yescharacteristic
25C2×D1304C2211yescharacteristic
26C2×#14579684C2211yescharacteristic
27C2×#14579684C2211yescharacteristic
28C2×#14579682C2211yescharacteristic
29C22×C163⋊C8C2211yescharacteristic, complemented
30#19104731—21no—
31#19104731C2211yescharacteristic, complemented
32#19104722C2211yescharacteristic
33#19104717—21no—
34#19104715C2211yescharacteristic
35#19104709—21no—
36#19104709C2211yescharacteristic, complemented
37#19104706—21no—
38#19104706—21no—
39#19104706C2211yescharacteristic
40#19104703—21no—
41Q8×Dic163C2211yescharacteristic, complemented
42#19104697C2211yescharacteristic
43C4×C22⋊D163—21no—
44C4×C22⋊D163C2211yescharacteristic
45C22⋊C4×D163—21no—
46#19104668C2211yescharacteristic, complemented
47Dic163⋊C8C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104657C2211yescharacteristic
50#19104656C2211yescharacteristic
51#19104655C2211yescharacteristic, complemented
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#14579700—22no—
63#14579700—41no—
64#14579699—22no—
65#14579699—41no—
66C2×Dic326C2311yescharacteristic
67C2×D652—21no—
68C2×D652C2311yescharacteristic
69C2×D652C2311yescharacteristic
70C2×C22⋊D163—21no—
71C2×C22⋊D163—21no—
72C2×C22⋊D163—21no—
73C2×C22⋊D163—21no—
74C2×C22⋊D163—21no—
75C2×C22⋊D163—21no—
76C2×C22⋊D163D411yescharacteristic
77D4×D163—22no—
78D4×D163—41no—
79C2×C4×D163—21no—
80C2×C4×D163—21no—
81C2×C4×D163—21no—
82C2×C4×D163C2311yescharacteristic
83#14579689—22no—
84#14579688—41no—
85#14579687—22no—
86#14579686D414yescomplemented
87D1304—22no—
88#14579684—22no—
89#14579684—22no—
90#14579682—22no—
91C2×C163⋊C8—22no—
92C2×C163⋊C8C2311yescharacteristic
93C2×C163⋊C8C2311yescharacteristic
94C4⋊Dic163—21no—
95C4⋊Dic163—21no—
96C4⋊Dic163C2311yescharacteristic
97Dic163⋊C4D411yescharacteristic
98C22⋊Dic163—21no—
99C22⋊Dic163—21no—
100C22⋊Dic163—21no—
101C22⋊Dic163D411yescharacteristic
102(C2×D163)⋊C4—21no—
103(C2×D163)⋊C4—21no—
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4D411yescharacteristic
106C4×Dic163—21no—
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—22no—
118D8×C163—41no—
119C2×C8×C163C2311yescharacteristic
120C2×C8×C163C2311yescharacteristic
121C4⋊C4×C163C2311yescharacteristic
122C22⋊C4×C163—21no—
123C22×D163—21no—
124C22×D163—21no—
125C22×D163—42no—
126C22×D163—42no—
127C22×D163—81no—
128C22×D163C2×D411yescharacteristic
129Dic326—21no—
130Dic326—21no—
131D652—21no—
132D652—21no—
133D652—21no—
134D652—42no—
135D652C2×D412yes—
136C22⋊D163—41no—
137C22⋊D163—42no—
138C22⋊D163—42no—
139C22⋊D163—42no—
140C22⋊D163—81no—
141C22⋊D163—81no—
142C22⋊D163—81no—
143C2×Dic163—21no—
144C2×Dic163—21no—
145C2×Dic163C2×D411yescharacteristic
146C2×Dic163C2×D411yescharacteristic
147C2×Dic163C2×D411yescharacteristic
148C4×D163—21no—
149C4×D163—21no—
150C4×D163—42no—
151C4×D163—42no—
152C4×D163—81no—
153C163⋊C8—21no—
154C163⋊C8C2×D412yes—
155C23×C163—21no—
156C23×C163—21no—
157C23×C163C2×D411yescharacteristic
158Q8×C163—21no—
159Q8×C163C2×D412yes—
160D4×C163—21no—
161D4×C163—21no—
162D4×C163—21no—
163D4×C163—21no—
164D4×C163—42no—
165D4×C163—81no—
166C2×C4×C163—21no—
167C2×C4×C163—21no—
168C2×C4×C163—21no—
169C2×C4×C163—21no—
170C2×C4×C163—42no—
171C2×C4×C163C2×D411yescharacteristic
172C2×C4×C163C2411yescharacteristic
173C8×C163—21no—
174C8×C163—21no—
175C22×C163—21no—
176C22×C163—21no—
177C22×C163—21no—
178C22×C163—41no—
179C22×C163—41no—
180C22×C163—41no—
181C22×C163—41no—
182C22×C163—41no—
183C22×C163—41no—
184C22×C163C22×D411yescharacteristic
185C2×D163—21no—
186C2×D163—22no—
187C2×D163—21no—
188C2×D163—21no—
189C2×D163—41no—
190C2×D163—41no—
191C2×D163—41no—
192C2×D163—41no—
193C2×D163—41no—
194C2×D163—84no—
195C4×C163—21no—
196C4×C163—22no—
197C4×C163—41no—
198C4×C163C22×D411yescharacteristic
199C4×C163H(5,𝔽2)11yescharacteristic
200Dic163—21no—
201Dic163—21no—
202Dic163—21no—
203Dic163—41no—
204Dic163—41no—
205C2×C163—41no—
206C2×C163—81no—
207C2×C163—81no—
208C2×C163D4211yescharacteristic
209C2×C163#57511yescharacteristic
210C2×C163#57611yescharacteristic
211D163—41no—
212D163—42no—
213D163—81no—
214C163#326811yescharacteristic, complemented, Sylow
215#3268—1631nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1631no—
217C2×#135—1631no—
218#550—1631no—
219#546—1631no—
220#496—1631no—
221#468—1631no—
222#466—1631no—
223#464—1631no—
224#459—1631no—
225#452—1631no—
226#449—1631no—
227#449D16311yescharacteristic, complemented
228#447—1631no—
229#440—1631no—
230#435—1631no—
231C2×#41—1631no—
232C2×#41—1631no—
233C2×#41C2×D16311yescharacteristic, complemented
234C22×D4—1631no—
235AGL(1,ℤ/8)—3264no—
236#135—3264no—
237C2×Q16—1631no—
238C2×SD16—1631no—
239C2×SD16—1631no—
240C2×SD16—1631no—
241C2×SD16C2×D16311yescharacteristic
242C2×D8—1631no—
243C2×D8—1631no—
244C2×D8—1631no—
245C2×D8C2×D16311yescharacteristic
246C2×M16—1631no—
247C22×C8—1631no—
248#127—3261no—
249#127—1631no—
250#124—1631no—
251#123—3261no—
252C22⋊Q8—1631no—
253#121—3261no—
254#121—3261no—
255#121—3261no—
256#121—1631no—
257#121—1631no—
258#121C2×D16311yescharacteristic
259#120—3261no—
260C4×Q8—1631no—
261C4×D4—3261no—
262C4×D4—1631no—
263C4×D4—1631no—
264C2×C22⋊C4—3261no—
265#107—1631no—
266C4⋊C8—1631no—
267#103—1631no—
268#103C2×D16311yescharacteristic, complemented
269#102—1631no—
270#102—1631no—
271#102—1631no—
272#102C2×D16311yescharacteristic, complemented
273C22⋊C8—1631no—
274C22⋊C8C2×D16311yescharacteristic, complemented
275#97—1631no—
276C24—3261no—
277#41—22no—
278#41—41no—
279#41—3262no—
280#41—3262no—
281#41—6521no—
282#41—6521no—
283C2×Q8—1631no—
284C2×Q8C22×D16311yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—3261no—
288C2×D4—3262no—
289C2×D4—3261no—
290C2×D4—3261no—
291C2×D4—3261no—
292C2×D4—3261no—
293C2×D4—3261no—
294C2×D4—3261no—
295C2×D4—6521no—
296C2×D4—1631no—
297C2×D4—1631no—
298C2×D4—1631no—
299C2×D4C22×D16311yescharacteristic
300C2×D4C22×D16311yescharacteristic
301C22×C4—21no—
302C22×C4—3261no—
303C22×C4—3261no—
304C22×C4—3261no—
305C22×C4—1631no—
306C22×C4C22×D16311yescharacteristic
307Q16—3262no—
308SD16—22no—
309SD16—3262no—
310SD16—6521no—
311SD16—1634no—
312D8—41no—
313D8—3262no—
314D8—3262no—
315D8—3262no—
316M16—3262no—
317C2×C8—3262no—
318C2×C8—1631no—
319C2×C8—1631no—
320C2×C8C22×D16311yescharacteristic
321C2×C8C22×D16311yescharacteristic
322C4⋊C4—3261no—
323C4⋊C4—3261no—
324C4⋊C4—1631no—
325C4⋊C4—1631no—
326C4⋊C4C22×D16311yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—3261no—
329C22⋊C4—3261no—
330C22⋊C4—3261no—
331C22⋊C4—3261no—
332C22⋊C4—3261no—
333C22⋊C4—3261no—
334C22⋊C4—1631no—
335C22⋊C4—1631no—
336C42—3261no—
337C42—1631no—
338C23—21no—
339C23—21no—
340C23—13041no—
341C23—3261no—
342C23—3261no—
343C23—6522no—
344C23—6522no—
345C23—1631no—
346C23D4×D16311yescharacteristic
347Q8—21no—
348Q8—3261no—
349Q8—3261no—
350Q8D4×D16312yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—42no—
356D4—81no—
357D4—13041no—
358D4—13041no—
359D4—13041no—
360D4—3261no—
361D4—3261no—
362D4—3261no—
363D4—6522no—
364D4—6521no—
365D4—6522no—
366D4—6522no—
367D4—6522no—
368D4—1632no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—13041no—
375C2×C4—3261no—
376C2×C4—3261no—
377C2×C4—3261no—
378C2×C4—3261no—
379C2×C4—6522no—
380C2×C4—6522no—
381C2×C4—1631no—
382C2×C4—1631no—
383C2×C4—1631no—
384C2×C4D4×D16311yescharacteristic
385C2×C4C23×D16311yescharacteristic
386C8—21no—
387C8—21no—
388C8—3261no—
389C8—1632no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—13044no—
400C22—3261no—
401C22—3261no—
402C22—3261no—
403C22—3262no—
404C22—6521no—
405C22—6521no—
406C22—6521no—
407C22—6521no—
408C22—6521no—
409C22C2×D4×D16311yescharacteristic
410C4—21no—
411C4—22no—
412C4—41no—
413C4—3261no—
414C4—3261no—
415C4—3261no—
416C4—6521no—
417C4—6521no—
418C4C2×D4×D16311yescharacteristic
419C4#1910478011yescharacteristic
420C2—41no—
421C2—81no—
422C2—81no—
423C2—13041no—
424C2—6521no—
425C2—6522no—
426C2#2378229711yescharacteristic
427C2#2378246211yescharacteristic
428C2#2378246911yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#449×C163C2
socleC22×C163C22×D4
hypercenter#449D163
hypocenterC163#3268
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×#19104762C2semidirect
2. C2×#19104755C2semidirect
3. #23782334C2semidirect
4. #23782312C2semidirect
5. #23782053C2semidirect
6. #23781910C2semidirect
7. #23781899C2semidirect
8. #23781885C2semidirect
9. #23781859C2semidirect
10. #23781830C2semidirect
11. #23781812C2semidirect
12. #23781797C2semidirect
13. #23781768C2semidirect
14. #23781733C2semidirect
15. #449×C163C2semidirect
21. C2×#14579689C22semidirect
22. C2×#14579688C22semidirect
23. C2×#14579687C22semidirect
29. C22×C163⋊C8C22semidirect
31. #19104731C22semidirect
36. #19104709C22semidirect
41. Q8×Dic163C22semidirect
46. #19104668C22semidirect
47. Dic163⋊C8C22semidirect
48. #19104660C22semidirect
51. #19104655C22semidirect
53. #19104645C22semidirect
54. C2×#41×C163C22semidirect
58. #103×C163C22semidirect
59. #102×C163C22semidirect
60. C22⋊C8×C163C22semidirect
86. #14579686D4semidirect
214. C163#3268semidirect
227. #449D163semidirect
233. C2×#41C2×D163semidirect
268. #103C2×D163semidirect
272. #102C2×D163semidirect
274. C22⋊C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047622
C2×#191047552
#237823342
#237823122
#237820532
#237819102
#237818992
#237818852
#237818592
#237818302
#237818122
#237817972
#237817682
#237817332
#449×C1632
#3268163