Finite Group Databasebeta

← #35226055

Subgroups — #35226055

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×#19104754C211yesmaximal, characteristic, complemented
2#23782314C211yesmaximal, characteristic, complemented
3#23782309C211yesmaximal, characteristic, complemented
4#23782056C211yesmaximal, characteristic, complemented
5#23781923C211yesmaximal, characteristic, complemented
6#23781891C211yesmaximal, characteristic, complemented
7#23781878C211yesmaximal, characteristic, complemented
8#23781868C211yesmaximal, characteristic, complemented
9#23781827C211yesmaximal, characteristic, complemented
10#23781824C211yesmaximal, characteristic, complemented
11#23781816C211yesmaximal, characteristic, complemented
12#23781807C211yesmaximal, characteristic, complemented
13#23781777C211yesmaximal, characteristic, complemented
14#23781734C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C2×#14579698C2211yescharacteristic, complemented
18#19104754—24no—
19C2×Dic652C2211yescharacteristic, complemented
20C2×#14579689C2211yescharacteristic, complemented
21C2×#14579687C2211yescharacteristic
22C2×#14579687C2211yescharacteristic
23C2×#14579687C2211yescharacteristic, complemented
24C2×C8×D163C2211yescharacteristic, complemented
25#19104733C2211yescharacteristic, complemented
26#19104722C2211yescharacteristic
27#19104718—21no—
28#19104716—21no—
29#19104716—21no—
30#19104714C2211yescharacteristic
31#19104714C2211yescharacteristic
32#19104711C2211yescharacteristic, complemented
33#19104707—21no—
34#19104707—21no—
35#19104706C2211yescharacteristic
36#19104704—21no—
37D4×Dic163C2211yescharacteristic
38D4×Dic163C2211yescharacteristic, complemented
39C4×C22⋊D163C2211yescharacteristic
40C2×Dic163⋊C4—21no—
41C2×C22⋊Dic163—21no—
42#19104667C2211yescharacteristic, complemented
43Dic163⋊C8C2211yescharacteristic, complemented
44#19104660C2211yescharacteristic, complemented
45#19104660C2211yescharacteristic, complemented
46#19104659C2211yescharacteristic, complemented
47#19104656C2211yescharacteristic
48#19104656C2211yescharacteristic
49#19104656C2211yescharacteristic, complemented
50C22⋊(C163⋊C8)C2211yescharacteristic
51(C2×D163)⋊C8C2211yescharacteristic, complemented
52#19104645C2211yescharacteristic, complemented
53C2×#41×C163C2211yescharacteristic
54C22×D4×C163C2211yescharacteristic
55AGL(1,ℤ/8)×C163—24no—
56C2×SD16×C163C2211yescharacteristic
57C2×SD16×C163C2211yescharacteristic
58C2×D8×C163C2211yescharacteristic
59C2×D8×C163C2211yescharacteristic
60C2×M16×C163C2211yescharacteristic
61#14579698—22no—
62#14579698—22no—
63#14579698—41no—
64#14579698—41no—
65C2×Dic326C2311yescharacteristic
66C2×Dic326C2311yescharacteristic
67C2×C22⋊D163—21no—
68C2×C22⋊D163—21no—
69C2×C22⋊D163—21no—
70C2×C22⋊D163D411yescharacteristic
71C22×Dic163—21no—
72C22×Dic163—21no—
73C22×Dic163—21no—
74C22×Dic163—21no—
75C2×C4×D163C2311yescharacteristic
76Dic652—22no—
77#14579689—41no—
78#14579687—22no—
79#14579687—22no—
80#14579687—41no—
81C2×C163⋊C8C2311yescharacteristic
82C2×C163⋊C8C2311yescharacteristic
83C8×D163—22no—
84C4⋊Dic163C2311yescharacteristic
85C4⋊Dic163C2311yescharacteristic
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88Dic163⋊C4—21no—
89Dic163⋊C4—21no—
90Dic163⋊C4—21no—
91Dic163⋊C4D411yescharacteristic
92C22⋊Dic163—21no—
93C22⋊Dic163—21no—
94C22⋊Dic163—21no—
95C22⋊Dic163—21no—
96C22⋊Dic163—21no—
97C22⋊Dic163—21no—
98C22⋊Dic163—21no—
99C22⋊Dic163D411yescharacteristic
100(C2×D163)⋊C4—21no—
101(C2×D163)⋊C4D411yescharacteristic
102C4×Dic163C2311yescharacteristic
103C24×C163—21no—
104#41×C163—22no—
105#41×C163—41no—
106C2×Q8×C163C2311yescharacteristic
107C2×D4×C163—22no—
108C2×D4×C163—21no—
109C2×D4×C163—21no—
110C2×D4×C163—41no—
111C2×D4×C163C2311yescharacteristic
112C2×D4×C163C2311yescharacteristic
113C2×D4×C163C2311yescharacteristic
114C22×C4×C163—21no—
115C22×C4×C163C2311yescharacteristic
116SD16×C163—22no—
117SD16×C163—22no—
118D8×C163—22no—
119D8×C163D414yescomplemented
120M16×C163—22no—
121C2×C8×C163C2311yescharacteristic
122C2×C8×C163C2311yescharacteristic
123C22×D163C2×D411yescharacteristic
124Dic326—21no—
125Dic326—21no—
126Dic326—21no—
127Dic326—21no—
128C22⋊D163—41no—
129C22⋊D163—42no—
130C22⋊D163—42no—
131C22⋊D163—81no—
132C2×Dic163—21no—
133C2×Dic163—21no—
134C2×Dic163—21no—
135C2×Dic163—21no—
136C2×Dic163—42no—
137C2×Dic163—42no—
138C2×Dic163—42no—
139C2×Dic163—81no—
140C2×Dic163C2×D411yescharacteristic
141C2×Dic163C2×D411yescharacteristic
142C2×Dic163C2×D411yescharacteristic
143C4×D163—21no—
144C4×D163—21no—
145C163⋊C8—21no—
146C163⋊C8—21no—
147C23×C163—21no—
148C23×C163—21no—
149C23×C163—21no—
150C23×C163—42no—
151C23×C163—42no—
152C23×C163—81no—
153C23×C163C2×D411yescharacteristic
154Q8×C163—21no—
155Q8×C163—21no—
156D4×C163—21no—
157D4×C163—21no—
158D4×C163—21no—
159D4×C163—21no—
160D4×C163—42no—
161D4×C163—42no—
162D4×C163C2×D412yes—
163D4×C163C2×D412yes—
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×C163C2×D412yes—
172C22×C163—21no—
173C22×C163—22no—
174C22×C163—22no—
175C22×C163—21no—
176C22×C163—21no—
177C22×C163—41no—
178C22×C163—41no—
179C22×C163—41no—
180C22×C163—41no—
181C22×C163—41no—
182C22×C163—41no—
183C22×C163—41no—
184C22×C163—84no—
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—21no—
195Dic163—21no—
196Dic163—41no—
197Dic163—41no—
198Dic163—41no—
199Dic163—41no—
200C2×C163—41no—
201C2×C163—42no—
202C2×C163—42no—
203C2×C163—81no—
204C2×C163D4211yescharacteristic
205C2×C163#57511yescharacteristic
206C2×C163#57711yescharacteristic
207D163—41no—
208C163#326711yescharacteristic, complemented, Sylow
209#3267—1631nomaximal, Sylow
210C2×AGL(1,ℤ/8)D16311yescharacteristic, complemented
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#449—1631no—
223#442—1631no—
224#435—1631no—
225C2×#41—1631no—
226C2×#41—1631no—
227C2×#41C2×D16311yescharacteristic
228C22×D4C2×D16311yescharacteristic
229AGL(1,ℤ/8)—24no—
230#135—3264no—
231C2×Q16—1631no—
232C2×Q16—1631no—
233C2×SD16—1631no—
234C2×SD16—1631no—
235C2×SD16—1631no—
236C2×SD16C2×D16311yescharacteristic
237C2×SD16C2×D16311yescharacteristic
238C2×D8C2×D16311yescharacteristic
239C2×D8C2×D16311yescharacteristic
240C2×M16C2×D16311yescharacteristic
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#121—1631no—
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#103—1631no—
264#102—1631no—
265#102—1631no—
266#102—1631no—
267C22⋊C8—1631no—
268C22⋊C8—1631no—
269#97—1631no—
270C24—21no—
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—22no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4—3261no—
285C2×D4—3261no—
286C2×D4—3261no—
287C2×D4—1631no—
288C2×D4C22×D16311yescharacteristic
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—22no—
301SD16—22no—
302SD16—3262no—
303SD16—3262no—
304SD16—6521no—
305D8—22no—
306D8C22⋊D16314yescomplemented
307M16—22no—
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⋊C4—1631no—
321C22⋊C4—3261no—
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—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—1631no—
339C23D4×D16311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—3261no—
343Q8—3261no—
344Q8—3261no—
345Q8—3261no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4—42no—
352D4—13041no—
353D4—6521no—
354D4—6522no—
355D4—6522no—
356D4C2×C22⋊D16312yes—
357D4C2×C22⋊D16312yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
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—3261no—
379C8—3261no—
380C8C2×C22⋊D16312yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—3261no—
395C22—3261no—
396C22—3261no—
397C22C2×D4×D16311yescharacteristic
398C4—21no—
399C4—41no—
400C4—3261no—
401C4—3261no—
402C4—3261no—
403C4—6521no—
404C4—6521no—
405C4—6521no—
406C4—6521no—
407C4C22×C22⋊D16311yescharacteristic
408C4#1910478011yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—81no—
413C2—6521no—
414C2D4×C22⋊D16311yescharacteristic
415C2#2378246111yescharacteristic
416C2#2378247511yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×AGL(1,ℤ/8)×C163C2
socleC22×C163C22×D4
hypercenterC2×AGL(1,ℤ/8)D163
hypocenterC163#3267
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19104754C2semidirect
2. #23782314C2semidirect
3. #23782309C2semidirect
4. #23782056C2semidirect
5. #23781923C2semidirect
6. #23781891C2semidirect
7. #23781878C2semidirect
8. #23781868C2semidirect
9. #23781827C2semidirect
10. #23781824C2semidirect
11. #23781816C2semidirect
12. #23781807C2semidirect
13. #23781777C2semidirect
14. #23781734C2semidirect
15. C2×AGL(1,ℤ/8)×C163C2semidirect
17. C2×#14579698C22semidirect
19. C2×Dic652C22semidirect
20. C2×#14579689C22semidirect
23. C2×#14579687C22semidirect
24. C2×C8×D163C22semidirect
25. #19104733C22semidirect
32. #19104711C22semidirect
38. D4×Dic163C22semidirect
42. #19104667C22semidirect
43. Dic163⋊C8C22semidirect
44. #19104660C22semidirect
45. #19104660C22semidirect
46. #19104659C22semidirect
49. #19104656C22semidirect
51. (C2×D163)⋊C8C22semidirect
52. #19104645C22semidirect
119. D8×C163D4semidirect
208. C163#3267semidirect
210. C2×AGL(1,ℤ/8)D163semidirect
306. D8C22⋊D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047542
#237823142
#237823092
#237820562
#237819232
#237818912
#237818782
#237818682
#237818272
#237818242
#237818162
#237818072
#237817772
#237817342
C2×AGL(1,ℤ/8)×C1632
#3267163