Finite Group Databasebeta

← #35226062

Subgroups — #35226062

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×#19104760C211yesmaximal, characteristic, complemented
2C2×#19104755C211yesmaximal, characteristic, complemented
3#23782311C211yesmaximal, characteristic, complemented
4#23782057C211yesmaximal, characteristic, complemented
5#23781921C211yesmaximal, characteristic, complemented
6#23781896C211yesmaximal, characteristic, complemented
7#23781880C211yesmaximal, characteristic, complemented
8#23781867C211yesmaximal, characteristic, complemented
9#23781827C211yesmaximal, characteristic, complemented
10#23781825C211yesmaximal, characteristic, complemented
11#23781815C211yesmaximal, characteristic, complemented
12#23781810C211yesmaximal, characteristic, complemented
13#23781775C211yesmaximal, characteristic, complemented
14#23781737C211yesmaximal, characteristic, complemented
15#547×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C22×D652C2211yescharacteristic
18#19104760—24no—
19#19104755—24no—
20C2×#14579689C2211yescharacteristic, complemented
21C2×#14579689C2211yescharacteristic, complemented
22C2×#14579687C2211yescharacteristic, complemented
23C2×#14579686C2211yescharacteristic
24C2×#14579686C2211yescharacteristic
25C2×#14579686C2211yescharacteristic
26C2×#14579686C2211yescharacteristic
27C2×#14579684C2211yescharacteristic
28C2×#14579684C2211yescharacteristic
29C2×#14579683C2211yescharacteristic
30C22×C163⋊C8C2211yescharacteristic, complemented
31#19104723C2211yescharacteristic
32#19104720—21no—
33#19104714C2211yescharacteristic
34#19104709C2211yescharacteristic
35#19104708—21no—
36#19104703—21no—
37C4×D652C2211yescharacteristic
38C22⋊C4×D163—21no—
39#19104666C2211yescharacteristic, complemented
40C4⋊(C163⋊C8)C2211yescharacteristic, complemented
41#19104659C2211yescharacteristic, complemented
42#19104659C2211yescharacteristic, complemented
43#19104658C2211yescharacteristic, complemented
44#19104656C2211yescharacteristic, complemented
45#19104655C2211yescharacteristic
46#19104655C2211yescharacteristic
47C22⋊(C163⋊C8)C2211yescharacteristic
48C22⋊(C163⋊C8)C2211yescharacteristic, complemented
49#19104646C2211yescharacteristic, complemented
50C2×#41×C163C2211yescharacteristic
51C2×#41×C163C2211yescharacteristic, complemented
52C4⋊Q8×C163C2211yescharacteristic, complemented
53#123×C163—21no—
54#123×C163—21no—
55C22⋊Q8×C163C2211yescharacteristic
56C22⋊Q8×C163C2211yescharacteristic, complemented
57#121×C163—21no—
58C4×D4×C163C2211yescharacteristic
59C4×D4×C163C2211yescharacteristic, complemented
60C2×C4⋊C4×C163—21no—
61C23×D163—21no—
62#14579699—22no—
63#14579699—41no—
64C2×Dic326C2311yescharacteristic
65C2×D652—22no—
66C2×D652—21no—
67C2×D652—41no—
68C2×D652C2311yescharacteristic
69C2×D652C2311yescharacteristic
70C2×C22⋊D163—21no—
71C2×C22⋊D163—21no—
72C2×C4×D163—21no—
73C2×C4×D163—21no—
74#14579689—22no—
75#14579689—41no—
76#14579687—22no—
77#14579686—22no—
78#14579686—22no—
79#14579686—22no—
80#14579686—41no—
81#14579684—22no—
82#14579684D414yescomplemented
83#14579683—22no—
84C2×C163⋊C8—22no—
85C2×C163⋊C8C2311yescharacteristic
86C2×C163⋊C8C2311yescharacteristic
87C2×C163⋊C8C2311yescharacteristic
88C2×C163⋊C8C2311yescharacteristic
89C4⋊Dic163C2311yescharacteristic
90Dic163⋊C4—21no—
91C22⋊Dic163—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4—21no—
96#41×C163—22no—
97#41×C163—22no—
98#41×C163—41no—
99#41×C163—41no—
100C2×Q8×C163C2311yescharacteristic
101C2×Q8×C163C2311yescharacteristic
102C2×D4×C163—21no—
103C2×D4×C163—21no—
104C2×D4×C163D411yescharacteristic
105C2×D4×C163C2311yescharacteristic
106C22×C4×C163—21no—
107C22×C4×C163—21no—
108C22×C4×C163—21no—
109C22×C4×C163C2311yescharacteristic
110C22×C4×C163C2311yescharacteristic
111C4⋊C4×C163—21no—
112C4⋊C4×C163—21no—
113C4⋊C4×C163—21no—
114C4⋊C4×C163—21no—
115C4⋊C4×C163D411yescharacteristic
116C4⋊C4×C163C2311yescharacteristic
117C22⋊C4×C163—21no—
118C22⋊C4×C163—21no—
119C22⋊C4×C163—21no—
120C22⋊C4×C163D411yescharacteristic
121C22⋊C4×C163D411yescharacteristic
122C42×C163C2311yescharacteristic
123C22×D163—21no—
124C22×D163—21no—
125C22×D163—42no—
126C22×D163—42no—
127C22×D163—81no—
128Dic326—21no—
129Dic326—21no—
130D652—21no—
131D652—21no—
132D652—21no—
133D652—42no—
134D652C2×D412yes—
135C22⋊D163—42no—
136C22⋊D163—81no—
137C2×Dic163—21no—
138C2×Dic163—21no—
139C4×D163—42no—
140C4×D163—81no—
141C163⋊C8—21no—
142C163⋊C8—21no—
143C163⋊C8—21no—
144C163⋊C8C2×D412yes—
145C23×C163—21no—
146C23×C163C2×D411yescharacteristic
147C23×C163C2×D411yescharacteristic
148Q8×C163—21no—
149Q8×C163—21no—
150Q8×C163—21no—
151Q8×C163—21no—
152D4×C163—21no—
153D4×C163—41no—
154D4×C163—42no—
155D4×C163—42no—
156D4×C163C2×D412yes—
157C2×C4×C163—21no—
158C2×C4×C163—21no—
159C2×C4×C163—21no—
160C2×C4×C163—21no—
161C2×C4×C163—21no—
162C2×C4×C163—21no—
163C2×C4×C163—21no—
164C2×C4×C163—42no—
165C2×C4×C163—42no—
166C2×C4×C163—42no—
167C2×C4×C163C2×D411yescharacteristic
168C2×C4×C163C2×D411yescharacteristic
169C2×C4×C163C2×D411yescharacteristic
170C2×C4×C163C2×D411yescharacteristic
171C2×C4×C163C2411yescharacteristic
172C22×C163—21no—
173C22×C163—21no—
174C22×C163—22no—
175C22×C163—21no—
176C22×C163—21no—
177C22×C163—21no—
178C22×C163—21no—
179C22×C163—41no—
180C22×C163—41no—
181C22×C163C22×D411yescharacteristic
182C2×D163—22no—
183C2×D163—41no—
184C2×D163—41no—
185C2×D163—41no—
186C2×D163—41no—
187C2×D163—41no—
188C2×D163—84no—
189C4×C163—21no—
190C4×C163—21no—
191C4×C163—21no—
192C4×C163—21no—
193C4×C163—41no—
194C4×C163—41no—
195C4×C163—41no—
196C4×C163C22×D411yescharacteristic
197C4×C163H(5,𝔽2)11yescharacteristic
198Dic163—41no—
199Dic163—41no—
200C2×C163—41no—
201C2×C163—41no—
202C2×C163—42no—
203C2×C163D4211yescharacteristic
204C2×C163#57511yescharacteristic
205C2×C163#57711yescharacteristic
206D163—42no—
207D163—81no—
208C163#326711yescharacteristic, complemented, Sylow
209#3267—1631nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1631no—
211C2×#135—1631no—
212#547D16311yescharacteristic, complemented
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×#41C2×D16311yescharacteristic
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×SD16—1631no—
238C2×D8—1631no—
239C2×D8—1631no—
240C2×M16—1631no—
241C22×C8—1631no—
242C4⋊Q8C2×D16311yescharacteristic, complemented
243#124—1631no—
244#123—21no—
245#123—21no—
246#123—3261no—
247C22⋊Q8—1631no—
248C22⋊Q8C2×D16311yescharacteristic
249C22⋊Q8C2×D16311yescharacteristic, complemented
250#121—21no—
251#121—3261no—
252#121—1631no—
253#120—3261no—
254C4×D4—1631no—
255C4×D4C2×D16311yescharacteristic
256C4×D4C2×D16311yescharacteristic, complemented
257C2×C4⋊C4—21no—
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—3261no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—3262no—
276#41—6521no—
277C2×Q8—1631no—
278C2×Q8C22×D16311yescharacteristic
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—21no—
293C22×C4—21no—
294C22×C4—3261no—
295C22×C4—3261no—
296C22×C4C22×D16311yescharacteristic
297C22×C4C22×D16311yescharacteristic
298Q16—3262no—
299Q16—6521no—
300SD16—3262no—
301SD16—3262no—
302SD16—3262no—
303SD16—3262no—
304SD16—6521no—
305D8—3262no—
306D8—1634no—
307M16—3262no—
308C2×C8—3262no—
309C2×C8—1631no—
310C2×C8—1631no—
311C2×C8—1631no—
312C2×C8—1631no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—3261no—
318C4⋊C4—1631no—
319C4⋊C4C22⋊D16311yescharacteristic
320C4⋊C4C22×D16311yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—3261no—
325C22⋊C4—3261no—
326C22⋊C4—3261no—
327C22⋊C4—3261no—
328C22⋊C4—3261no—
329C22⋊C4C22⋊D16311yescharacteristic
330C22⋊C4C22⋊D16311yescharacteristic
331C42C22×D16311yescharacteristic
332C23—21no—
333C23—13041no—
334C23—3261no—
335C23—3261no—
336C23—6522no—
337C23—6522no—
338C23C2×C22⋊D16311yescharacteristic
339C23C2×C22⋊D16311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—3261no—
345Q8—3261no—
346D4—21no—
347D4—41no—
348D4—42no—
349D4—42no—
350D4—13041no—
351D4—3261no—
352D4—3261no—
353D4—3261no—
354D4—6522no—
355D4—6522no—
356D4—1632no—
357D4D4×D16312yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—13041no—
369C2×C4—3261no—
370C2×C4—3261no—
371C2×C4—6522no—
372C2×C4C2×C22⋊D16311yescharacteristic
373C2×C4C2×C22⋊D16311yescharacteristic
374C2×C4C2×C22⋊D16311yescharacteristic
375C2×C4C2×C22⋊D16311yescharacteristic
376C2×C4C23×D16311yescharacteristic
377C8—3261no—
378C8—3261no—
379C8—3261no—
380C8—1632no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—41no—
389C22—41no—
390C22—13044no—
391C22—3262no—
392C22—6521no—
393C22—6521no—
394C22—6521no—
395C22—6521no—
396C22—6521no—
397C22C22×C22⋊D16311yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—6521no—
406C4—6521no—
407C4C2×D4×D16311yescharacteristic
408C4#1910478011yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—13041no—
413C2—6522no—
414C2D4×C22⋊D16311yescharacteristic
415C2#2378246311yescharacteristic
416C2#2378247311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#547×C163C2
socleC22×C163C22×D4
hypercenter#547D163
hypocenterC163#3267
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#19104760C2semidirect
2. C2×#19104755C2semidirect
3. #23782311C2semidirect
4. #23782057C2semidirect
5. #23781921C2semidirect
6. #23781896C2semidirect
7. #23781880C2semidirect
8. #23781867C2semidirect
9. #23781827C2semidirect
10. #23781825C2semidirect
11. #23781815C2semidirect
12. #23781810C2semidirect
13. #23781775C2semidirect
14. #23781737C2semidirect
15. #547×C163C2semidirect
20. C2×#14579689C22semidirect
21. C2×#14579689C22semidirect
22. C2×#14579687C22semidirect
30. C22×C163⋊C8C22semidirect
39. #19104666C22semidirect
40. C4⋊(C163⋊C8)C22semidirect
41. #19104659C22semidirect
42. #19104659C22semidirect
43. #19104658C22semidirect
44. #19104656C22semidirect
48. C22⋊(C163⋊C8)C22semidirect
49. #19104646C22semidirect
51. C2×#41×C163C22semidirect
52. C4⋊Q8×C163C22semidirect
56. C22⋊Q8×C163C22semidirect
59. C4×D4×C163C22semidirect
82. #14579684D4semidirect
208. C163#3267semidirect
212. #547D163semidirect
227. C2×#41C2×D163semidirect
242. C4⋊Q8C2×D163semidirect
249. C22⋊Q8C2×D163semidirect
256. C4×D4C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047602
C2×#191047552
#237823112
#237820572
#237819212
#237818962
#237818802
#237818672
#237818272
#237818252
#237818152
#237818102
#237817752
#237817372
#547×C1632
#3267163