Finite Group Databasebeta

← #35226248

Subgroups — #35226248

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×#19104757C211yesmaximal, characteristic, complemented
2C22×#14579688C211yesmaximal, characteristic, complemented
3#23782324C211yesmaximal, characteristic, complemented
4#23782311C211yesmaximal, characteristic, complemented
5#23782049C211yesmaximal, characteristic, complemented
6#23781894C211yesmaximal, characteristic, complemented
7#23781894C211yesmaximal, characteristic, complemented
8#23781888C211yesmaximal, characteristic, complemented
9#23781881C211yesmaximal, characteristic, complemented
10#23781829C211yesmaximal, characteristic, complemented
11#23781823C211yesmaximal, characteristic, complemented
12#23781812C211yesmaximal, characteristic, complemented
13#23781801C211yesmaximal, characteristic, complemented
14C4×#14579688C211yesmaximal, characteristic, complemented
15C8×D4×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C2×#14579699C2211yescharacteristic, complemented
18C22×Dic326C2211yescharacteristic
19C22×D652C2211yescharacteristic
20#19104757—24no—
21C2×Dic652C2211yescharacteristic, complemented
22C2×Dic652C2211yescharacteristic, complemented
23C2×#14579688—24no—
24C2×#14579688C2211yescharacteristic
25C2×#14579688C2211yescharacteristic
26C2×#14579688C2211yescharacteristic
27C2×#14579688C2211yescharacteristic
28C2×#14579688C2211yescharacteristic, complemented
29C2×D1304C2211yescharacteristic
30C2×D1304C2211yescharacteristic
31#19104723C2211yescharacteristic
32#19104723C2211yescharacteristic, complemented
33#19104721—21no—
34#19104720—21no—
35#19104714C2211yescharacteristic
36#19104714C2211yescharacteristic, complemented
37#19104713—21no—
38#19104709C2211yescharacteristic
39#19104709C2211yescharacteristic
40#19104708—21no—
41#19104707—21no—
42#19104703—21no—
43C4×Dic326C2211yescharacteristic, complemented
44C4×D652C2211yescharacteristic
45(C22⋊D163)⋊C4—21no—
46C22⋊C4×D163—21no—
47#19104667C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104660C2211yescharacteristic, complemented
50#19104660C2211yescharacteristic, complemented
51#19104657C2211yescharacteristic
52#19104657C2211yescharacteristic
53#19104657C2211yescharacteristic, complemented
54C22×C8×C163C2211yescharacteristic
55C22×C8×C163C2211yescharacteristic, complemented
56C4×D4×C163C2211yescharacteristic
57C4⋊C8×C163C2211yescharacteristic, complemented
58C22⋊C8×C163C2211yescharacteristic
59C22⋊C8×C163C2211yescharacteristic, complemented
60C4×C8×C163C2211yescharacteristic, complemented
61C23×D163—21no—
62#14579699—22no—
63#14579699—22no—
64#14579699—41no—
65#14579699—41no—
66C2×Dic326—22no—
67C2×Dic326—21no—
68C2×Dic326—41no—
69C2×Dic326C2311yescharacteristic
70C2×Dic326C2311yescharacteristic
71C2×Dic326C2311yescharacteristic
72C2×D652—22no—
73C2×D652—21no—
74C2×D652—41no—
75C2×D652C2311yescharacteristic
76C2×D652C2311yescharacteristic
77C2×D652C2311yescharacteristic
78C2×C22⋊D163—21no—
79C2×C22⋊D163—21no—
80C2×C22⋊D163—21no—
81C2×C22⋊D163—21no—
82C22×Dic163—21no—
83C2×C4×D163—21no—
84C2×C4×D163—21no—
85C2×C4×D163—21no—
86Dic652—22no—
87Dic652—41no—
88#14579688—22no—
89#14579688—22no—
90#14579688—22no—
91#14579688—22no—
92#14579688D414yescomplemented
93D1304—22no—
94D1304—41no—
95C4⋊Dic163C2311yescharacteristic
96C4⋊Dic163C2311yescharacteristic
97Dic163⋊C4—21no—
98Dic163⋊C4—21no—
99Dic163⋊C4—21no—
100C22⋊Dic163—21no—
101C22⋊Dic163—21no—
102(C2×D163)⋊C4—21no—
103(C2×D163)⋊C4—21no—
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4—21no—
106(C2×D163)⋊C4—21no—
107(C2×D163)⋊C4—21no—
108(C2×D163)⋊C4—21no—
109C4×Dic163—21no—
110C2×D4×C163D411yescharacteristic
111C22×C4×C163C2311yescharacteristic
112C22×C4×C163C2311yescharacteristic
113C2×C8×C163—22no—
114C2×C8×C163—22no—
115C2×C8×C163C2311yescharacteristic
116C2×C8×C163C2311yescharacteristic
117C2×C8×C163C2311yescharacteristic
118C2×C8×C163C2311yescharacteristic
119C4⋊C4×C163D411yescharacteristic
120C22⋊C4×C163D411yescharacteristic
121C22⋊C4×C163D411yescharacteristic
122C42×C163C2311yescharacteristic
123C22×D163—21no—
124C22×D163—21no—
125C22×D163—21no—
126C22×D163—42no—
127C22×D163—42no—
128C22×D163—81no—
129Dic326—21no—
130Dic326—21no—
131Dic326—21no—
132Dic326—21no—
133Dic326—21no—
134Dic326—42no—
135Dic326C2×D412yes—
136D652—21no—
137D652—21no—
138D652—21no—
139D652—21no—
140D652—21no—
141D652—42no—
142D652C2×D412yes—
143C22⋊D163—42no—
144C22⋊D163—42no—
145C22⋊D163—42no—
146C22⋊D163—81no—
147C2×Dic163—21no—
148C2×Dic163—21no—
149C2×Dic163—21no—
150C2×Dic163—21no—
151C2×Dic163—21no—
152C2×Dic163—42no—
153C4×D163—42no—
154C4×D163—42no—
155C4×D163—81no—
156C23×C163C2×D411yescharacteristic
157C23×C163C2×D411yescharacteristic
158D4×C163—22no—
159C2×C4×C163—21no—
160C2×C4×C163—21no—
161C2×C4×C163—21no—
162C2×C4×C163—21no—
163C2×C4×C163C2×D411yescharacteristic
164C2×C4×C163C2×D411yescharacteristic
165C2×C4×C163C2×D411yescharacteristic
166C2×C4×C163C2×D411yescharacteristic
167C2×C4×C163C2411yescharacteristic
168C8×C163—21no—
169C8×C163—21no—
170C8×C163—21no—
171C8×C163C2×D412yes—
172C22×C163—21no—
173C22×C163—21no—
174C22×C163—21no—
175C22×C163—21no—
176C22×C163—21no—
177C22×C163#13512yescomplemented
178C22×C163C22×D411yescharacteristic
179C2×D163—22no—
180C2×D163—41no—
181C2×D163—41no—
182C2×D163—41no—
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×C163C22×D411yescharacteristic
194C4×C163H(5,𝔽2)11yescharacteristic
195Dic163—22no—
196Dic163—41no—
197Dic163—41no—
198Dic163—41no—
199Dic163—41no—
200C2×C163—22no—
201C2×C163—41no—
202C2×C163D4211yescharacteristic
203C2×C163C2×#13511yescharacteristic
204C2×C163#57711yescharacteristic
205D163—42no—
206D163—81no—
207D163—81no—
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#447—1631no—
223C4×SD16—1631no—
224C8×D4D16311yescharacteristic, complemented
225C2×#41—1631no—
226C2×#41—1631no—
227C22×Q8—1631no—
228C22×D4—1631no—
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×D8—1631no—
239C2×D8—1631no—
240C22×C8C2×D16311yescharacteristic
241C22×C8C2×D16311yescharacteristic, complemented
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#121—1631no—
253#120—3261no—
254C4×Q8—1631no—
255C4×D4—3261no—
256C4×D4—1631no—
257C4×D4C2×D16311yescharacteristic
258C2×C22⋊C4—3261no—
259#106—1631no—
260C4⋊C8C2×D16311yescharacteristic, complemented
261#103—1631no—
262#103—1631no—
263#103—1631no—
264#102—1631no—
265#102—1631no—
266#102—1631no—
267C22⋊C8C2×D16311yescharacteristic
268C22⋊C8C2×D16311yescharacteristic, complemented
269C4×C8C2×D16311yescharacteristic, complemented
270C24—3261no—
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—3262no—
282C2×D4—3261no—
283C2×D4—3261no—
284C2×D4—3261no—
285C2×D4—3261no—
286C2×D4—3261no—
287C2×D4—6521no—
288C2×D4—1631no—
289C2×D4—1631no—
290C2×D4—1631no—
291C2×D4D65211yescharacteristic
292C22×C4—3261no—
293C22×C4—3261no—
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—1634no—
305D8—3262no—
306D8—6521no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8C22×D16311yescharacteristic
310C2×C8C22×D16311yescharacteristic
311C2×C8C22×D16311yescharacteristic
312C2×C8C22×D16311yescharacteristic
313C4⋊C4—3261no—
314C4⋊C4—3261no—
315C4⋊C4—3261no—
316C4⋊C4—1631no—
317C4⋊C4—1631no—
318C4⋊C4D65211yescharacteristic
319C22⋊C4—3261no—
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⋊C4D65211yescharacteristic
329C22⋊C4D65211yescharacteristic
330C42—3261no—
331C42C22×D16311yescharacteristic
332C23—13041no—
333C23—3261no—
334C23—3261no—
335C23—3261no—
336C23—6522no—
337C23—6522no—
338C23C2×D65211yescharacteristic
339C23C2×D65211yescharacteristic
340Q8—3261no—
341Q8—3261no—
342Q8—3261no—
343Q8—3261no—
344Q8—3261no—
345Q8—6522no—
346Q8—1632no—
347D4—22no—
348D4—13041no—
349D4—3261no—
350D4—3261no—
351D4—3261no—
352D4—3261no—
353D4—3261no—
354D4—6522no—
355D4—6522no—
356D4—6522no—
357D4—6522no—
358D4—1632no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
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×C4C2×D65211yescharacteristic
373C2×C4C2×D65211yescharacteristic
374C2×C4C2×D65211yescharacteristic
375C2×C4C2×D65211yescharacteristic
376C2×C4C23×D16311yescharacteristic
377C8—21no—
378C8—21no—
379C8—21no—
380C8D4×D16312yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—13044no—
387C22—3262no—
388C22—6521no—
389C22—6521no—
390C22—6521no—
391C22—6521no—
392C22—6521no—
393C22—6521no—
394C22—6521no—
395C22—6521no—
396C22#1910475712yescomplemented
397C22C22×D65211yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—3262no—
403C4—6521no—
404C4—6521no—
405C4—6521no—
406C4—6521no—
407C4C2×D4×D16311yescharacteristic
408C4#1910478111yescharacteristic
409C2—22no—
410C2—41no—
411C2—13041no—
412C2—13041no—
413C2—6522no—
414C2D4×D65211yescharacteristic
415C2C2×#1910475711yescharacteristic
416C2#2378247711yescharacteristic
subgroupG/H
centerC22C22×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC8×D4×C163C2
socleC22×C163C22×D4
hypercenterC8×D4D163
hypocenterC163#3282
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. C2×#19104757C2semidirect
2. C22×#14579688C2semidirect
3. #23782324C2semidirect
4. #23782311C2semidirect
5. #23782049C2semidirect
6. #23781894C2semidirect
7. #23781894C2semidirect
8. #23781888C2semidirect
9. #23781881C2semidirect
10. #23781829C2semidirect
11. #23781823C2semidirect
12. #23781812C2semidirect
13. #23781801C2semidirect
14. C4×#14579688C2semidirect
15. C8×D4×C163C2semidirect
17. C2×#14579699C22semidirect
21. C2×Dic652C22semidirect
22. C2×Dic652C22semidirect
28. C2×#14579688C22semidirect
32. #19104723C22semidirect
36. #19104714C22semidirect
43. C4×Dic326C22semidirect
47. #19104667C22semidirect
48. #19104660C22semidirect
49. #19104660C22semidirect
50. #19104660C22semidirect
53. #19104657C22semidirect
55. C22×C8×C163C22semidirect
57. C4⋊C8×C163C22semidirect
59. C22⋊C8×C163C22semidirect
60. C4×C8×C163C22semidirect
92. #14579688D4semidirect
177. C22×C163#135semidirect
208. C163#3282semidirect
224. C8×D4D163semidirect
241. C22×C8C2×D163semidirect
260. C4⋊C8C2×D163semidirect
268. C22⋊C8C2×D163semidirect
269. C4×C8C2×D163semidirect
396. C22#19104757semidirect

Maximal subgroups

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

subgroupindex
C2×#191047572
C22×#145796882
#237823242
#237823112
#237820492
#237818942
#237818882
#237818812
#237818292
#237818232
#237818122
#237818012
C4×#145796882
C8×D4×C1632
#3282163