Finite Group Databasebeta

← #35224170

Subgroups — #35224170

Order 20864.

492 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D163C211yesmaximal, characteristic, complemented
2C2×#19104741C211yesmaximal, characteristic, complemented
3#23782238C211yesmaximal, characteristic, complemented
4#23782231C211yesmaximal, characteristic
5#23781914C211yesmaximal, characteristic, complemented
6#23781913C211yesmaximal, characteristic, complemented
7#23781910C211yesmaximal, characteristic, complemented
8#23781909C211yesmaximal, characteristic, complemented
9#23781895C211yesmaximal, characteristic, complemented
10#23781894C211yesmaximal, characteristic, complemented
11#23781891C211yesmaximal, characteristic, complemented
12#23781700C211yesmaximal, characteristic, complemented
13#23781672C211yesmaximal, characteristic, complemented
14#23781668C211yesmaximal, characteristic, complemented
15#465×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C2×Q8×D163C2211yescharacteristic
18C2×D4×D163C2211yescharacteristic
19C2×#14579688C2211yescharacteristic
20C2×#14579687C2211yescharacteristic
21C2×#14579686C2211yescharacteristic
22SD16×D163—24no—
23#19104741—24no—
24C2×#14579683C2211yescharacteristic, complemented
25C2×#14579682C2211yescharacteristic
26C2×#14579682C2211yescharacteristic, complemented
27C2×C8×D163C2211yescharacteristic
28C2×C8×D163C2211yescharacteristic, complemented
29#19104734—21no—
30#19104731—21no—
31#19104721—21no—
32#19104721—21no—
33#19104720—21no—
34#19104717—21no—
35#19104715C2211yescharacteristic, complemented
36#19104714C2211yescharacteristic, complemented
37#19104711C2211yescharacteristic, complemented
38#19104710C2211yescharacteristic
39#19104709C2211yescharacteristic
40#19104706C2211yescharacteristic
41#19104705—21no—
42#19104703—21no—
43#19104693C2211yescharacteristic
44#19104693C2211yescharacteristic
45#19104668C2211yescharacteristic, complemented
46#19104668C2211yescharacteristic, complemented
47#19104667C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104659C2211yescharacteristic, complemented
50#19104658C2211yescharacteristic, complemented
51#19104657C2211yescharacteristic
52#19104656C2211yescharacteristic
53#19104655C2211yescharacteristic
54C2×SD16×C163C2211yescharacteristic
55C22×C8×C163C2211yescharacteristic, complemented
56C22⋊Q8×C163C2211yescharacteristic, complemented
57#121×C163C2211yescharacteristic
58#106×C163C2211yescharacteristic, complemented
59#103×C163C2211yescharacteristic, complemented
60#102×C163C2211yescharacteristic
61C23×D163—21no—
62#14579699—22no—
63#14579699—22no—
64C2×Dic326—21no—
65C2×Dic326D411yescharacteristic
66C2×Dic326C2311yescharacteristic
67Q8×D163—22no—
68Q8×D163—22no—
69C2×D652—21no—
70C2×D652D411yescharacteristic
71C2×D652C2311yescharacteristic
72C2×C22⋊D163—21no—
73C2×C22⋊D163—21no—
74C2×C22⋊D163—21no—
75C2×C22⋊D163D411yescharacteristic
76C2×C22⋊D163D411yescharacteristic
77D4×D163—22no—
78D4×D163—22no—
79C2×C4×D163—21no—
80C2×C4×D163C2311yescharacteristic
81C2×C4×D163C2311yescharacteristic
82#14579688—22no—
83#14579687—22no—
84#14579686—22no—
85#14579683—22no—
86#14579682—22no—
87#14579682—22no—
88C2×C163⋊C8C2311yescharacteristic
89C2×C163⋊C8C2311yescharacteristic
90C8×D163—22no—
91C8×D163D414yescomplemented
92C4⋊Dic163—21no—
93C4⋊Dic163C2311yescharacteristic
94C4⋊Dic163C2311yescharacteristic
95Dic163⋊C4—21no—
96Dic163⋊C4—21no—
97C22⋊Dic163—21no—
98C22⋊Dic163—21no—
99(C2×D163)⋊C4—21no—
100(C2×D163)⋊C4—21no—
101(C2×D163)⋊C4—21no—
102(C2×D163)⋊C4—21no—
103(C2×D163)⋊C4—21no—
104(C2×D163)⋊C4—21no—
105C4×Dic163—21no—
106C4×Dic163—21no—
107C2×Q8×C163C2311yescharacteristic
108C2×D4×C163—21no—
109C2×D4×C163C2311yescharacteristic
110C22×C4×C163C2311yescharacteristic
111SD16×C163—22no—
112C2×C8×C163—22no—
113C2×C8×C163C2311yescharacteristic
114C2×C8×C163C2311yescharacteristic
115C4⋊C4×C163—21no—
116C4⋊C4×C163C2311yescharacteristic
117C4⋊C4×C163C2311yescharacteristic
118C22⋊C4×C163—21no—
119C22⋊C4×C163—21no—
120C22×D163—21no—
121C22×D163—42no—
122C22×D163—81no—
123C22×D163—81no—
124C22×D163C2×D411yescharacteristic
125C22×D163C2×D411yescharacteristic
126Dic326—22no—
127Dic326—21no—
128Dic326—21no—
129Dic326—42no—
130D652—22no—
131D652—21no—
132D652—21no—
133D652—81no—
134C22⋊D163—22no—
135C22⋊D163—22no—
136C22⋊D163—42no—
137C22⋊D163—81no—
138C22⋊D163—81no—
139C2×Dic163—21no—
140C2×Dic163—21no—
141C2×Dic163—21no—
142C2×Dic163C2×D411yescharacteristic
143C2×Dic163C2×D411yescharacteristic
144C4×D163—21no—
145C4×D163—21no—
146C4×D163—42no—
147C4×D163C2×D412yes—
148C4×D163#4112yes—
149C163⋊C8—21no—
150C163⋊C8C2×D412yes—
151C23×C163—21no—
152C23×C163C2×D411yescharacteristic
153Q8×C163—21no—
154Q8×C163—21no—
155D4×C163—21no—
156D4×C163—21no—
157D4×C163—81no—
158C2×C4×C163—21no—
159C2×C4×C163—21no—
160C2×C4×C163—21no—
161C2×C4×C163—21no—
162C2×C4×C163—21no—
163C2×C4×C163C2×D411yescharacteristic
164C2×C4×C163C2411yescharacteristic
165C8×C163—21no—
166C8×C163C2×D412yes—
167C22×C163—21no—
168C22×C163—21no—
169C22×C163—21no—
170C22×C163—41no—
171C22×C163—41no—
172C22×C163—41no—
173C22×C163C22×D411yescharacteristic
174C2×D163—21no—
175C2×D163—21no—
176C2×D163—21no—
177C2×D163—21no—
178C2×D163—21no—
179C2×D163—41no—
180C2×D163—41no—
181C2×D163—41no—
182C2×D163—84no—
183C2×D163#12112yes—
184C4×C163—21no—
185C4×C163—41no—
186C4×C163—41no—
187C4×C163—41no—
188C4×C163C22×D411yescharacteristic
189C4×C163C2×#4111yescharacteristic
190Dic163—21no—
191Dic163—41no—
192Dic163—41no—
193Dic163—41no—
194Dic163#12112yes—
195C2×C163—41no—
196C2×C163—81no—
197C2×C163C2×#12111yescharacteristic
198C2×C163#57711yescharacteristic
199C2×C163#57711yescharacteristic
200D163—22no—
201D163—41no—
202D163—81no—
203C163#305711yescharacteristic, complemented, Sylow
204#3057—1631nomaximal, Sylow
205C22×SD16—1631no—
206C2×#131—1631no—
207#536—1631no—
208#535—1631no—
209#468—1631no—
210#468—1631no—
211#468—1631no—
212#468—1631no—
213#465—1631no—
214#465—1631no—
215#465—1631no—
216#465D16311yescharacteristic, complemented
217#427—1631no—
218#419—1631no—
219#418—1631no—
220C2×#41—1631no—
221C22×Q8—1631no—
222C22×D4—1631no—
223C2×SD16—3264no—
224C2×SD16—1631no—
225C2×SD16—1631no—
226C2×SD16—1631no—
227C2×SD16C2×D16311yescharacteristic
228#131—3264no—
229C2×M16—1631no—
230C2×M16—1631no—
231C2×M16—1631no—
232C22×C8—1631no—
233C22×C8—1631no—
234C22×C8C2×D16311yescharacteristic, complemented
235C4⋊Q8—3261no—
236#127—3261no—
237#124—3261no—
238#124—3261no—
239#123—3261no—
240#123—3261no—
241C22⋊Q8—1631no—
242C22⋊Q8—1631no—
243C22⋊Q8—1631no—
244C22⋊Q8C2×D16311yescharacteristic, complemented
245#121—1631no—
246#121—1631no—
247#121—1631no—
248#121C2×D16311yescharacteristic
249#120—3261no—
250#120—3261no—
251#117—1631no—
252#117—1631no—
253#107—1631no—
254#107—1631no—
255#106—1631no—
256#106C2×D16311yescharacteristic, complemented
257#103—1631no—
258#103—1631no—
259#103—1631no—
260#103C2×D16311yescharacteristic, complemented
261#102—1631no—
262#102—1631no—
263#102—1631no—
264#102C2×D16311yescharacteristic
265C24—3261no—
266#41—3262no—
267#41—3262no—
268C2×Q8—3261no—
269C2×Q8—3262no—
270C2×Q8—3262no—
271C2×Q8—1631no—
272C2×Q8—1631no—
273C2×Q8C22×D16311yescharacteristic
274C2×D4—21no—
275C2×D4—3261no—
276C2×D4—3261no—
277C2×D4—3261no—
278C2×D4—3262no—
279C2×D4—3262no—
280C2×D4—3261no—
281C2×D4—1631no—
282C2×D4—1631no—
283C2×D4—1631no—
284C2×D4—1631no—
285C2×D4C22×D16311yescharacteristic
286C22×C4—3261no—
287C22×C4—1631no—
288C22×C4—1631no—
289C22×C4C22×D16311yescharacteristic
290SD16—22no—
291SD16—3262no—
292SD16—3262no—
293SD16—3262no—
294M16—3262no—
295M16—3262no—
296M16—3262no—
297C2×C8—22no—
298C2×C8—3262no—
299C2×C8—1631no—
300C2×C8—1631no—
301C2×C8—1634no—
302C2×C8C22×D16311yescharacteristic
303C2×C8C22×D16311yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—3261no—
306C4⋊C4—3261no—
307C4⋊C4—3261no—
308C4⋊C4—1631no—
309C4⋊C4—1631no—
310C4⋊C4C22×D16311yescharacteristic
311C4⋊C4C22×D16311yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—3261no—
315C22⋊C4—3261no—
316C22⋊C4—3261no—
317C22⋊C4—3261no—
318C22⋊C4—3261no—
319C22⋊C4—3261no—
320C22⋊C4—3261no—
321C22⋊C4—3261no—
322C42—3261no—
323C42—3261no—
324C23—21no—
325C23—13041no—
326C23—13041no—
327C23—3261no—
328C23—6522no—
329C23—1631no—
330C23—1631no—
331C23D4×D16311yescharacteristic
332Q8—21no—
333Q8—21no—
334Q8—3262no—
335Q8—3261no—
336Q8—3261no—
337Q8—6522no—
338D4—21no—
339D4—21no—
340D4—81no—
341D4—13041no—
342D4—13041no—
343D4—13041no—
344D4—3262no—
345D4—3262no—
346D4—3262no—
347D4—3261no—
348D4—3261no—
349D4—6522no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—3261no—
356C2×C4—3261no—
357C2×C4—3261no—
358C2×C4—3261no—
359C2×C4—3261no—
360C2×C4—6522no—
361C2×C4—1631no—
362C2×C4—1631no—
363C2×C4—1632no—
364C2×C4—1632no—
365C2×C4D4×D16311yescharacteristic
366C2×C4C23×D16311yescharacteristic
367C8—21no—
368C8—3261no—
369C8—1632no—
370C8D4×D16312yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—13044no—
378C22—3261no—
379C22—3261no—
380C22—3261no—
381C22—3261no—
382C22—3261no—
383C22—6521no—
384C22—6521no—
385C22—6521no—
386C22—1632no—
387C22C2×D4×D16311yescharacteristic
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—3261no—
393C4—6521no—
394C4—6521no—
395C4—6521no—
396C4—1632no—
397C4C2×D4×D16311yescharacteristic
398C4#41×D16311yescharacteristic
399C2—41no—
400C2—81no—
401C2—13041no—
402C2—3262no—
403C2—6521no—
404C2#121×D16311yescharacteristic
405C2#2378247111yescharacteristic
406C2#2378247211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#465×C163C2
socleC22×C163C22×D4
prime radical#23782231C2
hypercenter#465D163
hypocenterC163#3057
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D163C2semidirect
2. C2×#19104741C2semidirect
3. #23782238C2semidirect
5. #23781914C2semidirect
6. #23781913C2semidirect
7. #23781910C2semidirect
8. #23781909C2semidirect
9. #23781895C2semidirect
10. #23781894C2semidirect
11. #23781891C2semidirect
12. #23781700C2semidirect
13. #23781672C2semidirect
14. #23781668C2semidirect
15. #465×C163C2semidirect
24. C2×#14579683C22semidirect
26. C2×#14579682C22semidirect
28. C2×C8×D163C22semidirect
35. #19104715C22semidirect
36. #19104714C22semidirect
37. #19104711C22semidirect
45. #19104668C22semidirect
46. #19104668C22semidirect
47. #19104667C22semidirect
48. #19104660C22semidirect
49. #19104659C22semidirect
50. #19104658C22semidirect
55. C22×C8×C163C22semidirect
56. C22⋊Q8×C163C22semidirect
58. #106×C163C22semidirect
59. #103×C163C22semidirect
91. C8×D163D4semidirect
203. C163#3057semidirect
216. #465D163semidirect
234. C22×C8C2×D163semidirect
244. C22⋊Q8C2×D163semidirect
256. #106C2×D163semidirect
260. #103C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1632
C2×#191047412
#237822382
#237822312
#237819142
#237819132
#237819102
#237819092
#237818952
#237818942
#237818912
#237817002
#237816722
#237816682
#465×C1632
#3057163