Finite Group Databasebeta

← #35224140

Subgroups — #35224140

Order 20864.

500 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×#19104756C211yesmaximal, characteristic, complemented
2C2×M16×D163C211yesmaximal, characteristic, complemented
3#23782178C211yesmaximal, characteristic, complemented
4#23782175C211yesmaximal, characteristic, complemented
5#23781922C211yesmaximal, characteristic, complemented
6#23781921C211yesmaximal, characteristic, complemented
7#23781917C211yesmaximal, characteristic, complemented
8#23781916C211yesmaximal, characteristic, complemented
9#23781912C211yesmaximal, characteristic, complemented
10#23781911C211yesmaximal, characteristic, complemented
11#23781908C211yesmaximal, characteristic, complemented
12#23781702C211yesmaximal, characteristic, complemented
13#23781660C211yesmaximal, characteristic, complemented
14#23781658C211yesmaximal, characteristic, complemented
15#468×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579698C2211yescharacteristic
18C22×C4×D163C2211yescharacteristic
19#19104756—24no—
20C2×#14579689C2211yescharacteristic, complemented
21C2×#14579688C2211yescharacteristic, complemented
22C2×#14579684C2211yescharacteristic, complemented
23C2×#14579683C2211yescharacteristic
24C2×#14579682C2211yescharacteristic
25C2×#14579682C2211yescharacteristic
26M16×D163—24no—
27C2×C8×D163C2211yescharacteristic
28C2×C8×D163C2211yescharacteristic
29#19104720—21no—
30#19104717—21no—
31#19104715C2211yescharacteristic
32#19104714C2211yescharacteristic
33#19104711C2211yescharacteristic
34#19104710C2211yescharacteristic
35#19104709C2211yescharacteristic
36#19104706C2211yescharacteristic
37#19104705—21no—
38#19104703—21no—
39D4×Dic163—21no—
40#19104697—21no—
41(C22⋊D163)⋊C4—21no—
42(C22⋊D163)⋊C4—21no—
43#19104693C2211yescharacteristic
44#19104693C2211yescharacteristic
45#19104669C2211yescharacteristic, complemented
46#19104666C2211yescharacteristic, complemented
47#19104666C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104659C2211yescharacteristic, complemented
50#19104658C2211yescharacteristic, complemented
51#19104657C2211yescharacteristic, complemented
52#19104656C2211yescharacteristic, complemented
53#19104655C2211yescharacteristic, complemented
54C2×SD16×C163C2211yescharacteristic, complemented
55C2×M16×C163C2211yescharacteristic
56C22⋊Q8×C163C2211yescharacteristic
57#121×C163C2211yescharacteristic
58#107×C163C2211yescharacteristic, complemented
59#103×C163C2211yescharacteristic, complemented
60#102×C163C2211yescharacteristic, complemented
61C23×D163D411yescharacteristic
62#14579700—22no—
63#14579700—22no—
64#14579698—22no—
65#14579698—22no—
66C2×Dic326C2311yescharacteristic
67C2×D652—21no—
68C2×D652—21no—
69C2×D652C2311yescharacteristic
70C2×C22⋊D163—21no—
71C2×C22⋊D163—21no—
72C2×C22⋊D163—21no—
73C22×Dic163—21no—
74C22×Dic163D411yescharacteristic
75C2×C4×D163—22no—
76C2×C4×D163—22no—
77C2×C4×D163—21no—
78C2×C4×D163D411yescharacteristic
79C2×C4×D163D411yescharacteristic
80C2×C4×D163C2311yescharacteristic
81C2×C4×D163C2311yescharacteristic
82#14579689—22no—
83#14579688—22no—
84#14579684—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—
112M16×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—22no—
121C22×D163—21no—
122C22×D163—41no—
123C22×D163—41no—
124C22×D163C2×D411yescharacteristic
125C22×D163C2×D411yescharacteristic
126Dic326—21no—
127Dic326—21no—
128D652—21no—
129D652—21no—
130D652—42no—
131D652—42no—
132C22⋊D163—42no—
133C22⋊D163—42no—
134C22⋊D163—42no—
135C2×Dic163—22no—
136C2×Dic163—21no—
137C2×Dic163—21no—
138C2×Dic163—21no—
139C2×Dic163—42no—
140C2×Dic163C2×D411yescharacteristic
141C2×Dic163C2×D411yescharacteristic
142C4×D163—22no—
143C4×D163—21no—
144C4×D163—21no—
145C4×D163—22no—
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—42no—
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—44no—
180C2×D163—41no—
181C2×D163—41no—
182C2×D163—41no—
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#57511yescharacteristic
199C2×C163#57511yescharacteristic
200D163—22no—
201D163—41no—
202D163—81no—
203C163#305411yescharacteristic, complemented, Sylow
204#3054—1631nomaximal, Sylow
205C2×#135—1631no—
206C22×M16—1631no—
207#525—1631no—
208#525—1631no—
209#470—1631no—
210#470—1631no—
211#469—1631no—
212#469—1631no—
213#468—1631no—
214#468—1631no—
215#468—1631no—
216#468D16311yescharacteristic, complemented
217#427—1631no—
218#416—1631no—
219#416—1631no—
220C2×#41—1631no—
221C2×#41—1631no—
222C23×C4—1631no—
223#135—3264no—
224C2×Q16—1631no—
225C2×SD16—1631no—
226C2×SD16C2×D16311yescharacteristic, complemented
227C2×D8—1631no—
228C2×M16—3264no—
229C2×M16—1631no—
230C2×M16—1631no—
231C2×M16—1631no—
232C2×M16C2×D16311yescharacteristic
233C22×C8—1631no—
234C22×C8—1631no—
235#123—3261no—
236#123—3261no—
237C22⋊Q8—1631no—
238C22⋊Q8—1631no—
239C22⋊Q8—1631no—
240C22⋊Q8C2×D16311yescharacteristic
241#121—1631no—
242#121—1631no—
243#121—1631no—
244#121C2×D16311yescharacteristic
245#120—3261no—
246#120—3261no—
247C4×D4—3261no—
248C4×D4—3261no—
249C4×D4—3261no—
250C4×D4—3261no—
251#117—1631no—
252#117—1631no—
253#107—1631no—
254#107C2×D16311yescharacteristic, complemented
255#106—1631no—
256#106—1631no—
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, complemented
265C24—1631no—
266#41—3262no—
267#41—3262no—
268#41—3262no—
269#41—3262no—
270C2×Q8—1631no—
271C2×Q8C22×D16311yescharacteristic
272C2×D4—21no—
273C2×D4—3261no—
274C2×D4—3261no—
275C2×D4—3261no—
276C2×D4—3261no—
277C2×D4—3261no—
278C2×D4—1631no—
279C2×D4C22×D16311yescharacteristic
280C22×C4—3262no—
281C22×C4—3262no—
282C22×C4—3261no—
283C22×C4—3261no—
284C22×C4—1631no—
285C22×C4—1631no—
286C22×C4—1631no—
287C22×C4—1631no—
288C22×C4—1631no—
289C22×C4C22×D16311yescharacteristic
290Q16—3262no—
291SD16—22no—
292SD16—3262no—
293D8—3262no—
294M16—22no—
295M16—3262no—
296M16—3262no—
297M16—3262no—
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—3262no—
326C23—3261no—
327C23—6521no—
328C23—6521no—
329C23—1631no—
330C23—1631no—
331C23D4×D16311yescharacteristic
332Q8—21no—
333Q8—21no—
334Q8—3261no—
335Q8—3261no—
336D4—21no—
337D4—21no—
338D4—42no—
339D4—3261no—
340D4—3261no—
341D4—6522no—
342D4—6522no—
343D4—6522no—
344D4—6522no—
345D4—6522no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—3262no—
352C2×C4—3262no—
353C2×C4—3262no—
354C2×C4—3261no—
355C2×C4—3261no—
356C2×C4—3261no—
357C2×C4—3261no—
358C2×C4—3261no—
359C2×C4—6522no—
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—3261no—
378C22—3261no—
379C22—3261no—
380C22—3261no—
381C22—3261no—
382C22—6524no—
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#2378246111yescharacteristic
406C2#2378246411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#468×C163C2
socleC22×C163C22×D4
hypercenter#468D163
hypocenterC163#3054
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#19104756C2semidirect
2. C2×M16×D163C2semidirect
3. #23782178C2semidirect
4. #23782175C2semidirect
5. #23781922C2semidirect
6. #23781921C2semidirect
7. #23781917C2semidirect
8. #23781916C2semidirect
9. #23781912C2semidirect
10. #23781911C2semidirect
11. #23781908C2semidirect
12. #23781702C2semidirect
13. #23781660C2semidirect
14. #23781658C2semidirect
15. #468×C163C2semidirect
20. C2×#14579689C22semidirect
21. C2×#14579688C22semidirect
22. C2×#14579684C22semidirect
45. #19104669C22semidirect
46. #19104666C22semidirect
47. #19104666C22semidirect
48. #19104660C22semidirect
49. #19104659C22semidirect
50. #19104658C22semidirect
51. #19104657C22semidirect
52. #19104656C22semidirect
53. #19104655C22semidirect
54. C2×SD16×C163C22semidirect
58. #107×C163C22semidirect
59. #103×C163C22semidirect
60. #102×C163C22semidirect
91. C8×D163D4semidirect
203. C163#3054semidirect
216. #468D163semidirect
226. C2×SD16C2×D163semidirect
254. #107C2×D163semidirect
260. #103C2×D163semidirect
264. #102C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047562
C2×M16×D1632
#237821782
#237821752
#237819222
#237819212
#237819172
#237819162
#237819122
#237819112
#237819082
#237817022
#237816602
#237816582
#468×C1632
#3054163