Finite Group Databasebeta

← #35223998

Subgroups — #35223998

Order 20864.

548 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104762C211yesmaximal, characteristic, complemented
2C2×#19104759C211yesmaximal, characteristic, complemented
3#23782214C211yesmaximal, characteristic, complemented
4C4×D4×D163C211yesmaximal, characteristic, complemented
5#23781881C211yesmaximal, characteristic, complemented
6#23781880C211yesmaximal, characteristic, complemented
7#23781877C211yesmaximal, characteristic, complemented
8#23781869C211yesmaximal, characteristic, complemented
9#23781868C211yesmaximal, characteristic, complemented
10#23781859C211yesmaximal, characteristic, complemented
11#23781858C211yesmaximal, characteristic, complemented
12#23781687C211yesmaximal, characteristic, complemented
13#23781665C211yesmaximal, characteristic, complemented
14#23781663C211yesmaximal, characteristic, complemented
15#463×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579698C2211yescharacteristic
18C2×D4×D163C2211yescharacteristic
19C22×C4×D163—21no—
20#19104762—24no—
21#19104759—24no—
22C2×#14579688C2211yescharacteristic
23C2×#14579687C2211yescharacteristic
24C2×#14579686C2211yescharacteristic
25C2×D1304C2211yescharacteristic
26C2×#14579684C2211yescharacteristic
27C2×#14579684C2211yescharacteristic
28C2×#14579682C2211yescharacteristic
29C2×#14579682C2211yescharacteristic
30#19104733C2211yescharacteristic, complemented
31#19104731C2211yescharacteristic, complemented
32#19104723C2211yescharacteristic, complemented
33#19104708—21no—
34#19104708—21no—
35C4×D652C2211yescharacteristic
36D4×Dic163C2211yescharacteristic
37#19104697C2211yescharacteristic
38C4×C22⋊D163—21no—
39(C22⋊D163)⋊C4—21no—
40(C22⋊D163)⋊C4—21no—
41(C22⋊D163)⋊C4—21no—
42C4⋊C4×D163C2211yescharacteristic
43C22⋊C4×D163—21no—
44C42×D163C2211yescharacteristic
45C4⋊(C163⋊C8)C2211yescharacteristic, complemented
46Dic163⋊C8C2211yescharacteristic, complemented
47Dic163⋊C8C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104659C2211yescharacteristic, complemented
50#19104658C2211yescharacteristic, complemented
51#19104657C2211yescharacteristic, complemented
52#19104656C2211yescharacteristic, complemented
53#19104655C2211yescharacteristic, complemented
54C2×SD16×C163C2211yescharacteristic
55C2×D8×C163C2211yescharacteristic
56#124×C163C2211yescharacteristic, complemented
57C4×D4×C163C2211yescharacteristic
58C4⋊C8×C163C2211yescharacteristic, complemented
59#103×C163C2211yescharacteristic, complemented
60#102×C163C2211yescharacteristic, complemented
61C23×D163—21no—
62#14579700—22no—
63#14579700—22no—
64#14579698—22no—
65#14579698—22no—
66C2×Dic326C2311yescharacteristic
67C2×D652—21no—
68C2×D652C2311yescharacteristic
69C2×D652C2311yescharacteristic
70C2×C22⋊D163—21no—
71C2×C22⋊D163—21no—
72C2×C22⋊D163—21no—
73C2×C22⋊D163—21no—
74D4×D163—22no—
75D4×D163D414yescomplemented
76C22×Dic163—21no—
77C22×Dic163—21no—
78C2×C4×D163—21no—
79C2×C4×D163—21no—
80C2×C4×D163—21no—
81C2×C4×D163—21no—
82C2×C4×D163—44no—
83C2×C4×D163D411yescharacteristic
84C2×C4×D163D411yescharacteristic
85C2×C4×D163C2311yescharacteristic
86#14579688—22no—
87#14579687—22no—
88#14579686—22no—
89D1304—22no—
90#14579684—22no—
91#14579684—22no—
92#14579682—22no—
93#14579682—22no—
94C2×C163⋊C8C2311yescharacteristic
95C2×C163⋊C8C2311yescharacteristic
96C4⋊Dic163C2311yescharacteristic
97Dic163⋊C4—21no—
98Dic163⋊C4—21no—
99Dic163⋊C4—21no—
100C22⋊Dic163—21no—
101(C2×D163)⋊C4—21no—
102(C2×D163)⋊C4—21no—
103(C2×D163)⋊C4—21no—
104(C2×D163)⋊C4—21no—
105C4×Dic163D411yescharacteristic
106C4×Dic163D411yescharacteristic
107C4×Dic163C2311yescharacteristic
108C2×Q8×C163C2311yescharacteristic
109C2×D4×C163C2311yescharacteristic
110C2×D4×C163C2311yescharacteristic
111C22×C4×C163—21no—
112SD16×C163—22no—
113D8×C163—22no—
114C2×C8×C163C2311yescharacteristic
115C2×C8×C163C2311yescharacteristic
116C4⋊C4×C163C2311yescharacteristic
117C22⋊C4×C163—21no—
118C22⋊C4×C163—21no—
119C22⋊C4×C163—21no—
120C42×C163C2311yescharacteristic
121C22×D163—21no—
122C22×D163—24no—
123C22×D163—21no—
124C22×D163—42no—
125C22×D163—42no—
126C22×D163C2×D411yescharacteristic
127Dic326—21no—
128Dic326—21no—
129D652—21no—
130D652—21no—
131D652—21no—
132D652—42no—
133D652C2×D412yes—
134C22⋊D163—24no—
135C22⋊D163—42no—
136C22⋊D163—42no—
137C22⋊D163—42no—
138C2×Dic163—21no—
139C2×Dic163—21no—
140C2×Dic163—42no—
141C2×Dic163—42no—
142C2×Dic163C2×D411yescharacteristic
143C2×Dic163C2×D411yescharacteristic
144C2×Dic163C2×D411yescharacteristic
145C4×D163—22no—
146C4×D163—41no—
147C4×D163—42no—
148C4×D163—42no—
149C4×D163—42no—
150C4×D163—42no—
151C4×D163C2×D412yes—
152C4×D163#4112yes—
153C163⋊C8—21no—
154C163⋊C8—21no—
155C23×C163—21no—
156C23×C163—21no—
157Q8×C163—21no—
158Q8×C163—21no—
159D4×C163—21no—
160D4×C163—21no—
161D4×C163—21no—
162D4×C163C2×D412yes—
163C2×C4×C163—21no—
164C2×C4×C163—21no—
165C2×C4×C163—42no—
166C2×C4×C163C2×D411yescharacteristic
167C2×C4×C163C2×D411yescharacteristic
168C2×C4×C163C2411yescharacteristic
169C8×C163—21no—
170C8×C163—21no—
171C22×C163—22no—
172C22×C163—41no—
173C22×C163—41no—
174C22×C163—41no—
175C22×C163—41no—
176C22×C163—41no—
177C22×C163C22×D411yescharacteristic
178C2×D163—21no—
179C2×D163—21no—
180C2×D163—22no—
181C2×D163—48no—
182C2×D163—41no—
183C2×D163—41no—
184C2×D163—41no—
185C2×D163—41no—
186C2×D163—41no—
187C2×D163#12112yes—
188C4×C163—21no—
189C4×C163—21no—
190C4×C163—41no—
191C4×C163—41no—
192C4×C163C22×D411yescharacteristic
193C4×C163C2×#4111yescharacteristic
194Dic163—21no—
195Dic163—41no—
196Dic163—41no—
197Dic163#12112yes—
198Dic163AGL(1,ℤ/8)12yescomplemented
199C2×C163—42no—
200C2×C163—81no—
201C2×C163C2×#12111yescharacteristic
202C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
203C2×C163#57511yescharacteristic
204D163—22no—
205D163—42no—
206D163—81no—
207C163#303611yescharacteristic, complemented, Sylow
208#3036—1631nomaximal, Sylow
209C2×AGL(1,ℤ/8)—1631no—
210C2×AGL(1,ℤ/8)—1631no—
211#532—1631no—
212C2×C4×D4—1631no—
213#463—1631no—
214#463—1631no—
215#463—1631no—
216#463D16311yescharacteristic, complemented
217#461—1631no—
218#461—1631no—
219#459—1631no—
220#459—1631no—
221#423—1631no—
222#417—1631no—
223#417—1631no—
224C2×#41—1631no—
225C2×#41—1631no—
226C22×D4—1631no—
227C23×C4—3261no—
228AGL(1,ℤ/8)—3264no—
229AGL(1,ℤ/8)—3264no—
230C2×SD16—1631no—
231C2×SD16—1631no—
232C2×SD16—1631no—
233C2×SD16C2×D16311yescharacteristic
234C2×D8—1631no—
235C2×D8—1631no—
236C2×D8—1631no—
237C2×D8C2×D16311yescharacteristic
238C2×M16—1631no—
239C2×M16—1631no—
240C4⋊Q8—1631no—
241#127—1631no—
242#124—1631no—
243#124C2×D16311yescharacteristic, complemented
244#121—3261no—
245#121—3261no—
246C4×D4—3261no—
247C4×D4—3261no—
248C4×D4—3261no—
249C4×D4—3261no—
250C4×D4—1631no—
251C4×D4—1631no—
252C4×D4—1631no—
253C4×D4C2×D16311yescharacteristic
254C2×C4⋊C4—1631no—
255C2×C22⋊C4—3261no—
256C2×C42—1631no—
257C4⋊C8—1631no—
258C4⋊C8—1631no—
259C4⋊C8—1631no—
260C4⋊C8C2×D16311yescharacteristic, complemented
261#103—1631no—
262#103—1631no—
263#103—1631no—
264#103C2×D16311yescharacteristic, complemented
265#102—1631no—
266#102—1631no—
267#102—1631no—
268#102C2×D16311yescharacteristic, complemented
269C24—3261no—
270#41—3262no—
271#41—3262no—
272#41—3262no—
273#41—3262no—
274C2×Q8—1631no—
275C2×Q8C22×D16311yescharacteristic
276C2×D4—3261no—
277C2×D4—3261no—
278C2×D4—3262no—
279C2×D4—3261no—
280C2×D4—3261no—
281C2×D4—3261no—
282C2×D4—1634no—
283C2×D4—1631no—
284C2×D4—1631no—
285C2×D4C22×D16311yescharacteristic
286C2×D4C22×D16311yescharacteristic
287C22×C4—21no—
288C22×C4—3261no—
289C22×C4—3261no—
290C22×C4—3261no—
291C22×C4—3261no—
292C22×C4—3261no—
293C22×C4—3261no—
294C22×C4—6524no—
295C22×C4—1631no—
296C22×C4—1631no—
297C22×C4—1631no—
298SD16—22no—
299SD16—3262no—
300SD16—3262no—
301SD16—3262no—
302D8—22no—
303D8—3262no—
304D8—3262no—
305D8—3262no—
306M16—3262no—
307M16—3262no—
308C2×C8—1631no—
309C2×C8—1631no—
310C2×C8C22×D16311yescharacteristic
311C2×C8C22×D16311yescharacteristic
312C4⋊C4—3261no—
313C4⋊C4—3261no—
314C4⋊C4—3261no—
315C4⋊C4—1631no—
316C4⋊C4C22×D16311yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—3261no—
321C22⋊C4—3261no—
322C22⋊C4—3261no—
323C22⋊C4—3261no—
324C22⋊C4—3261no—
325C42—1631no—
326C42—1631no—
327C42—1631no—
328C42C22×D16311yescharacteristic
329C23—21no—
330C23—21no—
331C23—3264no—
332C23—3261no—
333C23—3261no—
334C23—6522no—
335C23—6522no—
336C23—1631no—
337Q8—21no—
338Q8—21no—
339Q8—3261no—
340Q8—3261no—
341D4—21no—
342D4—21no—
343D4—21no—
344D4—3264no—
345D4—3261no—
346D4—3261no—
347D4—3261no—
348D4—6522no—
349D4—6522no—
350D4—6522no—
351D4—6522no—
352D4—1632no—
353D4D4×D16312yes—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—3262no—
358C2×C4—3261no—
359C2×C4—3261no—
360C2×C4—6521no—
361C2×C4—6522no—
362C2×C4—6522no—
363C2×C4—6522no—
364C2×C4—6522no—
365C2×C4—6522no—
366C2×C4—6522no—
367C2×C4—1631no—
368C2×C4—1631no—
369C2×C4—1631no—
370C2×C4—1632no—
371C2×C4—1632no—
372C2×C4D4×D16311yescharacteristic
373C2×C4D4×D16311yescharacteristic
374C2×C4C23×D16311yescharacteristic
375C8—21no—
376C8—21no—
377C8—3261no—
378C8—3261no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—3261no—
386C22—3261no—
387C22—3262no—
388C22—6528no—
389C22—6521no—
390C22—6521no—
391C22—6521no—
392C22—6521no—
393C22—6521no—
394C22—1632no—
395C22C2×D4×D16311yescharacteristic
396C4—21no—
397C4—21no—
398C4—41no—
399C4—41no—
400C4—3261no—
401C4—6521no—
402C4—6521no—
403C4—1632no—
404C4—1632no—
405C4C2×D4×D16311yescharacteristic
406C4#41×D16311yescharacteristic
407C2—42no—
408C2—81no—
409C2—13041no—
410C2—3262no—
411C2—6522no—
412C2#121×D16311yescharacteristic
413C2AGL(1,ℤ/8)×D16311yescharacteristic
414C2#2378246211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#463×C163C2
socleC22×C163C22×D4
hypercenter#463D163
hypocenterC163#3036
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#19104762C2semidirect
2. C2×#19104759C2semidirect
3. #23782214C2semidirect
4. C4×D4×D163C2semidirect
5. #23781881C2semidirect
6. #23781880C2semidirect
7. #23781877C2semidirect
8. #23781869C2semidirect
9. #23781868C2semidirect
10. #23781859C2semidirect
11. #23781858C2semidirect
12. #23781687C2semidirect
13. #23781665C2semidirect
14. #23781663C2semidirect
15. #463×C163C2semidirect
30. #19104733C22semidirect
31. #19104731C22semidirect
32. #19104723C22semidirect
45. C4⋊(C163⋊C8)C22semidirect
46. Dic163⋊C8C22semidirect
47. Dic163⋊C8C22semidirect
48. #19104660C22semidirect
49. #19104659C22semidirect
50. #19104658C22semidirect
51. #19104657C22semidirect
52. #19104656C22semidirect
53. #19104655C22semidirect
56. #124×C163C22semidirect
58. C4⋊C8×C163C22semidirect
59. #103×C163C22semidirect
60. #102×C163C22semidirect
75. D4×D163D4semidirect
198. Dic163AGL(1,ℤ/8)semidirect
207. C163#3036semidirect
216. #463D163semidirect
243. #124C2×D163semidirect
260. C4⋊C8C2×D163semidirect
264. #103C2×D163semidirect
268. #102C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047622
C2×#191047592
#237822142
C4×D4×D1632
#237818812
#237818802
#237818772
#237818692
#237818682
#237818592
#237818582
#237816872
#237816652
#237816632
#463×C1632
#3036163