Finite Group Databasebeta

← #35224153

Subgroups — #35224153

Order 20864.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104761C211yesmaximal, characteristic, complemented
2C2×M16×D163C211yesmaximal, characteristic, complemented
3#23782177C211yesmaximal, characteristic, complemented
4C2×#19104706C211yesmaximal, characteristic, complemented
5#23781917C212yesmaximal, complemented
6#23781910C212yesmaximal, complemented
7#23781900C212yesmaximal, complemented
8#23781893C212yesmaximal, complemented
9#23781708C211yesmaximal, characteristic, complemented
10#23781667C211yesmaximal, characteristic, complemented
11#418×C163C211yesmaximal, characteristic, complemented
12C2×#14579699C2211yescharacteristic
13C22×C22⋊D163—21no—
14C22×C4×D163C2211yescharacteristic
15#19104761—24no—
16C2×#14579687C2212yescomplemented
17C2×#14579684C2212yescomplemented
18C2×#14579683C2211yescharacteristic
19C2×#14579682C2212yescomplemented
20M16×D163—24no—
21C2×C8×D163C2212yescomplemented
22#19104719—21no—
23#19104715C2211yescharacteristic, complemented
24#19104715C2211yescharacteristic, complemented
25#19104710C2211yescharacteristic, complemented
26#19104710C2211yescharacteristic, complemented
27#19104706—21no—
28#19104706—21no—
29#19104706C2212yes—
30#19104706C2212yes—
31#19104705—21no—
32C4×D652—21no—
33C4×D652—21no—
34C2×C4⋊Dic163C2211yescharacteristic
35C2×C22⋊Dic163—21no—
36#19104668C2212yescomplemented
37#19104666C2212yescomplemented
38#19104660C2212yescomplemented
39#19104657C2212yescomplemented
40C22×D4×C163C2211yescharacteristic
41C2×M16×C163C2211yescharacteristic
42#117×C163C2211yescharacteristic
43#102×C163C2212yescomplemented
44#102×C163C2212yescomplemented
45C23×D163D411yescharacteristic
46#14579699—22no—
47#14579699—22no—
48C2×Dic326C2311yescharacteristic
49C2×D652—21no—
50C2×D652—21no—
51C2×D652C2311yescharacteristic
52C2×C22⋊D163—22no—
53C2×C22⋊D163—22no—
54C2×C22⋊D163—21no—
55C2×C22⋊D163—44no—
56C22×Dic163—21no—
57C22×Dic163D411yescharacteristic
58C2×C4×D163—22no—
59C2×C4×D163—21no—
60C2×C4×D163—22no—
61C2×C4×D163D411yescharacteristic
62C2×C4×D163D411yescharacteristic
63C2×C4×D163C2311yescharacteristic
64C2×C4×D163C2311yescharacteristic
65#14579687—24no—
66#14579684—24no—
67#14579683D414yescomplemented
68#14579682—24no—
69C2×C163⋊C8C2312yes—
70C8×D163—24no—
71C4⋊Dic163—21no—
72C4⋊Dic163—21no—
73C4⋊Dic163—21no—
74C4⋊Dic163C2312yes—
75C22⋊Dic163—22no—
76C22⋊Dic163—22no—
77(C2×D163)⋊C4—21no—
78(C2×D163)⋊C4—21no—
79(C2×D163)⋊C4—21no—
80(C2×D163)⋊C4—21no—
81C24×C163—21no—
82C2×D4×C163—22no—
83C2×D4×C163—21no—
84C2×D4×C163—22no—
85C2×D4×C163C2312yes—
86C22×C4×C163C2311yescharacteristic
87M16×C163—22no—
88C2×C8×C163C2312yes—
89C4⋊C4×C163C2311yescharacteristic
90C4⋊C4×C163C2311yescharacteristic
91C22⋊C4×C163—21no—
92C42×C163—21no—
93C22×D163—22no—
94C22×D163—22no—
95C22×D163—21no—
96C22×D163—41no—
97C22×D163C2×D411yescharacteristic
98C22×D163C2×D411yescharacteristic
99Dic326—21no—
100Dic326—21no—
101D652—21no—
102D652—21no—
103D652—42no—
104D652—42no—
105C22⋊D163—44no—
106C22⋊D163—42no—
107C22⋊D163—44no—
108C2×Dic163—22no—
109C2×Dic163—21no—
110C2×Dic163—22no—
111C2×Dic163—42no—
112C2×Dic163C2×D411yescharacteristic
113C2×Dic163C2×D411yescharacteristic
114C4×D163—22no—
115C4×D163—22no—
116C4×D163—21no—
117C4×D163—21no—
118C4×D163—21no—
119C4×D163—21no—
120C4×D163—42no—
121C163⋊C8C2×D414yescomplemented
122C23×C163—22no—
123C23×C163—42no—
124C23×C163—42no—
125C23×C163—42no—
126C23×C163C2×D411yescharacteristic
127D4×C163—22no—
128D4×C163—22no—
129D4×C163—42no—
130C2×C4×C163—21no—
131C2×C4×C163—21no—
132C2×C4×C163C2×D411yescharacteristic
133C2×C4×C163C2×D412yes—
134C2×C4×C163#4112yes—
135C2×C4×C163C2411yescharacteristic
136C8×C163—22no—
137C22×C163—21no—
138C22×C163—21no—
139C22×C163—42no—
140C22×C163—42no—
141C22×C163—42no—
142C22×C163—84no—
143C22×C163#12112yes—
144C22×C163C22×D411yescharacteristic
145C2×D163—21no—
146C2×D163—21no—
147C2×D163—21no—
148C2×D163—21no—
149C2×D163—21no—
150C2×D163—44no—
151C2×D163—41no—
152C2×D163—41no—
153C2×D163—41no—
154C2×D163AGL(1,ℤ/8)12yescomplemented
155C4×C163—41no—
156C4×C163—41no—
157C4×C163#12112yes—
158C4×C163C22×D411yescharacteristic
159C4×C163C2×#4111yescharacteristic
160Dic163—21no—
161Dic163—21no—
162Dic163—41no—
163Dic163—42no—
164C2×C163—22no—
165C2×C163—82no—
166C2×C163C2×#12111yescharacteristic
167C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C163#57511yescharacteristic
169D163—22no—
170D163—41no—
171D163—81no—
172C163#305511yescharacteristic, complemented, Sylow
173#3055—1631nomaximal, Sylow
174C2×AGL(1,ℤ/8)—1631no—
175C22×M16—1631no—
176#525—1631no—
177C2×#121—1631no—
178#469—1632no—
179#468—1632no—
180#466—1632no—
181#465—1632no—
182#428—1631no—
183#418D16311yescharacteristic, complemented
184#417—1631no—
185C2×#41—1631no—
186C22×D4—3261no—
187C22×D4C2×D16311yescharacteristic
188C23×C4—1631no—
189AGL(1,ℤ/8)—3264no—
190C2×SD16—1632no—
191C2×D8—1632no—
192C2×M16—3264no—
193C2×M16—1631no—
194C2×M16—1632no—
195C2×M16C2×D16311yescharacteristic
196C22×C8—1632no—
197#123—3261no—
198C22⋊Q8—1631no—
199C22⋊Q8—1631no—
200#121—3261no—
201#121—3261no—
202#121—1632no—
203#121—1632no—
204#121—1631no—
205#121—1631no—
206#120—3261no—
207C4×D4—3261no—
208C4×D4—3261no—
209#117C2×D16311yescharacteristic
210C2×C4⋊C4—1631no—
211C2×C22⋊C4—3261no—
212#107—1632no—
213#106—1632no—
214#103—1632no—
215#102—1632no—
216#102C2×D16312yescomplemented
217#102C2×D16312yescomplemented
218C24—21no—
219C24—1631no—
220#41—3262no—
221#41—3262no—
222C2×Q8—1631no—
223C2×D4—22no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—3261no—
227C2×D4—3261no—
228C2×D4—3262no—
229C2×D4—3262no—
230C2×D4—3261no—
231C2×D4—6524no—
232C2×D4—1631no—
233C2×D4C22×D16312yes—
234C22×C4—3262no—
235C22×C4—3262no—
236C22×C4—3261no—
237C22×C4—3261no—
238C22×C4—1631no—
239C22×C4—1631no—
240C22×C4—1631no—
241C22×C4—1631no—
242C22×C4—1631no—
243C22×C4C22×D16311yescharacteristic
244SD16—3264no—
245D8—3264no—
246M16—22no—
247M16—3264no—
248M16—1634no—
249C2×C8—3264no—
250C2×C8—1632no—
251C2×C8C22×D16312yes—
252C4⋊C4—3261no—
253C4⋊C4—3261no—
254C4⋊C4—3261no—
255C4⋊C4—1632no—
256C4⋊C4C22×D16311yescharacteristic
257C4⋊C4C22×D16311yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—3261no—
260C22⋊C4—3261no—
261C22⋊C4—3261no—
262C22⋊C4—3261no—
263C22⋊C4—3262no—
264C22⋊C4—3262no—
265C42—21no—
266C23—22no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—3262no—
271C23—3262no—
272C23—3261no—
273C23—6521no—
274C23—1631no—
275C23—1631no—
276C23D4×D16311yescharacteristic
277Q8—3261no—
278Q8—3261no—
279D4—22no—
280D4—22no—
281D4—42no—
282D4—3261no—
283D4—3261no—
284D4—6522no—
285D4—6522no—
286D4—6524no—
287D4—6524no—
288D4—6522no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—3262no—
292C2×C4—3261no—
293C2×C4—3261no—
294C2×C4—3261no—
295C2×C4—3261no—
296C2×C4—3262no—
297C2×C4—3262no—
298C2×C4—3261no—
299C2×C4—3262no—
300C2×C4—6522no—
301C2×C4—6522no—
302C2×C4—1631no—
303C2×C4—1631no—
304C2×C4D4×D16312yes—
305C2×C4D4×D16311yescharacteristic
306C2×C4#1457969912yes—
307C2×C4C23×D16311yescharacteristic
308C8—22no—
309C8—1634no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—84no—
316C22—3261no—
317C22—3261no—
318C22—3261no—
319C22—3261no—
320C22—3261no—
321C22—6524no—
322C22—6521no—
323C22—6521no—
324C22—6521no—
325C22—1632no—
326C22#1910470812yes—
327C22C2×D4×D16311yescharacteristic
328C4—41no—
329C4—41no—
330C4—3261no—
331C4—3261no—
332C4—6521no—
333C4—6522no—
334C4#1910470812yes—
335C4C2×D4×D16311yescharacteristic
336C4C2×#1457969911yescharacteristic
337C2—22no—
338C2—82no—
339C2—13041no—
340C2—3262no—
341C2—6521no—
342C2C2×#1910470811yescharacteristic
343C2AGL(1,ℤ/8)×D16311yescharacteristic
344C2#2378246111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#418×C163C2
socleC22×C163C22×D4
hypercenter#418D163
hypocenterC163#3055
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#19104761C2semidirect
2. C2×M16×D163C2semidirect
3. #23782177C2semidirect
4. C2×#19104706C2semidirect
5. #23781917C2semidirect
6. #23781910C2semidirect
7. #23781900C2semidirect
8. #23781893C2semidirect
9. #23781708C2semidirect
10. #23781667C2semidirect
11. #418×C163C2semidirect
16. C2×#14579687C22semidirect
17. C2×#14579684C22semidirect
19. C2×#14579682C22semidirect
21. C2×C8×D163C22semidirect
23. #19104715C22semidirect
24. #19104715C22semidirect
25. #19104710C22semidirect
26. #19104710C22semidirect
36. #19104668C22semidirect
37. #19104666C22semidirect
38. #19104660C22semidirect
39. #19104657C22semidirect
43. #102×C163C22semidirect
44. #102×C163C22semidirect
67. #14579683D4semidirect
121. C163⋊C8C2×D4semidirect
154. C2×D163AGL(1,ℤ/8)semidirect
172. C163#3055semidirect
183. #418D163semidirect
216. #102C2×D163semidirect
217. #102C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047612
C2×M16×D1632
#237821772
C2×#191047062
#237819172
#237819102
#237819002
#237818932
#237817082
#237816672
#418×C1632
#3055163