Finite Group Databasebeta

← #35224167

Subgroups — #35224167

Order 20864.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×#19104768C211yesmaximal, characteristic, complemented
2C2×M16×D163C211yesmaximal, characteristic, complemented
3#23782177C211yesmaximal, characteristic
4C2×#19104715C211yesmaximal, characteristic, complemented
5#23781924C212yesmaximal, complemented
6#23781912C212yesmaximal, complemented
7#23781907C212yesmaximal, complemented
8#23781895C212yesmaximal, complemented
9#23781673C211yesmaximal, characteristic, complemented
10#23781664C211yesmaximal, characteristic, complemented
11#428×C163C211yesmaximal, characteristic, complemented
12C2×#14579699C2211yescharacteristic
13C22×Dic326C2211yescharacteristic
14C22×C4×D163C2211yescharacteristic
15#19104768—24no—
16C2×Dic652C2212yescomplemented
17C2×#14579688C2212yescomplemented
18C2×#14579683C2211yescharacteristic
19C2×#14579682C2212yes—
20M16×D163—24no—
21C2×C8×D163C2212yes—
22#19104719—21no—
23#19104715—21no—
24#19104715—21no—
25#19104715C2212yes—
26#19104715C2211yescharacteristic
27#19104715C2212yes—
28#19104715C2211yescharacteristic
29#19104710C2211yescharacteristic
30#19104710C2211yescharacteristic
31#19104705—21no—
32C4×D652—21no—
33C4×D652—21no—
34C2×C4⋊Dic163—21no—
35C2×(C2×D163)⋊C4—21no—
36#19104658C2212yescomplemented
37#19104658C2212yescomplemented
38#19104658C2212yescomplemented
39#19104655C2212yescomplemented
40C2×M16×C163C2211yescharacteristic
41#117×C163C2211yescharacteristic
42C2×C4⋊C4×C163C2211yescharacteristic
43#107×C163C2212yescomplemented
44#106×C163C2212yescomplemented
45C23×D163D411yescharacteristic
46#14579699—22no—
47#14579699—22no—
48C2×Dic326—21no—
49C2×Dic326—22no—
50C2×Dic326—22no—
51C2×Dic326C2312yes—
52C2×Dic326C2311yescharacteristic
53C2×D652—21no—
54C2×D652—21no—
55C2×D652C2311yescharacteristic
56C2×C22⋊D163—21no—
57C22×Dic163—21no—
58C22×Dic163D411yescharacteristic
59C2×C4×D163—22no—
60C2×C4×D163—21no—
61C2×C4×D163—22no—
62C2×C4×D163D411yescharacteristic
63C2×C4×D163D411yescharacteristic
64C2×C4×D163C2311yescharacteristic
65C2×C4×D163C2311yescharacteristic
66Dic652—24no—
67#14579688—24no—
68#14579683—22no—
69#14579682—24no—
70C2×C163⋊C8C2312yes—
71C8×D163—24no—
72C4⋊Dic163—22no—
73C4⋊Dic163—22no—
74C4⋊Dic163—21no—
75C4⋊Dic163—21no—
76(C2×D163)⋊C4—22no—
77(C2×D163)⋊C4—22no—
78(C2×D163)⋊C4—21no—
79(C2×D163)⋊C4—21no—
80(C2×D163)⋊C4—21no—
81(C2×D163)⋊C4—21no—
82C22×C4×C163—21no—
83C22×C4×C163C2311yescharacteristic
84M16×C163D414yescomplemented
85C2×C8×C163C2312yes—
86C4⋊C4×C163—21no—
87C4⋊C4×C163C2311yescharacteristic
88C4⋊C4×C163C2312yes—
89C4⋊C4×C163C2311yescharacteristic
90C22⋊C4×C163—21no—
91C42×C163—21no—
92C22×D163—22no—
93C22×D163—22no—
94C22×D163—21no—
95C22×D163—41no—
96C22×D163C2×D411yescharacteristic
97C22×D163C2×D411yescharacteristic
98Dic326—22no—
99Dic326—21no—
100Dic326—22no—
101Dic326—21no—
102Dic326—42no—
103D652—21no—
104D652—21no—
105D652—42no—
106D652—42no—
107C22⋊D163—42no—
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⋊C8—22no—
122C23×C163C2×D411yescharacteristic
123C2×C4×C163—21no—
124C2×C4×C163—21no—
125C2×C4×C163—22no—
126C2×C4×C163—42no—
127C2×C4×C163C2×D411yescharacteristic
128C2×C4×C163C2×D412yes—
129C2×C4×C163#4112yes—
130C2×C4×C163C2411yescharacteristic
131C8×C163C2×D414yes—
132C22×C163—21no—
133C22×C163—21no—
134C22×C163#12112yes—
135C22×C163C22×D411yescharacteristic
136C2×D163—21no—
137C2×D163—21no—
138C2×D163—21no—
139C2×D163—21no—
140C2×D163—21no—
141C2×D163—44no—
142C2×D163—41no—
143C2×D163—41no—
144C2×D163—41no—
145C2×D163#13712yescomplemented
146C4×C163—41no—
147C4×C163—42no—
148C4×C163—41no—
149C4×C163#12112yes—
150C4×C163C22×D411yescharacteristic
151C4×C163C2×#4111yescharacteristic
152Dic163—21no—
153Dic163—21no—
154Dic163—41no—
155Dic163—42no—
156C2×C163—22no—
157C2×C163C2×#12111yescharacteristic
158C2×C163C2×#13711yescharacteristic
159C2×C163#57511yescharacteristic
160D163—22no—
161D163—41no—
162D163—81no—
163C163#305611yescharacteristic, complemented, Sylow
164#3056—1631nomaximal, Sylow
165C2×#137—1631no—
166C22×M16—1631no—
167#525—1631no—
168C2×C22⋊Q8—1631no—
169#470—1632no—
170#468—1632no—
171#467—1632no—
172#465—1632no—
173#428D16311yescharacteristic, complemented
174#419—1631no—
175#417—1631no—
176C2×#41—1631no—
177C22×Q8—1631no—
178C23×C4—1631no—
179#137—3264no—
180C2×Q16—1632no—
181C2×SD16—1632no—
182C2×M16—3264no—
183C2×M16—1631no—
184C2×M16—1632no—
185C2×M16C2×D16311yescharacteristic
186C22×C8—1632no—
187#123—3261no—
188C22⋊Q8—3261no—
189C22⋊Q8—3261no—
190C22⋊Q8—1632no—
191C22⋊Q8—1632no—
192C22⋊Q8—1631no—
193C22⋊Q8—1631no—
194#121—1631no—
195#121—1631no—
196#120—3261no—
197C4×D4—3261no—
198C4×D4—3261no—
199#117C2×D16311yescharacteristic
200C2×C4⋊C4—3261no—
201C2×C4⋊C4C2×D16311yescharacteristic
202C2×C22⋊C4—3261no—
203#107C2×D16312yescomplemented
204#106C2×D16312yescomplemented
205#103—1632no—
206#103—1632no—
207#103—1632no—
208#102—1632no—
209C24—1631no—
210#41—3262no—
211#41—3262no—
212C2×Q8—3261no—
213C2×Q8—3262no—
214C2×Q8—3262no—
215C2×Q8—1632no—
216C2×Q8—1631no—
217C2×D4—3261no—
218C2×D4—3261no—
219C2×D4—3261no—
220C2×D4—1631no—
221C22×C4—21no—
222C22×C4—3262no—
223C22×C4—3262no—
224C22×C4—3261no—
225C22×C4—3261no—
226C22×C4—1631no—
227C22×C4—1631no—
228C22×C4—1631no—
229C22×C4—1631no—
230C22×C4—1631no—
231C22×C4C22×D16311yescharacteristic
232Q16—3264no—
233SD16—3264no—
234M16—3262no—
235M16—3264no—
236M16D65214yescomplemented
237C2×C8—3264no—
238C2×C8—1632no—
239C2×C8C22×D16312yes—
240C4⋊C4—21no—
241C4⋊C4—3262no—
242C4⋊C4—3262no—
243C4⋊C4—3261no—
244C4⋊C4—3261no—
245C4⋊C4C22×D16312yes—
246C4⋊C4C22×D16311yescharacteristic
247C4⋊C4C22×D16311yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—3262no—
250C22⋊C4—3262no—
251C22⋊C4—3261no—
252C22⋊C4—3261no—
253C22⋊C4—3261no—
254C22⋊C4—3261no—
255C42—21no—
256C23—3262no—
257C23—3262no—
258C23—3261no—
259C23—6521no—
260C23—1631no—
261C23—1631no—
262C23D4×D16311yescharacteristic
263Q8—3262no—
264Q8—3261no—
265Q8—3262no—
266Q8—3261no—
267Q8—6522no—
268D4—3261no—
269D4—3261no—
270D4—6522no—
271D4—6522no—
272D4—6522no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—3262no—
278C2×C4—3261no—
279C2×C4—3261no—
280C2×C4—3261no—
281C2×C4—3261no—
282C2×C4—3262no—
283C2×C4—3262no—
284C2×C4—3261no—
285C2×C4—3262no—
286C2×C4—6522no—
287C2×C4—6522no—
288C2×C4—1631no—
289C2×C4—1631no—
290C2×C4D4×D16311yescharacteristic
291C2×C4C2×D65212yes—
292C2×C4#1457970012yes—
293C2×C4C23×D16311yescharacteristic
294C8—3262no—
295C8C2×D65214yes—
296C22—21no—
297C22—21no—
298C22—3261no—
299C22—3261no—
300C22—3261no—
301C22—3261no—
302C22—3261no—
303C22—6524no—
304C22—6521no—
305C22—6521no—
306C22—6521no—
307C22—1632no—
308C22#1910471012yes—
309C22C2×D4×D16311yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—3261no—
314C4—3261no—
315C4—6521no—
316C4—6522no—
317C4#1910471012yes—
318C4C22×D65211yescharacteristic
319C4C2×#1457970011yescharacteristic
320C2—22no—
321C2—13041no—
322C2—3262no—
323C2—6521no—
324C2C2×#1910471011yescharacteristic
325C2#137×D16311yescharacteristic
326C2#2378246111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#428×C163C2
socleC22×C163C22×D4
prime radical#23782177C2
hypercenter#428D163
hypocenterC163#3056
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#19104768C2semidirect
2. C2×M16×D163C2semidirect
4. C2×#19104715C2semidirect
5. #23781924C2semidirect
6. #23781912C2semidirect
7. #23781907C2semidirect
8. #23781895C2semidirect
9. #23781673C2semidirect
10. #23781664C2semidirect
11. #428×C163C2semidirect
16. C2×Dic652C22semidirect
17. C2×#14579688C22semidirect
36. #19104658C22semidirect
37. #19104658C22semidirect
38. #19104658C22semidirect
39. #19104655C22semidirect
43. #107×C163C22semidirect
44. #106×C163C22semidirect
84. M16×C163D4semidirect
145. C2×D163#137semidirect
163. C163#3056semidirect
173. #428D163semidirect
203. #107C2×D163semidirect
204. #106C2×D163semidirect
236. M16D652semidirect

Maximal subgroups

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

subgroupindex
C2×#191047682
C2×M16×D1632
#237821772
C2×#191047152
#237819242
#237819122
#237819072
#237818952
#237816732
#237816642
#428×C1632
#3056163