Finite Group Databasebeta

← #27181271

Subgroups — #27181271

Order 12928.

450 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#20265713C211yesmaximal, characteristic, complemented
2C2×D4×Dic101C211yesmaximal, characteristic, complemented
3#20265505C211yesmaximal, characteristic, complemented
4#20265491C211yesmaximal, characteristic, complemented
5#20265475C212yesmaximal, complemented
6#20265394C212yesmaximal, complemented
7#20265311C211yesmaximal, characteristic
8#20265286C212yesmaximal, complemented
9#20265278C212yesmaximal, complemented
10#20265231C211yesmaximal, characteristic, complemented
11#450×C101C211yesmaximal, characteristic, complemented
12C23×Dic101—21no—
13C2×#13108059C2212yes—
14#15692250—21no—
15#15692246C2211yescharacteristic, complemented
16#15692244C2211yescharacteristic, complemented
17#15692239—21no—
18#15692229C2211yescharacteristic
19Q8×Dic101C2211yescharacteristic, complemented
20D4×Dic101—21no—
21D4×Dic101—21no—
22D4×Dic101C2212yes—
23D4×Dic101C2212yes—
24#15692217C2211yescharacteristic, complemented
25#15692215—21no—
26#15692213—21no—
27C2×C4⋊Dic101C2211yescharacteristic
28C2×C22⋊Dic101—21no—
29C2×C4×Dic101C2211yescharacteristic
30#15692189C2212yescomplemented
31#15692186C2212yescomplemented
32#15692184C2212yescomplemented
33#15692179C2212yescomplemented
34#15692177C2212yescomplemented
35#15692176C2212yes—
36C22⋊#11717579C2211yescharacteristic
37#15692165C2212yescomplemented
38C22×D4×C101C2211yescharacteristic
39C2×SD16×C101C2212yes—
40C22⋊Q8×C101C2211yescharacteristic
41#102×C101C2212yes—
42C22⋊C8×C101C2211yescharacteristic
43C2×D404C2311yescharacteristic
44C2×C22⋊D101—21no—
45C22×Dic101—21no—
46C22×Dic101—22no—
47C22×Dic101—22no—
48C22×Dic101—44no—
49C22×Dic101D411yescharacteristic
50C22×Dic101D411yescharacteristic
51C2×C4×D101—21no—
52#13108059—24no—
53C2×#11717579C2312yes—
54C4⋊Dic101—21no—
55C4⋊Dic101—21no—
56C4⋊Dic101—21no—
57C4⋊Dic101C2311yescharacteristic
58C4⋊Dic101C2312yes—
59#13108053—21no—
60C22⋊Dic101—22no—
61C22⋊Dic101—22no—
62#13108051—21no—
63#13108051—21no—
64#13108051—21no—
65#13108051—21no—
66C4×Dic101—21no—
67C4×Dic101D411yescharacteristic
68C4×Dic101D411yescharacteristic
69C4×Dic101C2311yescharacteristic
70C4×Dic101C2311yescharacteristic
71C24×C101—21no—
72C2×Q8×C101C2311yescharacteristic
73C2×D4×C101—24no—
74C2×D4×C101—21no—
75C2×D4×C101C2312yes—
76C22×C4×C101C2311yescharacteristic
77SD16×C101—24no—
78C2×C8×C101C2312yes—
79C4⋊C4×C101—21no—
80C4⋊C4×C101C2311yescharacteristic
81C22⋊C4×C101—21no—
82C23×C101—24no—
83C23×C101—22no—
84C23×C101—42no—
85C23×C101—42no—
86C23×C101C2×D411yescharacteristic
87C22×D101—21no—
88Q8×C101—22no—
89D4×C101—24no—
90D4×C101—22no—
91D4×C101#4114yes—
92C2×C4×C101—22no—
93C2×C4×C101—21no—
94C2×C4×C101—21no—
95C2×C4×C101C2×D411yescharacteristic
96C2×C4×C101C2411yescharacteristic
97C22⋊D101—42no—
98C2×Dic101—22no—
99C2×Dic101—21no—
100C2×Dic101—22no—
101C2×Dic101—41no—
102C2×Dic101—44no—
103C2×Dic101—44no—
104C2×Dic101—42no—
105C2×Dic101C2×D411yescharacteristic
106C2×Dic101C2×D411yescharacteristic
107C2×Dic101C2×D411yescharacteristic
108C2×Dic101C2×D411yescharacteristic
109D404—22no—
110C4×D101—42no—
111C8×C101—22no—
112#11717579—22no—
113C22×C101—21no—
114C22×C101—21no—
115C22×C101—24no—
116C22×C101—48no—
117C22×C101—42no—
118C22×C101—42no—
119C22×C101AGL(1,ℤ/8)12yes—
120C22×C101C22×D411yescharacteristic
121C2×D101—41no—
122C2×D101—41no—
123C2×D101—41no—
124C4×C101—21no—
125C4×C101—41no—
126C4×C101—41no—
127C4×C101C2×#4112yes—
128Dic101—21no—
129Dic101—21no—
130Dic101—21no—
131Dic101—21no—
132Dic101—42no—
133Dic101—41no—
134C2×C101—22no—
135C2×C101—44no—
136C2×C101#52511yescharacteristic
137C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C101#57511yescharacteristic
139D101—81no—
140#3114—1011nomaximal, Sylow
141C101#311411yescharacteristic, complemented, Sylow
142#529—1011no—
143C2×C4×D4—1011no—
144#483—1011no—
145#480—1011no—
146#478—1012no—
147#463—1012no—
148#450D10111yescharacteristic, complemented
149#447—1011no—
150#442—1012no—
151#440—1012no—
152#432—1011no—
153C22×D4C2×D10111yescharacteristic
154C23×C4—2021no—
155C2×SD16C2×D10112yes—
156C2×D8—1012no—
157#126—2021no—
158#125—1011no—
159#124—1011no—
160#123—2021no—
161C22⋊Q8C2×D10111yescharacteristic
162#121—1011no—
163C4×Q8—1011no—
164C4×D4—1012no—
165C4×D4—1012no—
166C4×D4—1011no—
167C4×D4—2021no—
168C4×D4—2021no—
169C4×D4—2021no—
170#117—2021no—
171C2×C4⋊C4—1011no—
172C2×C22⋊C4—2021no—
173C2×C42—1011no—
174#107—1012no—
175#106—1012no—
176C4⋊C8—1012no—
177#103—1012no—
178#102—1012no—
179#102—1012no—
180#102C2×D10112yes—
181C22⋊C8—1011no—
182C22⋊C8C2×D10111yescharacteristic
183#97—1012no—
184C24—21no—
185C2×Q8C22×D10111yescharacteristic
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—1011no—
189C2×D4—2021no—
190C2×D4C22×D10112yes—
191C22×C4—1011no—
192C22×C4—1011no—
193C22×C4—4044no—
194C22×C4—2021no—
195C22×C4—2022no—
196C22×C4—2022no—
197C22×C4—2021no—
198C22×C4C22×D10111yescharacteristic
199SD16—24no—
200D8—2024no—
201C2×C8—1012no—
202C2×C8C22×D10112yes—
203C4⋊C4—21no—
204C4⋊C4—1011no—
205C4⋊C4—1012no—
206C4⋊C4—2021no—
207C4⋊C4—2021no—
208C4⋊C4—2021no—
209C4⋊C4—2021no—
210C4⋊C4C22×D10111yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—2021no—
213C22⋊C4—2021no—
214C22⋊C4—2021no—
215C22⋊C4—2022no—
216C22⋊C4—2022no—
217C22⋊C4—2021no—
218C42—1011no—
219C42—1011no—
220C42—1011no—
221C42—1011no—
222C42—2021no—
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—2021no—
228C23D4×D10111yescharacteristic
229Q8—22no—
230D4—24no—
231D4—22no—
232D4—4042no—
233D4—2022no—
234D4#1310807314yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—1011no—
239C2×C4—1011no—
240C2×C4—1011no—
241C2×C4—1011no—
242C2×C4—4042no—
243C2×C4—4042no—
244C2×C4—4044no—
245C2×C4—4044no—
246C2×C4—4041no—
247C2×C4—2021no—
248C2×C4—2022no—
249C2×C4—2022no—
250C2×C4D4×D10111yescharacteristic
251C2×C4C23×D10111yescharacteristic
252C8—22no—
253C8—2022no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—4041no—
261C22—4041no—
262C22—4041no—
263C22#1569228212yes—
264C22C2×D4×D10111yescharacteristic
265C4—21no—
266C4—41no—
267C4—41no—
268C4—4042no—
269C4—4041no—
270C4—2021no—
271C4—2021no—
272C4—2021no—
273C4—2021no—
274C4C2×#1310807312yes—
275C2—22no—
276C2—44no—
277C2—8081no—
278C2#2026568311yescharacteristic
279C2C2×#1569228211yescharacteristic
280C2#2026597311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#450×C101C2
socleC22×C101C22×D4
prime radical#20265311C2
hypercenter#450D101
hypocenterC101#3114
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #20265713C2semidirect
2. C2×D4×Dic101C2semidirect
3. #20265505C2semidirect
4. #20265491C2semidirect
5. #20265475C2semidirect
6. #20265394C2semidirect
8. #20265286C2semidirect
9. #20265278C2semidirect
10. #20265231C2semidirect
11. #450×C101C2semidirect
15. #15692246C22semidirect
16. #15692244C22semidirect
19. Q8×Dic101C22semidirect
24. #15692217C22semidirect
30. #15692189C22semidirect
31. #15692186C22semidirect
32. #15692184C22semidirect
33. #15692179C22semidirect
34. #15692177C22semidirect
37. #15692165C22semidirect
141. C101#3114semidirect
148. #450D101semidirect

Maximal subgroups

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

subgroupindex
#202657132
C2×D4×Dic1012
#202655052
#202654912
#202654752
#202653942
#202653112
#202652862
#202652782
#202652312
#450×C1012
#3114101