Finite Group Databasebeta

← #27181265

Subgroups — #27181265

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#20265708C211yesmaximal, characteristic, complemented
2C2×D4×Dic101C211yesmaximal, characteristic, complemented
3#20265508C211yesmaximal, characteristic, complemented
4#20265488C211yesmaximal, characteristic, complemented
5#20265479C212yesmaximal, complemented
6#20265390C212yesmaximal, complemented
7#20265328C211yesmaximal, characteristic, complemented
8#20265289C212yesmaximal, complemented
9#20265274C212yesmaximal, complemented
10#20265231C211yesmaximal, characteristic, complemented
11#447×C101C211yesmaximal, characteristic
12C23×Dic101—21no—
13C2×#13108062C2212yes—
14#15692248—21no—
15#15692246C2211yescharacteristic, complemented
16#15692242C2211yescharacteristic, complemented
17#15692236—21no—
18#15692234C2211yescharacteristic
19#15692220C2211yescharacteristic, complemented
20D4×Dic101—21no—
21D4×Dic101—21no—
22D4×Dic101—21no—
23D4×Dic101C2212yes—
24D4×Dic101C2212yes—
25D4×Dic101C2211yescharacteristic, complemented
26#15692211—21no—
27C2×C4⋊Dic101C2211yescharacteristic
28C2×C22⋊Dic101—21no—
29C2×C4×Dic101C2211yescharacteristic
30#15692188C2212yescomplemented
31#15692187C2212yescomplemented
32#15692184C2212yescomplemented
33#15692180C2212yescomplemented
34#15692176C2212yes—
35#15692176C2212yescomplemented
36C22⋊#11717579C2211yescharacteristic
37#15692165C2212yescomplemented
38C22×D4×C101C2211yescharacteristic
39C2×D8×C101C2212yes—
40#121×C101C2211yescharacteristic
41#102×C101C2212yes—
42C22⋊C8×C101C2211yescharacteristic
43C2×Dic202C2311yescharacteristic
44C22×Dic101—21no—
45C22×Dic101—22no—
46C22×Dic101—22no—
47C22×Dic101—21no—
48C22×Dic101—44no—
49C22×Dic101D411yescharacteristic
50C22×Dic101D411yescharacteristic
51#13108062—24no—
52C2×#11717579C2312yes—
53C4⋊Dic101—21no—
54C4⋊Dic101—21no—
55C4⋊Dic101C2311yescharacteristic
56C4⋊Dic101C2312yes—
57#13108053—21no—
58#13108053—21no—
59#13108053—21no—
60C22⋊Dic101—21no—
61C22⋊Dic101—22no—
62C22⋊Dic101—22no—
63C22⋊Dic101—21no—
64C22⋊Dic101—21no—
65C22⋊Dic101—21no—
66C4×Dic101—21no—
67C4×Dic101D411yescharacteristic
68C4×Dic101D411yescharacteristic
69C4×Dic101C2311yescharacteristic
70C4×Dic101C2311yescharacteristic
71C24×C101—21no—
72C2×D4×C101—24no—
73C2×D4×C101—21no—
74C2×D4×C101—21no—
75C2×D4×C101C2311yescharacteristic
76C2×D4×C101C2312yes—
77C22×C4×C101C2311yescharacteristic
78D8×C101—24no—
79C2×C8×C101C2312yes—
80C4⋊C4×C101C2311yescharacteristic
81C22⋊C4×C101—21no—
82C23×C101—24no—
83C23×C101—22no—
84C23×C101—21no—
85C23×C101—42no—
86C23×C101—42no—
87C23×C101C2×D411yescharacteristic
88D4×C101—24no—
89D4×C101—22no—
90D4×C101—22no—
91D4×C101—42no—
92D4×C101#4114yes—
93C2×C4×C101—22no—
94C2×C4×C101—21no—
95C2×C4×C101C2×D411yescharacteristic
96C2×C4×C101C2411yescharacteristic
97C2×Dic101—22no—
98C2×Dic101—21no—
99C2×Dic101—21no—
100C2×Dic101—22no—
101C2×Dic101—41no—
102C2×Dic101—42no—
103C2×Dic101—44no—
104C2×Dic101—44no—
105C2×Dic101—42no—
106C2×Dic101C2×D411yescharacteristic
107C2×Dic101C2×D411yescharacteristic
108C2×Dic101C2×D411yescharacteristic
109C2×Dic101C2×D411yescharacteristic
110Dic202—22no—
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×C101—41no—
120C22×C101—41no—
121C22×C101—41no—
122C22×C101AGL(1,ℤ/8)12yes—
123C22×C101C22×D411yescharacteristic
124C4×C101—21no—
125C4×C101—41no—
126C4×C101C2×#4112yes—
127Dic101—21no—
128Dic101—21no—
129Dic101—21no—
130Dic101—21no—
131Dic101—41no—
132Dic101—42no—
133Dic101—41no—
134C2×C101—22no—
135C2×C101—44no—
136C2×C101—81no—
137C2×C101#52511yescharacteristic
138C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
139C2×C101#57511yescharacteristic
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#450—1011no—
149#447D10111yescharacteristic
150#442—1012no—
151#440—1012no—
152#432—1011no—
153C22×D4C2×D10111yescharacteristic
154C23×C4—2021no—
155C2×SD16—1012no—
156C2×D8C2×D10112yes—
157#126—2021no—
158#125—1011no—
159#124—1011no—
160#123—2021no—
161C22⋊Q8—1011no—
162#121C2×D10111yescharacteristic
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×Q8—1011no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4C22×D10111yescharacteristic
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—2024no—
200D8—24no—
201C2×C8—1012no—
202C2×C8C22×D10112yes—
203C4⋊C4—1011no—
204C4⋊C4—1012no—
205C4⋊C4—2021no—
206C4⋊C4—2021no—
207C4⋊C4—2021no—
208C4⋊C4—2021no—
209C4⋊C4—2021no—
210C4⋊C4C22×D10111yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—2022no—
213C22⋊C4—2022no—
214C22⋊C4—2021no—
215C22⋊C4—2021no—
216C22⋊C4—2021no—
217C22⋊C4—2021no—
218C42—1011no—
219C42—1011no—
220C42—1011no—
221C42—1011no—
222C42—2021no—
223C23—24no—
224C23—22no—
225C23—21no—
226C23—42no—
227C23—42no—
228C23D4×D10111yescharacteristic
229Q8—2022no—
230D4—24no—
231D4—22no—
232D4—22no—
233D4—42no—
234D4#1310807314yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—1011no—
238C2×C4—1011no—
239C2×C4—1011no—
240C2×C4—1011no—
241C2×C4—4042no—
242C2×C4—4044no—
243C2×C4—4044no—
244C2×C4—4042no—
245C2×C4—4041no—
246C2×C4—2021no—
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—41no—
261C22—41no—
262C22—41no—
263C22#1569227912yes—
264C22C2×D4×D10111yescharacteristic
265C4—21no—
266C4—41no—
267C4—4041no—
268C4—4042no—
269C4—4041no—
270C4—2021no—
271C4—2021no—
272C4—2021no—
273C4—2021no—
274C4C2×#1310807312yes—
275C2—22no—
276C2—44no—
277C2—81no—
278C2#2026568311yescharacteristic
279C2C2×#1569227911yescharacteristic
280C2#2026597311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting, prime radical#447×C101C2
socleC22×C101C22×D4
hypercenter#447D101
hypocenterC101#3114
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #20265708C2semidirect
2. C2×D4×Dic101C2semidirect
3. #20265508C2semidirect
4. #20265488C2semidirect
5. #20265479C2semidirect
6. #20265390C2semidirect
7. #20265328C2semidirect
8. #20265289C2semidirect
9. #20265274C2semidirect
10. #20265231C2semidirect
15. #15692246C22semidirect
16. #15692242C22semidirect
19. #15692220C22semidirect
25. D4×Dic101C22semidirect
30. #15692188C22semidirect
31. #15692187C22semidirect
32. #15692184C22semidirect
33. #15692180C22semidirect
35. #15692176C22semidirect
37. #15692165C22semidirect
141. C101#3114semidirect

Maximal subgroups

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

subgroupindex
#202657082
C2×D4×Dic1012
#202655082
#202654882
#202654792
#202653902
#202653282
#202652892
#202652742
#202652312
#447×C1012
#3114101