Finite Group Databasebeta

← #27181270

Subgroups — #27181270

Order 12928.

450 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 129 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#20265711C211yesmaximal, characteristic, complemented
2#20265507C211yesmaximal, characteristic, complemented
3#20265490C211yesmaximal, characteristic, complemented
4#20265477C212yesmaximal, complemented
5#20265392C212yesmaximal, complemented
6#20265328C211yesmaximal, characteristic, complemented
7#20265311C211yesmaximal, characteristic
8#20265287C212yesmaximal, complemented
9#20265277C212yesmaximal, complemented
10#20265234C211yesmaximal, characteristic, complemented
11C2×C4×D4×C101C211yesmaximal, characteristic, complemented
12C2×#13108062C2212yes—
13C2×#13108059C2212yes—
14#15692249—21no—
15#15692247C2211yescharacteristic, complemented
16#15692243C2211yescharacteristic, complemented
17#15692238—21no—
18#15692234C2211yescharacteristic
19#15692229C2211yescharacteristic
20C4×Dic202C2211yescharacteristic, complemented
21C4×D404C2211yescharacteristic, complemented
22C4×C22⋊D101—21no—
23#15692212—21no—
24#15692188C2212yescomplemented
25#15692186C2212yescomplemented
26C4⋊#11717579C2212yescomplemented
27#15692178C2212yescomplemented
28#15692176C2212yes—
29#15692176C2212yes—
30#15692175C2212yescomplemented
31C22⋊#11717579C2211yescharacteristic
32C22⋊#11717579C2211yescharacteristic
33#15692166C2212yescomplemented
34C22×D4×C101C2211yescharacteristic
35C23×C4×C101—21no—
36C4×D4×C101—21no—
37C4×D4×C101—21no—
38C4×D4×C101C2212yes—
39C4×D4×C101C2212yes—
40C2×C4⋊C4×C101C2211yescharacteristic
41C2×C22⋊C4×C101—21no—
42C2×C42×C101C2211yescharacteristic
43C2×Dic202C2311yescharacteristic
44C2×D404C2311yescharacteristic
45C2×C22⋊D101—21no—
46C2×C4×D101—21no—
47#13108062—24no—
48#13108059—24no—
49C2×#11717579C2312yes—
50C2×#11717579C2312yes—
51C4⋊Dic101C2311yescharacteristic
52C4⋊Dic101C2311yescharacteristic
53#13108053—21no—
54#13108053—21no—
55#13108053—21no—
56#13108053—21no—
57C22⋊Dic101—21no—
58#13108051—21no—
59#13108051—21no—
60#13108051—21no—
61#13108051—21no—
62C4×Dic101—21no—
63C24×C101—21no—
64C2×D4×C101—24no—
65C2×D4×C101—21no—
66C2×D4×C101C2312yes—
67C22×C4×C101—21no—
68C22×C4×C101—22no—
69C22×C4×C101—22no—
70C22×C4×C101—44no—
71C22×C4×C101D411yescharacteristic
72C22×C4×C101D411yescharacteristic
73C22×C4×C101C2311yescharacteristic
74C4⋊C4×C101—21no—
75C4⋊C4×C101C2312yes—
76C22⋊C4×C101—22no—
77C22⋊C4×C101—22no—
78C42×C101D411yescharacteristic
79C42×C101D411yescharacteristic
80C42×C101C2311yescharacteristic
81C42×C101C2311yescharacteristic
82C23×C101—24no—
83C23×C101—22no—
84C23×C101—42no—
85C23×C101—42no—
86C23×C101C2×D411yescharacteristic
87C22×D101—21no—
88D4×C101—24no—
89D4×C101—22no—
90D4×C101#4114yes—
91C2×C4×C101—22no—
92C2×C4×C101—22no—
93C2×C4×C101—22no—
94C2×C4×C101—41no—
95C2×C4×C101—44no—
96C2×C4×C101—44no—
97C2×C4×C101—42no—
98C2×C4×C101C2×D411yescharacteristic
99C2×C4×C101C2×D411yescharacteristic
100C2×C4×C101C2×D411yescharacteristic
101C2×C4×C101C2×D411yescharacteristic
102C2×C4×C101C2×D411yescharacteristic
103C2×C4×C101C2411yescharacteristic
104C22⋊D101—42no—
105C2×Dic101—21no—
106C2×Dic101—21no—
107C2×Dic101—21no—
108D404—22no—
109C4×D101—42no—
110Dic202—22no—
111#11717579—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—21no—
126C4×C101—21no—
127C4×C101—21no—
128C4×C101—21no—
129C4×C101—42no—
130C4×C101C2×#4112yes—
131Dic101—41no—
132Dic101—41no—
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×D4D10111yescharacteristic, complemented
144#483—1011no—
145#480—1011no—
146#478—1012no—
147#463—1012no—
148#450—1011no—
149#447—1011no—
150#442—1012no—
151#440—1012no—
152#432—1011no—
153C22×D4C2×D10111yescharacteristic
154C23×C4—21no—
155C2×SD16—1012no—
156C2×D8—1012no—
157#126—2021no—
158#125—1011no—
159#124—1011no—
160#123—2021no—
161C22⋊Q8—1011no—
162#121—1011no—
163C4×Q8—1011no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4—1011no—
167C4×D4—2021no—
168C4×D4C2×D10112yes—
169C4×D4C2×D10112yes—
170#117—2021no—
171C2×C4⋊C4C2×D10111yescharacteristic
172C2×C22⋊C4—21no—
173C2×C42C2×D10111yescharacteristic
174#107—1012no—
175#106—1012no—
176C4⋊C8—1012no—
177#103—1012no—
178#102—1012no—
179#102—1012no—
180#102—1012no—
181C22⋊C8—1011no—
182C22⋊C8—1011no—
183#97—1012no—
184C24—21no—
185C2×Q8—1011no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—1011no—
189C2×D4—2021no—
190C2×D4C22×D10112yes—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—2021no—
196C22×C4C22⋊D10111yescharacteristic
197C22×C4C22⋊D10111yescharacteristic
198C22×C4C22×D10111yescharacteristic
199SD16—2024no—
200D8—2024no—
201C2×C8—1012no—
202C2×C8—1012no—
203C4⋊C4—21no—
204C4⋊C4—1011no—
205C4⋊C4—1011no—
206C4⋊C4—2021no—
207C4⋊C4—2021no—
208C4⋊C4—2021no—
209C4⋊C4—2021no—
210C4⋊C4C22×D10112yes—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—2021no—
214C22⋊C4—2021no—
215C22⋊C4—2021no—
216C22⋊C4—2021no—
217C22⋊C4—2021no—
218C42—2021no—
219C42C22⋊D10111yescharacteristic
220C42C22⋊D10111yescharacteristic
221C42C22×D10111yescharacteristic
222C42C22×D10111yescharacteristic
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—2021no—
228C23C2×C22⋊D10111yescharacteristic
229Q8—2022no—
230D4—24no—
231D4—22no—
232D4—4042no—
233D4—2022no—
234D4#1310807414yes—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—4042no—
243C2×C4—2021no—
244C2×C4—2021no—
245C2×C4—2021no—
246C2×C4C2×C22⋊D10111yescharacteristic
247C2×C4C2×C22⋊D10111yescharacteristic
248C2×C4C2×C22⋊D10111yescharacteristic
249C2×C4C2×C22⋊D10111yescharacteristic
250C2×C4C2×C22⋊D10111yescharacteristic
251C2×C4C23×D10111yescharacteristic
252C8—2022no—
253C8—2022no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—4041no—
261C22—4041no—
262C22—4041no—
263C22#1569228112yes—
264C22C22×C22⋊D10111yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—4041no—
272C4—4041no—
273C4—4041no—
274C4C2×#1310807412yes—
275C2—22no—
276C2—44no—
277C2—8081no—
278C2#2026568811yescharacteristic
279C2C2×#1569228111yescharacteristic
280C2#2026597511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×C4×D4×C101C2
socleC22×C101C22×D4
prime radical#20265311C2
hypercenterC2×C4×D4D101
hypocenterC101#3114
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #20265711C2semidirect
2. #20265507C2semidirect
3. #20265490C2semidirect
4. #20265477C2semidirect
5. #20265392C2semidirect
6. #20265328C2semidirect
8. #20265287C2semidirect
9. #20265277C2semidirect
10. #20265234C2semidirect
11. C2×C4×D4×C101C2semidirect
15. #15692247C22semidirect
16. #15692243C22semidirect
20. C4×Dic202C22semidirect
21. C4×D404C22semidirect
24. #15692188C22semidirect
25. #15692186C22semidirect
26. C4⋊#11717579C22semidirect
27. #15692178C22semidirect
30. #15692175C22semidirect
33. #15692166C22semidirect
141. C101#3114semidirect
143. C2×C4×D4D101semidirect

Maximal subgroups

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

subgroupindex
#202657112
#202655072
#202654902
#202654772
#202653922
#202653282
#202653112
#202652872
#202652772
#202652342
C2×C4×D4×C1012
#3114101