Finite Group Databasebeta

← #27181040

Subgroups — #27181040

Order 12928.

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

normal subgroups only

This group has 420 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#20265687C211yesmaximal, characteristic
2C2×#15692213C211yesmaximal, characteristic, complemented
3#20265520C212yesmaximal, complemented
4#20265513C211yesmaximal, characteristic, complemented
5#20265507C211yesmaximal, characteristic, complemented
6#20265499C212yesmaximal, complemented
7#20265493C211yesmaximal, characteristic, complemented
8#20265214C212yesmaximal, complemented
9#20265170C212yesmaximal, complemented
10C22⋊C8×D101C211yesmaximal, characteristic, complemented
11#480×C101C211yesmaximal, characteristic, complemented
12C2×#13108073C2211yescharacteristic
13C22×C4×D101C2211yescharacteristic
14C2×C8×D101C2212yes—
15#15692237—21no—
16#15692235C2211yescharacteristic
17#15692234C2211yescharacteristic
18#15692226C2211yescharacteristic
19#15692223—21no—
20D4×Dic101—21no—
21#15692215—21no—
22#15692213—21no—
23#15692213—21no—
24#15692213C2212yes—
25#15692213C2212yes—
26#15692213C2211yescharacteristic
27C2×C4⋊Dic101C2211yescharacteristic
28C2×#13108051—21no—
29C2×C4×Dic101—21no—
30#15692189C2212yescomplemented
31#15692186C2212yescomplemented
32#15692186C2212yescomplemented
33#15692180C2212yescomplemented
34#15692178C2212yescomplemented
35#15692176C2212yescomplemented
36C22⋊#11717579C2211yescharacteristic
37#15692167C2212yes—
38#121×C101C2211yescharacteristic
39C2×C4⋊C4×C101C2211yescharacteristic
40#107×C101C2212yescomplemented
41#102×C101C2212yescomplemented
42C22⋊C8×C101C2211yescharacteristic
43C23×D101D411yescharacteristic
44#13108073—24no—
45C2×Dic202C2311yescharacteristic
46C2×C22⋊D101—21no—
47C2×C22⋊D101—21no—
48C22×Dic101—21no—
49C22×Dic101—21no—
50C22×Dic101D411yescharacteristic
51C2×C4×D101—24no—
52C2×C4×D101D411yescharacteristic
53C2×C4×D101D411yescharacteristic
54C2×C4×D101C2311yescharacteristic
55C2×C4×D101C2311yescharacteristic
56C2×#11717579C2312yes—
57C8×D101—24no—
58C4⋊Dic101—21no—
59C4⋊Dic101—21no—
60C4⋊Dic101C2312yes—
61C4⋊Dic101C2311yescharacteristic
62#13108053—21no—
63C22⋊Dic101—21no—
64C22⋊Dic101—21no—
65#13108051—22no—
66#13108051—22no—
67#13108051—21no—
68#13108051—21no—
69C4×Dic101—22no—
70C4×Dic101—22no—
71C4×Dic101—21no—
72C2×D4×C101—21no—
73C2×D4×C101C2311yescharacteristic
74C22×C4×C101—21no—
75C22×C4×C101C2311yescharacteristic
76C2×C8×C101C2312yes—
77C4⋊C4×C101—21no—
78C4⋊C4×C101C2312yes—
79C4⋊C4×C101C2311yescharacteristic
80C22⋊C4×C101—21no—
81C23×C101—21no—
82C23×C101C2×D411yescharacteristic
83C22×D101—22no—
84C22×D101—22no—
85C22×D101—41no—
86C22×D101C2×D411yescharacteristic
87C22×D101C2×D411yescharacteristic
88D4×C101—22no—
89D4×C101—42no—
90C2×C4×C101—22no—
91C2×C4×C101—21no—
92C2×C4×C101—22no—
93C2×C4×C101—42no—
94C2×C4×C101C2×D411yescharacteristic
95C2×C4×C101C2411yescharacteristic
96C22⋊D101—42no—
97C22⋊D101—42no—
98C2×Dic101—22no—
99C2×Dic101—22no—
100C2×Dic101—21no—
101C2×Dic101—21no—
102C2×Dic101—42no—
103C2×Dic101—42no—
104C2×Dic101C2×D411yescharacteristic
105C2×Dic101C2×D411yescharacteristic
106C4×D101—22no—
107C4×D101—22no—
108C4×D101—22no—
109C4×D101#4114yes—
110Dic202—22no—
111C8×C101—22no—
112#11717579—22no—
113C22×C101—21no—
114C22×C101—21no—
115C22×C101—41no—
116C22×C101—41no—
117C22×C101—41no—
118C22×C101#13512yes—
119C22×C101C22×D411yescharacteristic
120C2×D101—21no—
121C2×D101—21no—
122C2×D101—21no—
123C2×D101—21no—
124C2×D101—21no—
125C2×D101—44no—
126C2×D101#12312yes—
127C4×C101—21no—
128C4×C101—42no—
129C4×C101—41no—
130C4×C101C2×#4112yes—
131Dic101—21no—
132Dic101—41no—
133Dic101—42no—
134Dic101—41no—
135Dic101#12312yes—
136C2×C101—22no—
137C2×C101—81no—
138C2×C101C2×#12311yescharacteristic
139C2×C101C2×#13511yescharacteristic
140C2×C101#57511yescharacteristic
141D101—22no—
142D101—41no—
143#3091—1011nomaximal, Sylow
144C101#309111yescharacteristic, complemented, Sylow
145#525—1011no—
146C2×#117—1011no—
147#485—1012no—
148#484—1011no—
149#483—1011no—
150#482—1012no—
151#481—1011no—
152#480D10111yescharacteristic, complemented
153#427—1012no—
154#416—1012no—
155C2×C22⋊C8—1011no—
156C2×#41—1011no—
157C23×C4—1011no—
158C22×C8—1012no—
159#123—2021no—
160C22⋊Q8—1011no—
161C22⋊Q8—1011no—
162#121—1011no—
163#121C2×D10111yescharacteristic
164#120—2021no—
165C4×D4—2021no—
166C4×D4—2021no—
167#117—1012no—
168#117—1012no—
169#117—1011no—
170#117—2021no—
171#117—2021no—
172C2×C4⋊C4—1011no—
173C2×C4⋊C4C2×D10111yescharacteristic
174C2×C22⋊C4—2021no—
175C2×C42—2021no—
176#107—1012no—
177#107C2×D10112yescomplemented
178#106—1012no—
179#106—1012no—
180#103—1012no—
181#103—1012no—
182#102—1012no—
183#102C2×D10112yescomplemented
184C22⋊C8—1011no—
185C22⋊C8—1012no—
186C22⋊C8C2×D10111yescharacteristic
187C24—1011no—
188#41—2024no—
189C2×Q8—1011no—
190C2×D4—21no—
191C2×D4—2021no—
192C2×D4—2021no—
193C2×D4C22×D10111yescharacteristic
194C22×C4—21no—
195C22×C4—1011no—
196C22×C4—1011no—
197C22×C4—1011no—
198C22×C4—1011no—
199C22×C4—1011no—
200C22×C4—2024no—
201C22×C4—2021no—
202C22×C4—2021no—
203C22×C4C22×D10111yescharacteristic
204C2×C8—1012no—
205C2×C8—2024no—
206C2×C8C22×D10112yes—
207C4⋊C4—21no—
208C4⋊C4—1012no—
209C4⋊C4—1011no—
210C4⋊C4—2021no—
211C4⋊C4—2021no—
212C4⋊C4—2021no—
213C4⋊C4C22×D10112yes—
214C4⋊C4C22×D10111yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—2022no—
217C22⋊C4—2022no—
218C22⋊C4—2021no—
219C22⋊C4—2021no—
220C22⋊C4—2021no—
221C22⋊C4—2021no—
222C42—2022no—
223C42—2022no—
224C42—2021no—
225C23—21no—
226C23—1011no—
227C23—1011no—
228C23—4041no—
229C23—2022no—
230C23—2022no—
231C23D4×D10111yescharacteristic
232Q8—2022no—
233D4—22no—
234D4—42no—
235D4—4042no—
236D4—4042no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—42no—
241C2×C4—1011no—
242C2×C4—1014no—
243C2×C4—1011no—
244C2×C4—4042no—
245C2×C4—4042no—
246C2×C4—2022no—
247C2×C4—2022no—
248C2×C4—2022no—
249C2×C4—2022no—
250C2×C4—2021no—
251C2×C4—2021no—
252C2×C4—2022no—
253C2×C4D4×D10111yescharacteristic
254C2×C4C23×D10111yescharacteristic
255C8—22no—
256C8—2022no—
257C22—21no—
258C22—21no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—1012no—
263C22—4044no—
264C22—2021no—
265C22—2021no—
266C22—2021no—
267C22—2021no—
268C22—2021no—
269C22#1569227412yes—
270C22C2×D4×D10111yescharacteristic
271C4—21no—
272C4—42no—
273C4—41no—
274C4—1012no—
275C4—4041no—
276C4—4042no—
277C4—4041no—
278C4—2021no—
279C4#41×D10112yes—
280C2—22no—
281C2—81no—
282C2—4041no—
283C2—2022no—
284C2#123×D10111yescharacteristic
285C2C2×#1569227411yescharacteristic
286C2#2026597611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#480×C101C2
socleC22×C101C22×D4
prime radical#20265687C2
hypercenter#480D101
hypocenterC101#3091
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×#15692213C2semidirect
3. #20265520C2semidirect
4. #20265513C2semidirect
5. #20265507C2semidirect
6. #20265499C2semidirect
7. #20265493C2semidirect
8. #20265214C2semidirect
9. #20265170C2semidirect
10. C22⋊C8×D101C2semidirect
11. #480×C101C2semidirect
30. #15692189C22semidirect
31. #15692186C22semidirect
32. #15692186C22semidirect
33. #15692180C22semidirect
34. #15692178C22semidirect
35. #15692176C22semidirect
40. #107×C101C22semidirect
41. #102×C101C22semidirect
144. C101#3091semidirect
152. #480D101semidirect
177. #107C2×D101semidirect
183. #102C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202656872
C2×#156922132
#202655202
#202655132
#202655072
#202654992
#202654932
#202652142
#202651702
C22⋊C8×D1012
#480×C1012
#3091101