Finite Group Databasebeta

← #27181069

Subgroups — #27181069

Order 12928.

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

normal subgroups only

This group has 462 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#121×D101C211yesmaximal, characteristic
2C2×#15692213C211yesmaximal, characteristic, complemented
3#20265503C212yesmaximal, complemented
4#20265499C212yesmaximal, complemented
5#20265495C211yesmaximal, characteristic, complemented
6#20265491C211yesmaximal, characteristic, complemented
7#20265488C211yesmaximal, characteristic, complemented
8#20265216C212yesmaximal, complemented
9#20265180C212yesmaximal, complemented
10#20265139C211yesmaximal, characteristic, complemented
11#481×C101C211yesmaximal, characteristic, complemented
12C2×D4×D101—21no—
13C2×D4×D101C2211yescharacteristic
14C22×C4×D101C2211yescharacteristic
15C2×#13108057C2212yes—
16#15692230C2211yescharacteristic
17#15692229C2211yescharacteristic
18#15692228—21no—
19#15692227—21no—
20#15692226C2211yescharacteristic
21#15692213—21no—
22#15692213—21no—
23#15692213C2212yes—
24#15692213C2212yescomplemented
25C2×C4⋊Dic101C2211yescharacteristic, complemented
26C4⋊C4×D101C2211yescharacteristic
27C2×#13108051—21no—
28C22⋊C4×D101—21no—
29C2×C4×Dic101—21no—
30#15692189C2212yescomplemented
31#15692188C2212yescomplemented
32#15692186C2212yescomplemented
33#15692177C2212yescomplemented
34#15692176C2212yescomplemented
35#15692175C2212yescomplemented
36C22⋊#11717579C2211yescharacteristic, complemented
37#15692167C2212yescomplemented
38#121×C101C2211yescharacteristic
39C2×C4⋊C4×C101C2211yescharacteristic, complemented
40#106×C101C2212yescomplemented
41#102×C101C2212yescomplemented
42C22⋊C8×C101C2211yescharacteristic, complemented
43C23×D101—21no—
44C23×D101D411yescharacteristic
45C2×D404—21no—
46C2×D404C2311yescharacteristic
47C2×C22⋊D101—21no—
48C2×C22⋊D101—21no—
49C2×C22⋊D101—21no—
50D4×D101—24no—
51D4×D101—44no—
52C22×Dic101—21no—
53C22×Dic101D411yescharacteristic
54C2×C4×D101—24no—
55C2×C4×D101—21no—
56C2×C4×D101D411yescharacteristic
57C2×C4×D101D411yescharacteristic
58C2×C4×D101C2311yescharacteristic
59C2×C4×D101C2311yescharacteristic
60#13108057—24no—
61C2×#11717579C2312yes—
62C4⋊Dic101—21no—
63C4⋊Dic101C2312yes—
64C4⋊Dic101C2311yescharacteristic
65#13108053—21no—
66C22⋊Dic101—21no—
67#13108051—22no—
68#13108051—22no—
69#13108051—21no—
70#13108051—21no—
71C4×Dic101—22no—
72C4×Dic101—22no—
73C2×D4×C101—21no—
74C2×D4×C101C2311yescharacteristic
75C22×C4×C101—21no—
76C22×C4×C101C2311yescharacteristic
77C2×C8×C101C2312yes—
78C4⋊C4×C101—21no—
79C4⋊C4×C101C2312yes—
80C4⋊C4×C101C2311yescharacteristic
81C22⋊C4×C101—21no—
82C23×C101—21no—
83C23×C101C2×D411yescharacteristic
84C22×D101—22no—
85C22×D101—22no—
86C22×D101—21no—
87C22×D101—41no—
88C22×D101—42no—
89C22×D101—42no—
90C22×D101—42no—
91C22×D101C2×D411yescharacteristic
92C22×D101C2×D411yescharacteristic
93D4×C101—22no—
94D4×C101—42no—
95C2×C4×C101—22no—
96C2×C4×C101—21no—
97C2×C4×C101—22no—
98C2×C4×C101—42no—
99C2×C4×C101C2×D411yescharacteristic
100C2×C4×C101C2411yescharacteristic
101C22⋊D101—42no—
102C22⋊D101—42no—
103C22⋊D101—42no—
104C2×Dic101—22no—
105C2×Dic101—22no—
106C2×Dic101—21no—
107C2×Dic101—42no—
108C2×Dic101C2×D411yescharacteristic
109C2×Dic101C2×D411yescharacteristic
110D404—22no—
111D404—42no—
112C4×D101—22no—
113C4×D101—22no—
114C4×D101—22no—
115C4×D101—42no—
116C4×D101#4114yes—
117C8×C101—22no—
118#11717579—22no—
119C22×C101—21no—
120C22×C101—21no—
121C22×C101—41no—
122C22×C101—41no—
123C22×C101—41no—
124C22×C101AGL(1,ℤ/8)12yes—
125C22×C101C22×D411yescharacteristic
126C2×D101—21no—
127C2×D101—21no—
128C2×D101—21no—
129C2×D101—21no—
130C2×D101—21no—
131C2×D101—44no—
132C2×D101—41no—
133C2×D101—41no—
134C2×D101—41no—
135C2×D101—84no—
136C2×D101#12312yes—
137C4×C101—21no—
138C4×C101—41no—
139C4×C101—42no—
140C4×C101C2×#4112yes—
141Dic101—21no—
142Dic101—41no—
143Dic101—42no—
144Dic101#12312yes—
145C2×C101—22no—
146C2×C101—81no—
147C2×C101C2×#12311yescharacteristic
148C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
149C2×C101#57511yescharacteristic
150D101—22no—
151D101—41no—
152D101—81no—
153#3094—1011nomaximal, Sylow
154C101#309411yescharacteristic, complemented, Sylow
155C2×#121—1011no—
156C2×#117—1011no—
157#482—1012no—
158#482—1012no—
159#481—1011no—
160#481D10111yescharacteristic, complemented
161#480—1011no—
162#480—1011no—
163#428—1012no—
164#418—1012no—
165#407—1011no—
166C22×D4—1011no—
167C22×D4—2021no—
168C23×C4—1011no—
169C2×M16—1012no—
170#121—1011no—
171#121—1011no—
172#121—1011no—
173#121—2021no—
174#121—2021no—
175#121C2×D10111yescharacteristic
176#117—1012no—
177#117—1012no—
178#117—2021no—
179#117—2021no—
180C2×C4⋊C4—1011no—
181C2×C4⋊C4—1011no—
182C2×C4⋊C4C2×D10111yescharacteristic, complemented
183C2×C22⋊C4—2021no—
184C2×C22⋊C4—2021no—
185C2×C42—2021no—
186#107—1012no—
187#107—1012no—
188#106—1012no—
189#106C2×D10112yescomplemented
190#102—1012no—
191#102—1012no—
192#102—1012no—
193#102C2×D10112yescomplemented
194C22⋊C8—1011no—
195C22⋊C8—1012no—
196C22⋊C8C2×D10111yescharacteristic, complemented
197C24—1011no—
198C24—2021no—
199C2×D4—21no—
200C2×D4—1011no—
201C2×D4—4044no—
202C2×D4—2021no—
203C2×D4—2024no—
204C2×D4—2021no—
205C2×D4—2021no—
206C2×D4—2021no—
207C2×D4C22×D10111yescharacteristic
208C22×C4—21no—
209C22×C4—1011no—
210C22×C4—1011no—
211C22×C4—1011no—
212C22×C4—1011no—
213C22×C4—1011no—
214C22×C4—2024no—
215C22×C4—2021no—
216C22×C4—2021no—
217C22×C4C22×D10111yescharacteristic
218M16—2024no—
219C2×C8—1012no—
220C2×C8C22×D10112yes—
221C4⋊C4—21no—
222C4⋊C4—1012no—
223C4⋊C4—1011no—
224C4⋊C4—2021no—
225C4⋊C4—2021no—
226C4⋊C4C22×D10112yes—
227C4⋊C4C22×D10111yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—2021no—
230C22⋊C4—2022no—
231C22⋊C4—2022no—
232C22⋊C4—2021no—
233C22⋊C4—2021no—
234C42—2022no—
235C42—2022no—
236C23—21no—
237C23—1011no—
238C23—1011no—
239C23—4041no—
240C23—4042no—
241C23—4042no—
242C23—4042no—
243C23—2022no—
244C23—2021no—
245C23—2022no—
246C23D4×D10111yescharacteristic
247D4—22no—
248D4—42no—
249D4—4042no—
250D4—4042no—
251D4—4042no—
252D4—4042no—
253D4—2022no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—1011no—
259C2×C4—1014no—
260C2×C4—1011no—
261C2×C4—4042no—
262C2×C4—4042no—
263C2×C4—2022no—
264C2×C4—2022no—
265C2×C4—2022no—
266C2×C4—2022no—
267C2×C4—2021no—
268C2×C4—2022no—
269C2×C4D4×D10111yescharacteristic
270C2×C4C23×D10111yescharacteristic
271C8—22no—
272C8—2022no—
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—8084no—
279C22—1012no—
280C22—4044no—
281C22—4041no—
282C22—4041no—
283C22—4041no—
284C22—2021no—
285C22—2021no—
286C22—2021no—
287C22—2021no—
288C22—2021no—
289C22#1569228212yes—
290C22C2×D4×D10111yescharacteristic
291C4—21no—
292C4—41no—
293C4—42no—
294C4—1012no—
295C4—4041no—
296C4—4042no—
297C4—2021no—
298C4#41×D10112yes—
299C2—22no—
300C2—81no—
301C2—8081no—
302C2—4041no—
303C2—2022no—
304C2#123×D10111yescharacteristic
305C2C2×#1569228211yescharacteristic
306C2#2026597311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#481×C101C2
socleC22×C101C22×D4
prime radical#121×D101C2
hypercenter#481D101
hypocenterC101#3094
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×#15692213C2semidirect
3. #20265503C2semidirect
4. #20265499C2semidirect
5. #20265495C2semidirect
6. #20265491C2semidirect
7. #20265488C2semidirect
8. #20265216C2semidirect
9. #20265180C2semidirect
10. #20265139C2semidirect
11. #481×C101C2semidirect
24. #15692213C22semidirect
25. C2×C4⋊Dic101C22semidirect
30. #15692189C22semidirect
31. #15692188C22semidirect
32. #15692186C22semidirect
33. #15692177C22semidirect
34. #15692176C22semidirect
35. #15692175C22semidirect
36. C22⋊#11717579C22semidirect
37. #15692167C22semidirect
39. C2×C4⋊C4×C101C22semidirect
40. #106×C101C22semidirect
41. #102×C101C22semidirect
42. C22⋊C8×C101C22semidirect
154. C101#3094semidirect
160. #481D101semidirect
182. C2×C4⋊C4C2×D101semidirect
189. #106C2×D101semidirect
193. #102C2×D101semidirect
196. C22⋊C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#121×D1012
C2×#156922132
#202655032
#202654992
#202654952
#202654912
#202654882
#202652162
#202651802
#202651392
#481×C1012
#3094101