Finite Group Databasebeta

← #27182957

Subgroups — #27182957

Order 12928.

462 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 139 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#20265884C211yesmaximal, characteristic, complemented
2D4×D404C211yesmaximal, characteristic
3#20265524C211yesmaximal, characteristic, complemented
4#20265496C212yesmaximal, complemented
5#20265468C211yesmaximal, characteristic, complemented
6#20265406C212yesmaximal, complemented
7#20265378C211yesmaximal, characteristic, complemented
8#20265330C212yesmaximal, complemented
9C4×#13108063C211yesmaximal, characteristic, complemented
10C2×#15692177C212yesmaximal, complemented
11C8×D4×C101C211yesmaximal, characteristic, complemented
12C22×D404C2212yes—
13C2×D4×D101—21no—
14C2×#13108063C2211yescharacteristic
15C2×#13108063C2212yescomplemented
16#15692255C2211yescharacteristic, complemented
17#15692252C2211yescharacteristic
18#15692246—21no—
19#15692234C2212yescomplemented
20#15692232—22no—
21#15692230—21no—
22#15692229C2212yes—
23#15692225—22no—
24C4×Dic202C2211yescharacteristic, complemented
25C4×D404C2211yescharacteristic
26D4×Dic101—21no—
27C2×C4⋊Dic101C2212yescomplemented
28#15692187C2211yescharacteristic, complemented
29#15692187C2212yescomplemented
30#15692180C2211yescharacteristic, complemented
31#15692177C2211yescharacteristic
32#15692177C2212yes—
33#15692177C2212yescomplemented
34#15692177C2212yescomplemented
35#15692177C2212yescomplemented
36C22×C8×C101C2212yescomplemented
37C4×D4×C101C2211yescharacteristic
38C4⋊C8×C101C2211yescharacteristic, complemented
39C22⋊C8×C101C2212yescomplemented
40C4×C8×C101C2211yescharacteristic, complemented
41C23×D101—22no—
42C2×Dic202—21no—
43C2×Dic202C2311yescharacteristic
44C2×D404—24no—
45C2×D404—21no—
46C2×D404—22no—
47C2×D404—21no—
48C2×D404—42no—
49C2×D404C2312yes—
50C2×D404C2311yescharacteristic
51C2×C22⋊D101—22no—
52D4×D101—44no—
53C22×Dic101—22no—
54C2×C4×D101—21no—
55#13108063—22no—
56#13108063—42no—
57C4⋊Dic101—22no—
58C4⋊Dic101—21no—
59C4⋊Dic101C2311yescharacteristic
60C4⋊Dic101C2312yes—
61C4⋊Dic101C2311yescharacteristic
62#13108053—21no—
63#13108053—22no—
64C22⋊Dic101—22no—
65#13108051—21no—
66#13108051—22no—
67C4×Dic101—21no—
68C2×D4×C101D411yescharacteristic
69C22×C4×C101C2312yes—
70C2×C8×C101—42no—
71C2×C8×C101C2311yescharacteristic
72C2×C8×C101C2311yescharacteristic
73C2×C8×C101C2312yes—
74C4⋊C4×C101D411yescharacteristic
75C22⋊C4×C101D412yes—
76C42×C101C2311yescharacteristic
77C23×C101C2×D412yes—
78C22×D101—21no—
79C22×D101—22no—
80C22×D101—44no—
81C22×D101—44no—
82C22×D101—82no—
83D4×C101SD1614yescomplemented
84C2×C4×C101—22no—
85C2×C4×C101—22no—
86C2×C4×C101C2×D411yescharacteristic
87C2×C4×C101C2×D412yes—
88C2×C4×C101C2×D411yescharacteristic
89C2×C4×C101C2411yescharacteristic
90C22⋊D101—44no—
91C2×Dic101—21no—
92C2×Dic101—21no—
93C2×Dic101—21no—
94C2×Dic101—22no—
95C2×Dic101—82no—
96D404—22no—
97D404—21no—
98D404—22no—
99D404—44no—
100D404—42no—
101D404—81no—
102D404#4112yes—
103C4×D101—42no—
104Dic202—21no—
105Dic202—21no—
106Dic202—81no—
107C8×C101—21no—
108C8×C101—21no—
109C8×C101—22no—
110C22×C101—22no—
111C22×C101—22no—
112C22×C101C2×SD1614yes—
113C22×C101C22×D411yescharacteristic
114C2×D101—22no—
115C2×D101—41no—
116C2×D101—41no—
117C2×D101—42no—
118C2×D101—42no—
119C2×D101—42no—
120C2×D101—88no—
121C4×C101—22no—
122C4×C101—21no—
123C4×C101C2×SD1612yes—
124C4×C101C2×#4111yescharacteristic
125C4×C101H(5,𝔽2)11yescharacteristic
126Dic101—41no—
127Dic101—41no—
128Dic101—42no—
129Dic101—41no—
130C2×C101—24no—
131C2×C101#54611yescharacteristic
132C2×C101C22×SD1611yescharacteristic
133C2×C101#57611yescharacteristic
134D101—42no—
135D101—82no—
136#3295—1011nomaximal, Sylow
137C101#329511yescharacteristic, complemented, Sylow
138#554—1011no—
139D42—1011no—
140#486—1011no—
141#481—1012no—
142#476—1011no—
143#465—1012no—
144#460—1011no—
145#450—1012no—
146C4×SD16—1011no—
147C8×D4D10111yescharacteristic, complemented
148C2×#102—1012no—
149C22×D4—1012no—
150C22×D4—2021no—
151C2×SD16—1012no—
152C2×SD16—1011no—
153C22×C8C2×D10112yescomplemented
154C4⋊Q8—1011no—
155#127—1011no—
156#125—2021no—
157C22⋊Q8—1012no—
158C22⋊Q8—2022no—
159#121—1012no—
160#121—2021no—
161#120—2022no—
162C4×Q8—1011no—
163C4×D4—1011no—
164C4×D4—2021no—
165C4×D4C2×D10111yescharacteristic
166C2×C4⋊C4—1012no—
167#106—1011no—
168#106—1012no—
169C4⋊C8C2×D10111yescharacteristic, complemented
170#103—1011no—
171#102—1011no—
172#102—1012no—
173#102—1012no—
174#102—1012no—
175#102—1012no—
176C22⋊C8C2×D10112yescomplemented
177C4×C8C2×D10111yescharacteristic, complemented
178C24—2022no—
179C2×Q8—1011no—
180C2×Q8—2021no—
181C2×D4—1011no—
182C2×D4—1012no—
183C2×D4—4042no—
184C2×D4—4044no—
185C2×D4—2024no—
186C2×D4—2022no—
187C2×D4—2021no—
188C2×D4—2022no—
189C2×D4—2021no—
190C2×D4D40411yescharacteristic
191C22×C4—2022no—
192C22×C4—2021no—
193C22×C4C22×D10112yes—
194SD16—4042no—
195SD16—2022no—
196C2×C8—42no—
197C2×C8C22×D10111yescharacteristic
198C2×C8C22×D10111yescharacteristic
199C2×C8C22×D10112yes—
200C4⋊C4—1011no—
201C4⋊C4—1012no—
202C4⋊C4—1011no—
203C4⋊C4—2021no—
204C4⋊C4—2022no—
205C4⋊C4—2022no—
206C4⋊C4—2021no—
207C4⋊C4D40411yescharacteristic
208C22⋊C4—2022no—
209C22⋊C4—2021no—
210C22⋊C4—2022no—
211C22⋊C4D40412yes—
212C42—2021no—
213C42C22×D10111yescharacteristic
214C23—8082no—
215C23—4044no—
216C23—4044no—
217C23—2022no—
218C23—2021no—
219C23C2×D40412yes—
220Q8—8081no—
221Q8—2021no—
222Q8—2021no—
223D4—8081no—
224D4—1012no—
225D4—4044no—
226D4—4044no—
227D4—4042no—
228D4—2022no—
229D4—2022no—
230D4—2021no—
231D4#1310806314yescomplemented
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—8082no—
235C2×C4—4042no—
236C2×C4—2021no—
237C2×C4—2021no—
238C2×C4—2021no—
239C2×C4—2022no—
240C2×C4C2×D40411yescharacteristic
241C2×C4C2×D40412yes—
242C2×C4C2×D40411yescharacteristic
243C2×C4C23×D10111yescharacteristic
244C8—21no—
245C8—21no—
246C8—22no—
247C22—22no—
248C22—22no—
249C22—8088no—
250C22—4042no—
251C22—4042no—
252C22—4042no—
253C22—4041no—
254C22—4041no—
255C22—2022no—
256C22C2×#1310806314yes—
257C22C22×D40411yescharacteristic
258C4—22no—
259C4—21no—
260C4—4041no—
261C4—4041no—
262C4—4041no—
263C4—4042no—
264C4C2×#1310806312yes—
265C4#41×D10111yescharacteristic
266C4#1569230011yescharacteristic
267C2—24no—
268C2—8082no—
269C2—4042no—
270C2#2026582211yescharacteristic
271C2C22×#1310806311yescharacteristic
272C2#2026598211yescharacteristic
subgroupG/H
centerC22C22×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC8×D4×C101C2
socleC22×C101C22×D4
prime radicalD4×D404C2
hypercenterC8×D4D101
hypocenterC101#3295
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #20265884C2semidirect
3. #20265524C2semidirect
4. #20265496C2semidirect
5. #20265468C2semidirect
6. #20265406C2semidirect
7. #20265378C2semidirect
8. #20265330C2semidirect
9. C4×#13108063C2semidirect
10. C2×#15692177C2semidirect
11. C8×D4×C101C2semidirect
15. C2×#13108063C22semidirect
16. #15692255C22semidirect
19. #15692234C22semidirect
24. C4×Dic202C22semidirect
27. C2×C4⋊Dic101C22semidirect
28. #15692187C22semidirect
29. #15692187C22semidirect
30. #15692180C22semidirect
33. #15692177C22semidirect
34. #15692177C22semidirect
35. #15692177C22semidirect
36. C22×C8×C101C22semidirect
38. C4⋊C8×C101C22semidirect
39. C22⋊C8×C101C22semidirect
40. C4×C8×C101C22semidirect
83. D4×C101SD16semidirect
137. C101#3295semidirect
147. C8×D4D101semidirect
153. C22×C8C2×D101semidirect
169. C4⋊C8C2×D101semidirect
176. C22⋊C8C2×D101semidirect
177. C4×C8C2×D101semidirect
231. D4#13108063semidirect

Maximal subgroups

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

subgroupindex
#202658842
D4×D4042
#202655242
#202654962
#202654682
#202654062
#202653782
#202653302
C4×#131080632
C2×#156921772
C8×D4×C1012
#3295101