Finite Group Databasebeta

← #27182958

Subgroups — #27182958

Order 12928.

462 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 125 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#20265887C211yesmaximal, characteristic, complemented
2D4×D404C211yesmaximal, characteristic
3#20265524C211yesmaximal, characteristic, complemented
4#20265497C212yesmaximal, complemented
5#20265467C211yesmaximal, characteristic, complemented
6#20265409C212yesmaximal, complemented
7#20265377C211yesmaximal, characteristic, complemented
8#20265326C212yesmaximal, complemented
9C8×D404C211yesmaximal, characteristic, complemented
10#102×D101C212yesmaximal, complemented
11C4×SD16×C101C211yesmaximal, characteristic, complemented
12C22×D404—21no—
13C2×D4×D101C2212yes—
14C2×#13108061C2212yescomplemented
15C2×C8×D101C2212yescomplemented
16#15692255C2211yescharacteristic, complemented
17#15692252C2211yescharacteristic
18#15692247—21no—
19#15692233—22no—
20#15692231C2212yescomplemented
21#15692230C2212yes—
22#15692229—21no—
23#15692225—22no—
24C4×D404—21no—
25C4×D404C2211yescharacteristic
26C4⋊C4×D101C2212yescomplemented
27#15692186C2212yescomplemented
28C4⋊#11717579C2211yescharacteristic, complemented
29#15692177C2212yescomplemented
30#15692177C2212yescomplemented
31#15692176C2212yes—
32#15692175C2212yescomplemented
33#15692167C2212yescomplemented
34C2×SD16×C101C2211yescharacteristic
35C4×Q8×C101C2211yescharacteristic, complemented
36C4×D4×C101C2211yescharacteristic
37#106×C101C2211yescharacteristic, complemented
38#103×C101C2211yescharacteristic, complemented
39#102×C101C2211yescharacteristic
40C4×C8×C101C2211yescharacteristic, complemented
41C23×D101—22no—
42C2×Dic202—21no—
43C2×D404—22no—
44C2×D404—21no—
45C2×D404—21no—
46C2×D404—44no—
47C2×D404D411yescharacteristic
48C2×D404C2312yes—
49C2×C22⋊D101—22no—
50D4×D101—24no—
51D4×D101—42no—
52C2×C4×D101—22no—
53C2×C4×D101C2312yes—
54#13108061—42no—
55C2×#11717579C2312yes—
56C8×D101—42no—
57C4⋊Dic101—21no—
58C4⋊Dic101D411yescharacteristic
59C4⋊Dic101C2312yes—
60#13108053—22no—
61#13108053—22no—
62#13108051—22no—
63#13108051—22no—
64#13108051D412yes—
65C2×Q8×C101C2311yescharacteristic
66C2×D4×C101C2311yescharacteristic
67C22×C4×C101—21no—
68SD16×C101—22no—
69C2×C8×C101C2311yescharacteristic
70C2×C8×C101C2311yescharacteristic
71C4⋊C4×C101—21no—
72C4⋊C4×C101C2311yescharacteristic
73C4⋊C4×C101C2311yescharacteristic
74C22⋊C4×C101—21no—
75C42×C101—21no—
76C42×C101C2311yescharacteristic
77C23×C101—21no—
78C22×D101—22no—
79C22×D101—44no—
80C22×D101—44no—
81C22×D101—82no—
82C22×D101C2×D412yes—
83Q8×C101—21no—
84Q8×C101—21no—
85D4×C101—21no—
86D4×C101#4112yes—
87C2×C4×C101—21no—
88C2×C4×C101—21no—
89C2×C4×C101—21no—
90C2×C4×C101—42no—
91C2×C4×C101C2×D411yescharacteristic
92C2×C4×C101C2×D411yescharacteristic
93C2×C4×C101C2411yescharacteristic
94C22⋊D101—44no—
95C2×Dic101—22no—
96C2×Dic101C2×D412yes—
97D404—22no—
98D404—22no—
99D404—44no—
100D404—42no—
101D404—81no—
102D404SD1614yescomplemented
103C4×D101—22no—
104C4×D101—22no—
105C4×D101—82no—
106Dic202—81no—
107C8×C101—21no—
108C8×C101—21no—
109#11717579—22no—
110C22×C101—22no—
111C22×C101—41no—
112C22×C101—41no—
113C22×C101C22×D411yescharacteristic
114C2×D101—22no—
115C2×D101—22no—
116C2×D101—42no—
117C2×D101—42no—
118C2×D101—42no—
119C2×D101—88no—
120C2×D101C2×SD1614yes—
121C4×C101—21no—
122C4×C101—41no—
123C4×C101—41no—
124C4×C101—41no—
125C4×C101C2×SD1612yes—
126C4×C101C2×#4111yescharacteristic
127C4×C101H(5,𝔽2)11yescharacteristic
128Dic101—22no—
129Dic101—42no—
130C2×C101—42no—
131C2×C101#54611yescharacteristic
132C2×C101C22×SD1611yescharacteristic
133C2×C101#57611yescharacteristic
134D101—24no—
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×SD16D10111yescharacteristic, complemented
147C8×D4—1011no—
148C2×#102—1012no—
149C22×D4—1012no—
150C22×D4—2021no—
151C2×SD16—1012no—
152C2×SD16C2×D10111yescharacteristic
153C22×C8—1012no—
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×Q8C2×D10111yescharacteristic, complemented
163C4×D4—1011no—
164C4×D4—2021no—
165C4×D4C2×D10111yescharacteristic
166C2×C4⋊C4—1012no—
167#106—1012no—
168#106C2×D10111yescharacteristic, complemented
169C4⋊C8—1011no—
170#103C2×D10111yescharacteristic, complemented
171#102—1012no—
172#102—1012no—
173#102—1012no—
174#102—1012no—
175#102C2×D10111yescharacteristic
176C22⋊C8—1012no—
177C4×C8C2×D10111yescharacteristic, complemented
178C24—2022no—
179C2×Q8—2021no—
180C2×Q8C22×D10111yescharacteristic
181C2×D4—1011no—
182C2×D4—1012no—
183C2×D4—4044no—
184C2×D4—4042no—
185C2×D4—2022no—
186C2×D4—2022no—
187C2×D4—2024no—
188C2×D4—2021no—
189C2×D4—2021no—
190C2×D4C22×D10111yescharacteristic
191C22×C4—21no—
192C22×C4—1012no—
193C22×C4—2022no—
194SD16—22no—
195SD16—4042no—
196C2×C8—1012no—
197C2×C8—4042no—
198C2×C8C22×D10111yescharacteristic
199C2×C8C22×D10111yescharacteristic
200C4⋊C4—21no—
201C4⋊C4—1011no—
202C4⋊C4—1012no—
203C4⋊C4—2022no—
204C4⋊C4—2022no—
205C4⋊C4—2021no—
206C4⋊C4C22×D10111yescharacteristic
207C4⋊C4C22×D10111yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—1012no—
210C22⋊C4—2022no—
211C22⋊C4—2022no—
212C42—21no—
213C42C22×D10111yescharacteristic
214C23—21no—
215C23—8082no—
216C23—1012no—
217C23—4044no—
218C23—4044no—
219C23—2022no—
220Q8—21no—
221Q8—21no—
222Q8—8081no—
223D4—21no—
224D4—8081no—
225D4—1014no—
226D4—4044no—
227D4—4044no—
228D4—4042no—
229D4—2022no—
230D4—2022no—
231D4#1310807412yes—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—8082no—
237C2×C4—1012no—
238C2×C4—2022no—
239C2×C4—2022no—
240C2×C4—2022no—
241C2×C4D4×D10111yescharacteristic
242C2×C4D4×D10111yescharacteristic
243C2×C4C23×D10111yescharacteristic
244C8—21no—
245C8—21no—
246C8—2022no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—8088no—
251C22—1014no—
252C22—4042no—
253C22—4042no—
254C22—4042no—
255C22—2022no—
256C22—2022no—
257C22C2×D4×D10111yescharacteristic
258C4—21no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—4042no—
263C4—2022no—
264C4SD16×D10112yes—
265C4C2×#1310807411yescharacteristic
266C4#1569230111yescharacteristic
267C2—42no—
268C2—8082no—
269C2—2024no—
270C2#2026582311yescharacteristic
271C2C2×SD16×D10111yescharacteristic
272C2#2026597911yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC4×SD16×C101C2
socleC22×C101C22×D4
prime radicalD4×D404C2
hypercenterC4×SD16D101
hypocenterC101#3295
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #20265887C2semidirect
3. #20265524C2semidirect
4. #20265497C2semidirect
5. #20265467C2semidirect
6. #20265409C2semidirect
7. #20265377C2semidirect
8. #20265326C2semidirect
9. C8×D404C2semidirect
10. #102×D101C2semidirect
11. C4×SD16×C101C2semidirect
14. C2×#13108061C22semidirect
15. C2×C8×D101C22semidirect
16. #15692255C22semidirect
20. #15692231C22semidirect
26. C4⋊C4×D101C22semidirect
27. #15692186C22semidirect
28. C4⋊#11717579C22semidirect
29. #15692177C22semidirect
30. #15692177C22semidirect
32. #15692175C22semidirect
33. #15692167C22semidirect
35. C4×Q8×C101C22semidirect
37. #106×C101C22semidirect
38. #103×C101C22semidirect
40. C4×C8×C101C22semidirect
102. D404SD16semidirect
137. C101#3295semidirect
146. C4×SD16D101semidirect
162. C4×Q8C2×D101semidirect
168. #106C2×D101semidirect
170. #103C2×D101semidirect
177. C4×C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202658872
D4×D4042
#202655242
#202654972
#202654672
#202654092
#202653772
#202653262
C8×D4042
#102×D1012
C4×SD16×C1012
#3295101