Finite Group Databasebeta

← #27182945

Subgroups — #27182945

Order 12928.

494 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 141 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#20265831C211yesmaximal, characteristic, complemented
2#20265523C211yesmaximal, characteristic, complemented
3#20265488C212yesmaximal, complemented
4#20265457C211yesmaximal, characteristic, complemented
5#20265411C212yesmaximal, complemented
6#20265369C211yesmaximal, characteristic, complemented
7#20265311C212yesmaximal, complemented
8C4×#13108059C211yesmaximal, characteristic, complemented
9D4×#11717579C211yesmaximal, characteristic, complemented
10C2×#15692176C212yesmaximal, complemented
11D42×C101C211yesmaximal, characteristic, complemented
12C2×#13108073—21no—
13C2×#13108059C2211yescharacteristic
14C2×#13108059C2212yescomplemented
15C22×#11717579C2212yescomplemented
16#15692255C2211yescharacteristic, complemented
17#15692239—22no—
18#15692235—21no—
19#15692229C2212yescomplemented
20C4×D404C2211yescharacteristic, complemented
21C2×C4⋊Dic101C2212yescomplemented
22#15692188C2211yescharacteristic, complemented
23#15692188C2212yescomplemented
24C4⋊#11717579C2211yescharacteristic, complemented
25#15692176C2211yescharacteristic
26#15692176C2212yes—
27#15692176C2212yescomplemented
28#15692176C2212yescomplemented
29#15692176C2212yescomplemented
30#15692175C2211yescharacteristic, complemented
31C22⋊#11717579C2212yescomplemented
32C4×#11717579C2211yescharacteristic, complemented
33C22×D4×C101—21no—
34C22×D4×C101C2212yes—
35#127×C101C2211yescharacteristic
36#121×C101—21no—
37#121×C101C2212yes—
38#120×C101—22no—
39C4×D4×C101C2211yescharacteristic
40C4×D4×C101C2211yescharacteristic
41#13108073—44no—
42C2×Dic202—21no—
43C2×D404C2311yescharacteristic
44C2×C22⋊D101—22no—
45C22×Dic101—22no—
46C2×C4×D101—21no—
47#13108059—22no—
48#13108059—42no—
49C2×#11717579—42no—
50C2×#11717579C2311yescharacteristic
51C2×#11717579C2311yescharacteristic
52C2×#11717579C2312yes—
53C4⋊Dic101—22no—
54C4⋊Dic101—21no—
55C4⋊Dic101C2311yescharacteristic
56C4⋊Dic101C2312yes—
57#13108051—21no—
58#13108051—22no—
59C24×C101—22no—
60C2×D4×C101—24no—
61C2×D4×C101—22no—
62C2×D4×C101—22no—
63C2×D4×C101—21no—
64C2×D4×C101—21no—
65C2×D4×C101—42no—
66C2×D4×C101—44no—
67C2×D4×C101D411yescharacteristic
68C2×D4×C101C2312yes—
69C2×D4×C101C2311yescharacteristic
70C22×C4×C101—21no—
71C22×C4×C101C2312yes—
72C4⋊C4×C101D411yescharacteristic
73C4⋊C4×C101C2311yescharacteristic
74C22⋊C4×C101—21no—
75C22⋊C4×C101—22no—
76C22⋊C4×C101D412yes—
77C42×C101C2311yescharacteristic
78C23×C101—21no—
79C23×C101—22no—
80C23×C101—44no—
81C23×C101—44no—
82C23×C101—82no—
83C23×C101C2×D412yes—
84C22×D101—21no—
85D4×C101—22no—
86D4×C101—21no—
87D4×C101—22no—
88D4×C101—44no—
89D4×C101—44no—
90D4×C101—42no—
91D4×C101—81no—
92D4×C101D814yescomplemented
93D4×C101#4112yescomplemented
94C2×C4×C101—22no—
95C2×C4×C101—22no—
96C2×C4×C101—21no—
97C2×C4×C101—42no—
98C2×C4×C101C2×D411yescharacteristic
99C2×C4×C101C2×D412yes—
100C2×C4×C101C2×D411yescharacteristic
101C2×C4×C101C2411yescharacteristic
102C22⋊D101—44no—
103C2×Dic101—21no—
104C2×Dic101—22no—
105C2×Dic101—44no—
106D404—21no—
107D404—21no—
108C4×D101—42no—
109Dic202—42no—
110#11717579—21no—
111#11717579—21no—
112#11717579—22no—
113C22×C101—22no—
114C22×C101—22no—
115C22×C101—22no—
116C22×C101—41no—
117C22×C101—41no—
118C22×C101—42no—
119C22×C101—42no—
120C22×C101—42no—
121C22×C101—88no—
122C22×C101C2×D814yes—
123C22×C101C22×D411yescharacteristic
124C2×D101—41no—
125C2×D101—41no—
126C2×D101—41no—
127C4×C101—22no—
128C4×C101—21no—
129C4×C101—41no—
130C4×C101C2×D812yes—
131C4×C101C2×#4111yescharacteristic
132C4×C101H(5,𝔽2)11yescharacteristic
133Dic101—42no—
134Dic101—41no—
135C2×C101—24no—
136C2×C101—42no—
137C2×C101—82no—
138C2×C101#54611yescharacteristic
139C2×C101C22×D811yescharacteristic
140C2×C101#57711yescharacteristic
141D101—81no—
142#3294—1011nomaximal, Sylow
143C101#329411yescharacteristic, complemented, Sylow
144#547—1011no—
145D42D10111yescharacteristic, complemented
146#486—1011no—
147#480—1012no—
148#474—1011no—
149#466—1012no—
150#459—1011no—
151#447—1012no—
152C4×D8—1011no—
153C8×D4—1011no—
154C2×#102—1012no—
155C2×#41—2021no—
156C22×D4—21no—
157C22×D4C2×D10112yes—
158C2×D8—1012no—
159C2×D8—1011no—
160C22×C8—1012no—
161C4⋊Q8—1011no—
162#127C2×D10111yescharacteristic
163#123—2022no—
164C22⋊Q8—2021no—
165#121—21no—
166#121—1012no—
167#121C2×D10112yes—
168#120—22no—
169C4×D4—1011no—
170C4×D4C2×D10111yescharacteristic
171C4×D4C2×D10111yescharacteristic
172C2×C4⋊C4—1012no—
173#107—1011no—
174#107—1012no—
175C4⋊C8—1011no—
176#102—1011no—
177#102—1012no—
178#102—1012no—
179#102—1012no—
180#102—1012no—
181#102—1011no—
182C22⋊C8—1012no—
183C4×C8—1011no—
184C24—22no—
185#41—4044no—
186C2×Q8—2021no—
187C2×D4—24no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—42no—
193C2×D4—44no—
194C2×D4—1011no—
195C2×D4—2022no—
196C2×D4C22⋊D10111yescharacteristic
197C2×D4C22×D10112yes—
198C2×D4C22×D10111yescharacteristic
199C22×C4—21no—
200C22×C4—2022no—
201C22×C4—2021no—
202C22×C4C22×D10112yes—
203D8—4042no—
204D8—2022no—
205C2×C8—1011no—
206C2×C8—1011no—
207C2×C8—1012no—
208C2×C8—4042no—
209C4⋊C4—1011no—
210C4⋊C4—1012no—
211C4⋊C4—2022no—
212C4⋊C4—2021no—
213C4⋊C4C22⋊D10111yescharacteristic
214C4⋊C4C22×D10111yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—2021no—
218C22⋊C4—2022no—
219C22⋊C4C22⋊D10112yes—
220C42C22×D10111yescharacteristic
221C23—21no—
222C23—22no—
223C23—44no—
224C23—44no—
225C23—82no—
226C23—2021no—
227C23C2×C22⋊D10112yes—
228Q8—4042no—
229D4—22no—
230D4—21no—
231D4—22no—
232D4—44no—
233D4—44no—
234D4—42no—
235D4—81no—
236D4—4044no—
237D4—2021no—
238D4—2021no—
239D4#1310805914yescomplemented
240D4#1310807312yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—42no—
245C2×C4—4044no—
246C2×C4—4042no—
247C2×C4—2021no—
248C2×C4—2022no—
249C2×C4C2×C22⋊D10111yescharacteristic
250C2×C4C2×C22⋊D10112yes—
251C2×C4C2×C22⋊D10111yescharacteristic
252C2×C4C23×D10111yescharacteristic
253C8—2021no—
254C8—2021no—
255C8—2022no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—88no—
265C22—4041no—
266C22—4041no—
267C22—4041no—
268C22C2×#1310805914yes—
269C22C22×C22⋊D10111yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—4041no—
274C4—4042no—
275C4C2×#1310805912yes—
276C4C2×#1310807311yescharacteristic
277C4#1569230011yescharacteristic
278C2—24no—
279C2—42no—
280C2—82no—
281C2—8081no—
282C2#2026582111yescharacteristic
283C2C22×#1310805911yescharacteristic
284C2#2026598811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingD42×C101C2
socleC22×C101C22×D4
hypercenterD42D101
hypocenterC101#3294
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #20265831C2semidirect
2. #20265523C2semidirect
3. #20265488C2semidirect
4. #20265457C2semidirect
5. #20265411C2semidirect
6. #20265369C2semidirect
7. #20265311C2semidirect
8. C4×#13108059C2semidirect
9. D4×#11717579C2semidirect
10. C2×#15692176C2semidirect
11. D42×C101C2semidirect
14. C2×#13108059C22semidirect
15. C22×#11717579C22semidirect
16. #15692255C22semidirect
19. #15692229C22semidirect
20. C4×D404C22semidirect
21. C2×C4⋊Dic101C22semidirect
22. #15692188C22semidirect
23. #15692188C22semidirect
24. C4⋊#11717579C22semidirect
27. #15692176C22semidirect
28. #15692176C22semidirect
29. #15692176C22semidirect
30. #15692175C22semidirect
31. C22⋊#11717579C22semidirect
32. C4×#11717579C22semidirect
92. D4×C101D8semidirect
93. D4×C101#41semidirect
143. C101#3294semidirect
145. D42D101semidirect
239. D4#13108059semidirect
240. D4#13108073semidirect

Maximal subgroups

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

subgroupindex
#202658312
#202655232
#202654882
#202654572
#202654112
#202653692
#202653112
C4×#131080592
D4×#117175792
C2×#156921762
D42×C1012
#3294101