Finite Group Databasebeta

← #27182947

Subgroups — #27182947

Order 12928.

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

normal subgroups only

This group has 494 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#20265832C211yesmaximal, characteristic, complemented
2D4×D404C211yesmaximal, characteristic, complemented
3#20265524C211yesmaximal, characteristic, complemented
4#20265492C212yesmaximal, complemented
5#20265457C211yesmaximal, characteristic, complemented
6#20265412C212yesmaximal, complemented
7#20265372C211yesmaximal, characteristic, complemented
8#20265309C212yesmaximal, complemented
9C8×D404C211yesmaximal, characteristic, complemented
10#102×D101C212yesmaximal, complemented
11C4×D8×C101C211yesmaximal, characteristic, complemented
12C2×#13108074—21no—
13C22×D404—21no—
14C2×D4×D101C2212yes—
15C2×#13108059C2212yescomplemented
16C2×C8×D101C2212yescomplemented
17#15692255C2211yescharacteristic, complemented
18#15692252C2211yescharacteristic
19#15692237—22no—
20#15692234—21no—
21#15692230C2212yes—
22#15692229—21no—
23#15692226C2212yescomplemented
24#15692225—22no—
25C4×D404C2211yescharacteristic
26C4⋊C4×D101C2212yescomplemented
27#15692188C2212yescomplemented
28C4⋊#11717579C2211yescharacteristic, complemented
29#15692177C2212yescomplemented
30#15692177C2212yescomplemented
31#15692176C2212yes—
32#15692175C2212yescomplemented
33#15692167C2212yescomplemented
34C2×D8×C101C2211yescharacteristic
35C4×D4×C101C2211yescharacteristic
36C4×D4×C101C2211yescharacteristic, complemented
37#107×C101C2211yescharacteristic, complemented
38#102×C101C2211yescharacteristic
39#102×C101C2211yescharacteristic, complemented
40C4×C8×C101C2211yescharacteristic, complemented
41C23×D101—22no—
42#13108074—44no—
43C2×Dic202—21no—
44C2×D404—22no—
45C2×D404—21no—
46C2×D404—21no—
47C2×D404—44no—
48C2×D404D411yescharacteristic
49C2×D404C2312yes—
50C2×C22⋊D101—22no—
51C2×C22⋊D101—22no—
52D4×D101—24no—
53D4×D101—42no—
54C2×C4×D101—22no—
55C2×C4×D101C2312yes—
56#13108059—42no—
57C2×#11717579C2312yes—
58C8×D101—42no—
59C4⋊Dic101—21no—
60C4⋊Dic101D411yescharacteristic
61C4⋊Dic101C2312yes—
62#13108053—22no—
63C22⋊Dic101—22no—
64#13108051—22no—
65#13108051D412yes—
66C2×D4×C101C2311yescharacteristic
67C2×D4×C101C2311yescharacteristic
68C22×C4×C101—21no—
69C22×C4×C101—21no—
70D8×C101—22no—
71C2×C8×C101C2311yescharacteristic
72C2×C8×C101C2311yescharacteristic
73C4⋊C4×C101C2311yescharacteristic
74C4⋊C4×C101C2311yescharacteristic
75C22⋊C4×C101—21no—
76C22⋊C4×C101—21no—
77C42×C101C2311yescharacteristic
78C23×C101—21no—
79C23×C101—21no—
80C22×D101—22no—
81C22×D101—44no—
82C22×D101—44no—
83C22×D101—82no—
84C22×D101C2×D412yes—
85D4×C101—21no—
86D4×C101—21no—
87D4×C101—21no—
88D4×C101#4112yescomplemented
89C2×C4×C101—21no—
90C2×C4×C101—21no—
91C2×C4×C101—42no—
92C2×C4×C101—42no—
93C2×C4×C101C2×D411yescharacteristic
94C2×C4×C101C2×D411yescharacteristic
95C2×C4×C101C2411yescharacteristic
96C22⋊D101—44no—
97C22⋊D101—44no—
98C2×Dic101—22no—
99C2×Dic101C2×D412yes—
100D404—22no—
101D404—22no—
102D404—44no—
103D404—42no—
104D404—81no—
105D404D814yescomplemented
106C4×D101—22no—
107C4×D101—22no—
108C4×D101—44no—
109Dic202—42no—
110C8×C101—21no—
111C8×C101—21no—
112#11717579—22no—
113C22×C101—22no—
114C22×C101—41no—
115C22×C101—41no—
116C22×C101—41no—
117C22×C101—41no—
118C22×C101—41no—
119C22×C101C22×D411yescharacteristic
120C2×D101—22no—
121C2×D101—22no—
122C2×D101—42no—
123C2×D101—42no—
124C2×D101—42no—
125C2×D101—88no—
126C2×D101C2×D814yes—
127C4×C101—21no—
128C4×C101—41no—
129C4×C101—41no—
130C4×C101C2×D812yes—
131C4×C101C2×#4111yescharacteristic
132C4×C101H(5,𝔽2)11yescharacteristic
133Dic101—22no—
134Dic101—42no—
135C2×C101—42no—
136C2×C101—81no—
137C2×C101#54611yescharacteristic
138C2×C101C22×D811yescharacteristic
139C2×C101#57711yescharacteristic
140D101—24no—
141D101—82no—
142#3294—1011nomaximal, Sylow
143C101#329411yescharacteristic, complemented, Sylow
144#547—1011no—
145D42—1011no—
146#486—1011no—
147#480—1012no—
148#474—1011no—
149#466—1012no—
150#459—1011no—
151#447—1012no—
152C4×D8D10111yescharacteristic, complemented
153C8×D4—1011no—
154C2×#102—1012no—
155C2×#41—2021no—
156C22×D4—1012no—
157C22×D4—2021no—
158C2×D8—1012no—
159C2×D8C2×D10111yescharacteristic
160C22×C8—1012no—
161C4⋊Q8—1011no—
162#127—1011no—
163#123—2022no—
164C22⋊Q8—2021no—
165#121—1012no—
166#121—1012no—
167#121—2021no—
168#120—2022no—
169C4×D4—1011no—
170C4×D4C2×D10111yescharacteristic
171C4×D4C2×D10111yescharacteristic, complemented
172C2×C4⋊C4—1012no—
173#107—1012no—
174#107C2×D10111yescharacteristic, complemented
175C4⋊C8—1011no—
176#102—1012no—
177#102—1012no—
178#102—1012no—
179#102—1012no—
180#102C2×D10111yescharacteristic
181#102C2×D10111yescharacteristic, complemented
182C22⋊C8—1012no—
183C4×C8C2×D10111yescharacteristic, complemented
184C24—2022no—
185#41—4044no—
186C2×Q8—2021no—
187C2×D4—1011no—
188C2×D4—1012no—
189C2×D4—4044no—
190C2×D4—4042no—
191C2×D4—2022no—
192C2×D4—2022no—
193C2×D4—2022no—
194C2×D4—2024no—
195C2×D4—2021no—
196C2×D4—2021no—
197C2×D4C22×D10111yescharacteristic
198C2×D4C22×D10111yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—1012no—
202C22×C4—2022no—
203D8—22no—
204D8—4042no—
205C2×C8—1012no—
206C2×C8—4042no—
207C2×C8C22×D10111yescharacteristic
208C2×C8C22×D10111yescharacteristic
209C4⋊C4—1011no—
210C4⋊C4—1012no—
211C4⋊C4—2022no—
212C4⋊C4—2021no—
213C4⋊C4C22×D10111yescharacteristic
214C4⋊C4C22×D10111yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—1012no—
218C22⋊C4—2022no—
219C22⋊C4—2022no—
220C42C22×D10111yescharacteristic
221C23—21no—
222C23—21no—
223C23—8082no—
224C23—1012no—
225C23—4044no—
226C23—4044no—
227C23—2022no—
228Q8—4042no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—8081no—
233D4—1014no—
234D4—4044no—
235D4—4044no—
236D4—4044no—
237D4—4042no—
238D4—2022no—
239D4—2022no—
240D4#1310807412yescomplemented
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—1012no—
246C2×C4—4044no—
247C2×C4—2022no—
248C2×C4—2022no—
249C2×C4—2022no—
250C2×C4D4×D10111yescharacteristic
251C2×C4D4×D10111yescharacteristic
252C2×C4C23×D10111yescharacteristic
253C8—21no—
254C8—21no—
255C8—2022no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—8088no—
263C22—1014no—
264C22—4042no—
265C22—4042no—
266C22—4042no—
267C22—2022no—
268C22—2022no—
269C22C2×D4×D10111yescharacteristic
270C4—21no—
271C4—41no—
272C4—41no—
273C4—4042no—
274C4—2022no—
275C4D8×D10112yes—
276C4C2×#1310807411yescharacteristic
277C4#1569230111yescharacteristic
278C2—42no—
279C2—81no—
280C2—8082no—
281C2—2024no—
282C2#2026582311yescharacteristic
283C2C2×D8×D10111yescharacteristic
284C2#2026598411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC4×D8×C101C2
socleC22×C101C22×D4
hypercenterC4×D8D101
hypocenterC101#3294
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #20265832C2semidirect
2. D4×D404C2semidirect
3. #20265524C2semidirect
4. #20265492C2semidirect
5. #20265457C2semidirect
6. #20265412C2semidirect
7. #20265372C2semidirect
8. #20265309C2semidirect
9. C8×D404C2semidirect
10. #102×D101C2semidirect
11. C4×D8×C101C2semidirect
15. C2×#13108059C22semidirect
16. C2×C8×D101C22semidirect
17. #15692255C22semidirect
23. #15692226C22semidirect
26. C4⋊C4×D101C22semidirect
27. #15692188C22semidirect
28. C4⋊#11717579C22semidirect
29. #15692177C22semidirect
30. #15692177C22semidirect
32. #15692175C22semidirect
33. #15692167C22semidirect
36. C4×D4×C101C22semidirect
37. #107×C101C22semidirect
39. #102×C101C22semidirect
40. C4×C8×C101C22semidirect
88. D4×C101#41semidirect
105. D404D8semidirect
143. C101#3294semidirect
152. C4×D8D101semidirect
171. C4×D4C2×D101semidirect
174. #107C2×D101semidirect
181. #102C2×D101semidirect
183. C4×C8C2×D101semidirect
240. D4#13108074semidirect

Maximal subgroups

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

subgroupindex
#202658322
D4×D4042
#202655242
#202654922
#202654572
#202654122
#202653722
#202653092
C8×D4042
#102×D1012
C4×D8×C1012
#3294101