Finite Group Databasebeta

← #27182946

Subgroups — #27182946

Order 12928.

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

normal subgroups only

This group has 494 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#20265831C211yesmaximal, characteristic, complemented
2D4×D404C211yesmaximal, characteristic, complemented
3#20265524C211yesmaximal, characteristic, complemented
4#20265491C212yesmaximal, complemented
5#20265458C211yesmaximal, characteristic, complemented
6#20265413C212yesmaximal, complemented
7#20265373C211yesmaximal, characteristic, complemented
8#20265313C212yesmaximal, complemented
9C4×D808C211yesmaximal, characteristic, complemented
10C2×#15692177C212yesmaximal, complemented
11C8×D4×C101C211yesmaximal, characteristic, complemented
12C2×#13108073—21no—
13C22×D404C2212yes—
14C2×D4×D101—21no—
15C2×D808C2211yescharacteristic
16C2×D808C2212yescomplemented
17#15692255C2211yescharacteristic, complemented
18#15692252C2211yescharacteristic
19#15692239—22no—
20#15692235—21no—
21#15692230—21no—
22#15692229C2212yes—
23#15692229C2212yescomplemented
24#15692225—22no—
25C4×D404C2211yescharacteristic
26C4×D404C2211yescharacteristic, complemented
27C2×C4⋊Dic101C2212yescomplemented
28#15692189C2211yescharacteristic, complemented
29#15692189C2212yescomplemented
30#15692177C2211yescharacteristic
31#15692177C2212yes—
32#15692177C2212yescomplemented
33#15692177C2212yescomplemented
34#15692177C2211yescharacteristic, complemented
35#15692177C2212yescomplemented
36C22×C8×C101C2212yescomplemented
37C4×D4×C101C2211yescharacteristic
38C4⋊C8×C101C2211yescharacteristic, complemented
39C22⋊C8×C101C2212yescomplemented
40C4×C8×C101C2211yescharacteristic, complemented
41C23×D101—22no—
42#13108073—44no—
43C2×Dic202—21no—
44C2×D404—24no—
45C2×D404—21no—
46C2×D404—22no—
47C2×D404—21no—
48C2×D404—42no—
49C2×D404C2312yes—
50C2×D404C2311yescharacteristic
51C2×D404C2311yescharacteristic
52C2×C22⋊D101—22no—
53C2×C22⋊D101—22no—
54D4×D101—44no—
55C22×Dic101—22no—
56C2×C4×D101—21no—
57C2×C4×D101—21no—
58D808—22no—
59D808—42no—
60C4⋊Dic101—22no—
61C4⋊Dic101—21no—
62C4⋊Dic101C2311yescharacteristic
63C4⋊Dic101C2312yes—
64C4⋊Dic101C2311yescharacteristic
65#13108051—21no—
66#13108051—21no—
67#13108051—22no—
68#13108051—22no—
69C2×D4×C101D411yescharacteristic
70C22×C4×C101C2312yes—
71C2×C8×C101—42no—
72C2×C8×C101C2311yescharacteristic
73C2×C8×C101C2311yescharacteristic
74C2×C8×C101C2312yes—
75C4⋊C4×C101D411yescharacteristic
76C22⋊C4×C101D412yes—
77C42×C101C2311yescharacteristic
78C23×C101C2×D412yes—
79C22×D101—21no—
80C22×D101—21no—
81C22×D101—22no—
82C22×D101—44no—
83C22×D101—44no—
84C22×D101—82no—
85D4×C101D814yescomplemented
86C2×C4×C101—22no—
87C2×C4×C101—22no—
88C2×C4×C101C2×D411yescharacteristic
89C2×C4×C101C2×D412yes—
90C2×C4×C101C2×D411yescharacteristic
91C2×C4×C101C2411yescharacteristic
92C22⋊D101—44no—
93C22⋊D101—44no—
94C2×Dic101—21no—
95C2×Dic101—21no—
96C2×Dic101—22no—
97C2×Dic101—44no—
98D404—21no—
99D404—22no—
100D404—21no—
101D404—21no—
102D404—22no—
103D404—44no—
104D404—42no—
105D404—81no—
106D404#4112yescomplemented
107C4×D101—42no—
108C4×D101—42no—
109Dic202—42no—
110C8×C101—21no—
111C8×C101—21no—
112C8×C101—22no—
113C22×C101—22no—
114C22×C101—22no—
115C22×C101C2×D814yes—
116C22×C101C22×D411yescharacteristic
117C2×D101—22no—
118C2×D101—41no—
119C2×D101—41no—
120C2×D101—42no—
121C2×D101—41no—
122C2×D101—41no—
123C2×D101—42no—
124C2×D101—42no—
125C2×D101—41no—
126C2×D101—88no—
127C4×C101—22no—
128C4×C101—21no—
129C4×C101C2×D812yes—
130C4×C101C2×#4111yescharacteristic
131C4×C101H(5,𝔽2)11yescharacteristic
132Dic101—42no—
133Dic101—41no—
134Dic101—41no—
135C2×C101—24no—
136C2×C101#54611yescharacteristic
137C2×C101C22×D811yescharacteristic
138C2×C101#57711yescharacteristic
139D101—42no—
140D101—81no—
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×D8—1011no—
153C8×D4D10111yescharacteristic, complemented
154C2×#102—1012no—
155C2×#41—2021no—
156C22×D4—1012no—
157C22×D4—2021no—
158C2×D8—1012no—
159C2×D8—1011no—
160C22×C8C2×D10112yescomplemented
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×D4—1011no—
171C4×D4C2×D10111yescharacteristic
172C2×C4⋊C4—1012no—
173#107—1011no—
174#107—1012no—
175C4⋊C8C2×D10111yescharacteristic, complemented
176#102—1011no—
177#102—1012no—
178#102—1012no—
179#102—1012no—
180#102—1011no—
181#102—1012no—
182C22⋊C8C2×D10112yescomplemented
183C4×C8C2×D10111yescharacteristic, complemented
184C24—2022no—
185#41—4044no—
186C2×Q8—2021no—
187C2×D4—1011no—
188C2×D4—1012no—
189C2×D4—1011no—
190C2×D4—4042no—
191C2×D4—4044no—
192C2×D4—2024no—
193C2×D4—2022no—
194C2×D4—2022no—
195C2×D4—2021no—
196C2×D4—2022no—
197C2×D4—2021no—
198C2×D4D40411yescharacteristic
199C22×C4—2022no—
200C22×C4—2021no—
201C22×C4—2021no—
202C22×C4C22×D10112yes—
203D8—4042no—
204D8—2022no—
205C2×C8—42no—
206C2×C8C22×D10111yescharacteristic
207C2×C8C22×D10111yescharacteristic
208C2×C8C22×D10112yes—
209C4⋊C4—1011no—
210C4⋊C4—1012no—
211C4⋊C4—1011no—
212C4⋊C4—2022no—
213C4⋊C4—2021no—
214C4⋊C4D40411yescharacteristic
215C22⋊C4—2021no—
216C22⋊C4—2021no—
217C22⋊C4—2022no—
218C22⋊C4—2022no—
219C22⋊C4D40412yes—
220C42C22×D10111yescharacteristic
221C23—8082no—
222C23—4044no—
223C23—4044no—
224C23—2021no—
225C23—2022no—
226C23—2021no—
227C23C2×D40412yes—
228Q8—4042no—
229D4—8081no—
230D4—1012no—
231D4—4044no—
232D4—4044no—
233D4—4044no—
234D4—4042no—
235D4—2021no—
236D4—2022no—
237D4—2021no—
238D4—2022no—
239D4—2021no—
240D4D80814yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—4044no—
244C2×C4—4042no—
245C2×C4—4042no—
246C2×C4—2021no—
247C2×C4—2021no—
248C2×C4—2022no—
249C2×C4C2×D40411yescharacteristic
250C2×C4C2×D40412yes—
251C2×C4C2×D40411yescharacteristic
252C2×C4C23×D10111yescharacteristic
253C8—21no—
254C8—21no—
255C8—22no—
256C22—22no—
257C22—22no—
258C22—8088no—
259C22—4042no—
260C22—4041no—
261C22—4042no—
262C22—4041no—
263C22—4041no—
264C22—4042no—
265C22—4041no—
266C22—4041no—
267C22—2022no—
268C22C2×D80814yes—
269C22C22×D40411yescharacteristic
270C4—21no—
271C4—22no—
272C4—4041no—
273C4—4041no—
274C4—4042no—
275C4C2×D80812yes—
276C4#41×D10111yescharacteristic
277C4#1569230011yescharacteristic
278C2—24no—
279C2—8082no—
280C2—8081no—
281C2—4042no—
282C2#2026582211yescharacteristic
283C2C22×D80811yescharacteristic
284C2#2026598911yescharacteristic
subgroupG/H
centerC22C22×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC8×D4×C101C2
socleC22×C101C22×D4
hypercenterC8×D4D101
hypocenterC101#3294
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. #20265831C2semidirect
2. D4×D404C2semidirect
3. #20265524C2semidirect
4. #20265491C2semidirect
5. #20265458C2semidirect
6. #20265413C2semidirect
7. #20265373C2semidirect
8. #20265313C2semidirect
9. C4×D808C2semidirect
10. C2×#15692177C2semidirect
11. C8×D4×C101C2semidirect
16. C2×D808C22semidirect
17. #15692255C22semidirect
23. #15692229C22semidirect
26. C4×D404C22semidirect
27. C2×C4⋊Dic101C22semidirect
28. #15692189C22semidirect
29. #15692189C22semidirect
32. #15692177C22semidirect
33. #15692177C22semidirect
34. #15692177C22semidirect
35. #15692177C22semidirect
36. C22×C8×C101C22semidirect
38. C4⋊C8×C101C22semidirect
39. C22⋊C8×C101C22semidirect
40. C4×C8×C101C22semidirect
85. D4×C101D8semidirect
106. D404#41semidirect
143. C101#3294semidirect
153. C8×D4D101semidirect
160. C22×C8C2×D101semidirect
175. C4⋊C8C2×D101semidirect
182. C22⋊C8C2×D101semidirect
183. C4×C8C2×D101semidirect
240. D4D808semidirect

Maximal subgroups

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

subgroupindex
#202658312
D4×D4042
#202655242
#202654912
#202654582
#202654132
#202653732
#202653132
C4×D8082
C2×#156921772
C8×D4×C1012
#3294101