Finite Group Databasebeta

← #27182948

Subgroups — #27182948

Order 12928.

494 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 123 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
2#20265812C211yesmaximal, characteristic, complemented
3#20265523C211yesmaximal, characteristic, complemented
4#20265489C212yesmaximal, complemented
5#20265458C211yesmaximal, characteristic, complemented
6#20265410C212yesmaximal, complemented
7#20265309C212yesmaximal, complemented
8C4×#13108059C211yesmaximal, characteristic, complemented
9#20265242C211yesmaximal, characteristic, complemented
10#102×D101C212yesmaximal, complemented
11#459×C101C211yesmaximal, characteristic, complemented
12C2×#13108074—21no—
13C2×D4×D101—21no—
14C2×D4×D101C2212yes—
15C2×#13108059C2211yescharacteristic
16C2×C8×D101C2212yescomplemented
17#15692255C2211yescharacteristic, complemented
18#15692237—22no—
19#15692234—21no—
20#15692226—21no—
21#15692226C2212yes—
22#15692226C2212yescomplemented
23#15692223—22no—
24C4×D404C2211yescharacteristic
25C4×D404C2211yescharacteristic
26C4⋊C4×D101C2212yescomplemented
27#15692189C2212yescomplemented
28#15692188C2211yescharacteristic, complemented
29#15692177C2212yes—
30#15692176C2212yescomplemented
31#15692176C2211yescharacteristic, complemented
32#15692176C2212yescomplemented
33#15692175C2211yescharacteristic
34#15692167C2212yescomplemented
35C4×#11717579C2211yescharacteristic, complemented
36C2×D8×C101C2212yescomplemented
37#127×C101C2211yescharacteristic
38C4×D4×C101C2211yescharacteristic, complemented
39C4⋊C8×C101C2211yescharacteristic, complemented
40#102×C101C2212yescomplemented
41C23×D101—22no—
42#13108074—44no—
43C2×Dic202—21no—
44C2×D404D411yescharacteristic
45C2×D404C2311yescharacteristic
46C2×C22⋊D101—22no—
47C2×C22⋊D101—22no—
48C2×C22⋊D101—22no—
49D4×D101—24no—
50D4×D101—42no—
51D4×D101—44no—
52C2×C4×D101—22no—
53C2×C4×D101—21no—
54C2×C4×D101C2312yes—
55#13108059—22no—
56C2×#11717579C2311yescharacteristic
57C2×#11717579C2311yescharacteristic
58C8×D101—42no—
59C4⋊Dic101—21no—
60C4⋊Dic101D411yescharacteristic
61C4⋊Dic101C2312yes—
62C4⋊Dic101C2311yescharacteristic
63#13108053—22no—
64C22⋊Dic101—22no—
65C22⋊Dic101—22no—
66#13108051—21no—
67#13108051D412yes—
68C2×D4×C101—21no—
69C2×D4×C101—21no—
70C2×D4×C101C2312yes—
71C2×D4×C101C2311yescharacteristic
72C22×C4×C101—21no—
73D8×C101—42no—
74C2×C8×C101C2312yes—
75C4⋊C4×C101C2311yescharacteristic
76C22⋊C4×C101—21no—
77C42×C101C2311yescharacteristic
78C23×C101—21no—
79C23×C101—22no—
80C22×D101—21no—
81C22×D101—44no—
82C22×D101—44no—
83C22×D101—82no—
84C22×D101C2×D412yes—
85D4×C101—21no—
86D4×C101—22no—
87D4×C101—21no—
88D4×C101—22no—
89D4×C101—42no—
90D4×C101—81no—
91C2×C4×C101—21no—
92C2×C4×C101—42no—
93C2×C4×C101C2×D411yescharacteristic
94C2×C4×C101C2×D411yescharacteristic
95C2×C4×C101C2411yescharacteristic
96C22⋊D101—44no—
97C22⋊D101—44no—
98C22⋊D101—44no—
99C2×Dic101—21no—
100C2×Dic101—22no—
101C2×Dic101C2×D412yes—
102D404—21no—
103D404D814yescomplemented
104D404#4112yescomplemented
105C4×D101—22no—
106C4×D101—22no—
107C4×D101—44no—
108C4×D101—42no—
109Dic202—42no—
110C8×C101—22no—
111#11717579—21no—
112#11717579—21no—
113C22×C101—42no—
114C22×C101—41no—
115C22×C101—41no—
116C22×C101—42no—
117C22×C101—42no—
118C22×C101—41no—
119C22×C101C22×D411yescharacteristic
120C2×D101—22no—
121C2×D101—22no—
122C2×D101—22no—
123C2×D101—41no—
124C2×D101—41no—
125C2×D101—88no—
126C2×D101C2×D814yes—
127C4×C101—21no—
128C4×C101—41no—
129C4×C101C2×D812yes—
130C4×C101C2×#4111yescharacteristic
131C4×C101H(5,𝔽2)11yescharacteristic
132Dic101—22no—
133Dic101—42no—
134Dic101—41no—
135C2×C101—82no—
136C2×C101—81no—
137C2×C101#54611yescharacteristic
138C2×C101C22×D811yescharacteristic
139C2×C101#57711yescharacteristic
140D101—24no—
141D101—42no—
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#459D10111yescharacteristic, complemented
151#447—1012no—
152C4×D8—1011no—
153C8×D4—1011no—
154C2×#102—1012no—
155C2×#41—2021no—
156C22×D4—1012no—
157C22×D4—2021no—
158C2×D8—1011no—
159C2×D8C2×D10112yescomplemented
160C22×C8—1012no—
161C4⋊Q8—1011no—
162#127C2×D10111yescharacteristic
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, complemented
172C2×C4⋊C4—1012no—
173#107—1012no—
174#107—1011no—
175C4⋊C8C2×D10111yescharacteristic, complemented
176#102—1012no—
177#102—1011no—
178#102—1012no—
179#102—1012no—
180#102—1011no—
181#102C2×D10112yescomplemented
182C22⋊C8—1012no—
183C4×C8—1011no—
184C24—2022no—
185#41—4044no—
186C2×Q8—2021no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—1011no—
190C2×D4—1011no—
191C2×D4—4044no—
192C2×D4—4042no—
193C2×D4—2022no—
194C2×D4—2024no—
195C2×D4—2022no—
196C2×D4—2022no—
197C2×D4C22×D10112yes—
198C2×D4C22×D10111yescharacteristic
199C22×C4—21no—
200C22×C4—1012no—
201C22×C4—2022no—
202C22×C4—2021no—
203D8—42no—
204D8—2022no—
205C2×C8—1011no—
206C2×C8—1011no—
207C2×C8—4042no—
208C2×C8C22×D10112yes—
209C4⋊C4—1011no—
210C4⋊C4—1012no—
211C4⋊C4—1011no—
212C4⋊C4—2022no—
213C4⋊C4—2021no—
214C4⋊C4C22×D10111yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—1012no—
217C22⋊C4—2022no—
218C22⋊C4—2022no—
219C22⋊C4—2021no—
220C42C22×D10111yescharacteristic
221C23—21no—
222C23—22no—
223C23—8082no—
224C23—1012no—
225C23—4044no—
226C23—4044no—
227C23—2021no—
228Q8—4042no—
229D4—21no—
230D4—22no—
231D4—21no—
232D4—22no—
233D4—42no—
234D4—81no—
235D4—1012no—
236D4—1014no—
237D4—4044no—
238D4—4044no—
239D4—4044no—
240D4—2021no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—1012no—
244C2×C4—4044no—
245C2×C4—4042no—
246C2×C4—2021no—
247C2×C4—2022no—
248C2×C4—2022no—
249C2×C4—2022no—
250C2×C4D4×D10111yescharacteristic
251C2×C4D4×D10111yescharacteristic
252C2×C4C23×D10111yescharacteristic
253C8—22no—
254C8—2021no—
255C8—2021no—
256C22—42no—
257C22—41no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—8088no—
263C22—1014no—
264C22—4041no—
265C22—4041no—
266C22—2022no—
267C22—2022no—
268C22—2022no—
269C22C2×D4×D10111yescharacteristic
270C4—21no—
271C4—41no—
272C4—4041no—
273C4—4042no—
274C4—2022no—
275C4D8×D10112yes—
276C4#41×D10111yescharacteristic
277C4#1569230011yescharacteristic
278C2—82no—
279C2—81no—
280C2—4042no—
281C2—2024no—
282C2#2026582411yescharacteristic
283C2C2×D8×D10111yescharacteristic
284C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#459×C101C2
socleC22×C101C22×D4
hypercenter#459D101
hypocenterC101#3294
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #20265832C2semidirect
2. #20265812C2semidirect
3. #20265523C2semidirect
4. #20265489C2semidirect
5. #20265458C2semidirect
6. #20265410C2semidirect
7. #20265309C2semidirect
8. C4×#13108059C2semidirect
9. #20265242C2semidirect
10. #102×D101C2semidirect
11. #459×C101C2semidirect
16. C2×C8×D101C22semidirect
17. #15692255C22semidirect
22. #15692226C22semidirect
26. C4⋊C4×D101C22semidirect
27. #15692189C22semidirect
28. #15692188C22semidirect
30. #15692176C22semidirect
31. #15692176C22semidirect
32. #15692176C22semidirect
34. #15692167C22semidirect
35. C4×#11717579C22semidirect
36. C2×D8×C101C22semidirect
38. C4×D4×C101C22semidirect
39. C4⋊C8×C101C22semidirect
40. #102×C101C22semidirect
103. D404D8semidirect
104. D404#41semidirect
143. C101#3294semidirect
150. #459D101semidirect
159. C2×D8C2×D101semidirect
171. C4×D4C2×D101semidirect
175. C4⋊C8C2×D101semidirect
181. #102C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202658322
#202658122
#202655232
#202654892
#202654582
#202654102
#202653092
C4×#131080592
#202652422
#102×D1012
#459×C1012
#3294101