Finite Group Databasebeta

← #27182950

Subgroups — #27182950

Order 12928.

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

normal subgroups only

This group has 462 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
1D4×D404C211yesmaximal, characteristic
2#20265525C211yesmaximal, characteristic, complemented
3#20265495C212yesmaximal, complemented
4#20265464C211yesmaximal, characteristic, complemented
5#20265408C212yesmaximal, complemented
6#20265377C211yesmaximal, characteristic, complemented
7#20265327C212yesmaximal, complemented
8C4×#13108061C211yesmaximal, characteristic, complemented
9D4×#11717579C211yesmaximal, characteristic, complemented
10C2×#15692175C212yesmaximal, complemented
11#554×C101C211yesmaximal, characteristic, complemented
12C22×D404C2212yes—
13C2×D4×D101—21no—
14C2×#13108061C2211yescharacteristic
15C2×#13108061C2212yescomplemented
16C22×#11717579C2212yescomplemented
17#15692252C2211yescharacteristic
18#15692230—21no—
19#15692229C2212yes—
20#15692225—22no—
21C4×D404C2211yescharacteristic
22#15692186C2211yescharacteristic, complemented
23#15692186C2212yescomplemented
24C4⋊#11717579C2211yescharacteristic, complemented
25#15692179C2211yescharacteristic, complemented
26#15692175C2211yescharacteristic
27#15692175C2212yes—
28#15692175C2212yescomplemented
29#15692175C2212yescomplemented
30#15692175C2212yescomplemented
31C22⋊#11717579C2212yescomplemented
32C4×#11717579C2211yescharacteristic, complemented
33C4⋊Q8×C101C2211yescharacteristic, complemented
34#125×C101—21no—
35C22⋊Q8×C101—22no—
36C22⋊Q8×C101C2212yescomplemented
37C4×Q8×C101C2211yescharacteristic, complemented
38C4×D4×C101—21no—
39C4×D4×C101C2211yescharacteristic
40C2×C4⋊C4×C101C2212yescomplemented
41C23×D101—22no—
42C2×D404—24no—
43C2×D404—21no—
44C2×D404—22no—
45C2×D404—21no—
46C2×D404—42no—
47C2×D404C2312yes—
48C2×D404C2311yescharacteristic
49C2×C22⋊D101—22no—
50D4×D101—44no—
51C2×C4×D101—21no—
52#13108061—22no—
53#13108061—42no—
54C2×#11717579—42no—
55C2×#11717579C2311yescharacteristic
56C2×#11717579C2311yescharacteristic
57C2×#11717579C2312yes—
58C4⋊Dic101C2311yescharacteristic
59#13108051—21no—
60#13108051—22no—
61C2×Q8×C101—21no—
62C2×Q8×C101C2311yescharacteristic
63C2×D4×C101D411yescharacteristic
64C22×C4×C101—22no—
65C22×C4×C101C2312yes—
66C4⋊C4×C101—22no—
67C4⋊C4×C101—21no—
68C4⋊C4×C101—21no—
69C4⋊C4×C101—22no—
70C4⋊C4×C101D411yescharacteristic
71C4⋊C4×C101C2311yescharacteristic
72C4⋊C4×C101C2312yes—
73C22⋊C4×C101—22no—
74C22⋊C4×C101D412yes—
75C42×C101—21no—
76C42×C101C2311yescharacteristic
77C23×C101C2×D412yes—
78C22×D101—21no—
79C22×D101—22no—
80C22×D101—44no—
81C22×D101—44no—
82C22×D101—82no—
83Q8×C101—21no—
84Q8×C101—21no—
85Q8×C101—81no—
86D4×C101SD1614yescomplemented
87C2×C4×C101—22no—
88C2×C4×C101—22no—
89C2×C4×C101—21no—
90C2×C4×C101—21no—
91C2×C4×C101—22no—
92C2×C4×C101—82no—
93C2×C4×C101C2×D411yescharacteristic
94C2×C4×C101C2×D412yes—
95C2×C4×C101C2×D411yescharacteristic
96C2×C4×C101C2411yescharacteristic
97C22⋊D101—44no—
98C2×Dic101—21no—
99D404—22no—
100D404—21no—
101D404—22no—
102D404—44no—
103D404—42no—
104D404—81no—
105D404#4112yes—
106C4×D101—42no—
107#11717579—21no—
108#11717579—21no—
109#11717579—22no—
110C22×C101—22no—
111C22×C101—22no—
112C22×C101C2×SD1614yes—
113C22×C101C22×D411yescharacteristic
114C2×D101—22no—
115C2×D101—41no—
116C2×D101—41no—
117C2×D101—42no—
118C2×D101—42no—
119C2×D101—42no—
120C2×D101—88no—
121C4×C101—22no—
122C4×C101—21no—
123C4×C101—41no—
124C4×C101—41no—
125C4×C101—42no—
126C4×C101C2×SD1612yes—
127C4×C101C2×#4111yescharacteristic
128C4×C101H(5,𝔽2)11yescharacteristic
129Dic101—41no—
130C2×C101—24no—
131C2×C101#54611yescharacteristic
132C2×C101C22×SD1611yescharacteristic
133C2×C101#57611yescharacteristic
134D101—42no—
135D101—82no—
136#3295—1011nomaximal, Sylow
137C101#329511yescharacteristic, complemented, Sylow
138#554D10111yescharacteristic, complemented
139D42—1011no—
140#486—1011no—
141#481—1012no—
142#476—1011no—
143#465—1012no—
144#460—1011no—
145#450—1012no—
146C4×SD16—1011no—
147C8×D4—1011no—
148C2×#102—1012no—
149C22×D4—1012no—
150C22×D4—2021no—
151C2×SD16—1012no—
152C2×SD16—1011no—
153C22×C8—1012no—
154C4⋊Q8C2×D10111yescharacteristic, complemented
155#127—1011no—
156#125—21no—
157C22⋊Q8—22no—
158C22⋊Q8C2×D10112yescomplemented
159#121—1012no—
160#121—2021no—
161#120—2022no—
162C4×Q8C2×D10111yescharacteristic, complemented
163C4×D4—21no—
164C4×D4—1011no—
165C4×D4C2×D10111yescharacteristic
166C2×C4⋊C4C2×D10112yescomplemented
167#106—1011no—
168#106—1012no—
169C4⋊C8—1011no—
170#103—1011no—
171#102—1011no—
172#102—1012no—
173#102—1012no—
174#102—1012no—
175#102—1012no—
176C22⋊C8—1012no—
177C4×C8—1011no—
178C24—2022no—
179C2×Q8—21no—
180C2×Q8C22×D10111yescharacteristic
181C2×D4—1011no—
182C2×D4—1012no—
183C2×D4—4042no—
184C2×D4—4044no—
185C2×D4—2024no—
186C2×D4—2022no—
187C2×D4—2021no—
188C2×D4—2022no—
189C2×D4—2021no—
190C2×D4C22⋊D10111yescharacteristic
191C22×C4—22no—
192C22×C4—2021no—
193C22×C4C22×D10112yes—
194SD16—4042no—
195SD16—2022no—
196C2×C8—1011no—
197C2×C8—1011no—
198C2×C8—1012no—
199C2×C8—4042no—
200C4⋊C4—22no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—22no—
204C4⋊C4—1011no—
205C4⋊C4C22⋊D10111yescharacteristic
206C4⋊C4C22×D10111yescharacteristic
207C4⋊C4C22×D10112yes—
208C22⋊C4—22no—
209C22⋊C4—2021no—
210C22⋊C4—2022no—
211C22⋊C4C22⋊D10112yes—
212C42—21no—
213C42C22×D10111yescharacteristic
214C23—8082no—
215C23—4044no—
216C23—4044no—
217C23—2022no—
218C23—2021no—
219C23C2×C22⋊D10112yes—
220Q8—21no—
221Q8—21no—
222Q8—81no—
223D4—8081no—
224D4—1012no—
225D4—4044no—
226D4—4044no—
227D4—4042no—
228D4—2022no—
229D4—2022no—
230D4—2021no—
231D4#1310806114yescomplemented
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—82no—
238C2×C4—4042no—
239C2×C4—2021no—
240C2×C4C2×C22⋊D10111yescharacteristic
241C2×C4C2×C22⋊D10112yes—
242C2×C4C2×C22⋊D10111yescharacteristic
243C2×C4C23×D10111yescharacteristic
244C8—2021no—
245C8—2021no—
246C8—2022no—
247C22—22no—
248C22—22no—
249C22—8088no—
250C22—4042no—
251C22—4042no—
252C22—4042no—
253C22—4041no—
254C22—4041no—
255C22—2022no—
256C22C2×#1310806114yes—
257C22C22×C22⋊D10111yescharacteristic
258C4—22no—
259C4—21no—
260C4—41no—
261C4—41no—
262C4—42no—
263C4—4041no—
264C4C2×#1310806112yes—
265C4#41×D10111yescharacteristic
266C4#1569230111yescharacteristic
267C2—24no—
268C2—8082no—
269C2—4042no—
270C2#2026581511yescharacteristic
271C2C22×#1310806111yescharacteristic
272C2#2026598011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#554×C101C2
socleC22×C101C22×D4
prime radicalD4×D404C2
hypercenter#554D101
hypocenterC101#3295
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #20265525C2semidirect
3. #20265495C2semidirect
4. #20265464C2semidirect
5. #20265408C2semidirect
6. #20265377C2semidirect
7. #20265327C2semidirect
8. C4×#13108061C2semidirect
9. D4×#11717579C2semidirect
10. C2×#15692175C2semidirect
11. #554×C101C2semidirect
15. C2×#13108061C22semidirect
16. C22×#11717579C22semidirect
22. #15692186C22semidirect
23. #15692186C22semidirect
24. C4⋊#11717579C22semidirect
25. #15692179C22semidirect
28. #15692175C22semidirect
29. #15692175C22semidirect
30. #15692175C22semidirect
31. C22⋊#11717579C22semidirect
32. C4×#11717579C22semidirect
33. C4⋊Q8×C101C22semidirect
36. C22⋊Q8×C101C22semidirect
37. C4×Q8×C101C22semidirect
40. C2×C4⋊C4×C101C22semidirect
86. D4×C101SD16semidirect
137. C101#3295semidirect
138. #554D101semidirect
154. C4⋊Q8C2×D101semidirect
158. C22⋊Q8C2×D101semidirect
162. C4×Q8C2×D101semidirect
166. C2×C4⋊C4C2×D101semidirect
231. D4#13108061semidirect

Maximal subgroups

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

subgroupindex
D4×D4042
#202655252
#202654952
#202654642
#202654082
#202653772
#202653272
C4×#131080612
D4×#117175792
C2×#156921752
#554×C1012
#3295101