Finite Group Databasebeta

← #27181072

Subgroups — #27181072

Order 12928.

462 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#15692229C211yesmaximal, characteristic
2#20265501C214yesmaximal, complemented
3#20265495C212yesmaximal, complemented
4#20265490C212yesmaximal, complemented
5#20265220C212yesmaximal, complemented
6#20265181C212yesmaximal, complemented
7#20265141C211yesmaximal, characteristic, complemented
8C2×#117×C101C211yesmaximal, characteristic, complemented
9C22×D404C2211yescharacteristic
10C22×C22⋊D101—21no—
11C2×#13108058C2212yes—
12#15692229—21no—
13#15692229—21no—
14#15692229C2212yes—
15#15692229C2212yes—
16C2×C4⋊Dic101C2211yescharacteristic
17C2×#13108051—21no—
18#15692188C2214yescomplemented
19#15692186C2214yescomplemented
20#15692175C2214yescomplemented
21#15692175C2214yescomplemented
22C22⋊#11717579C2212yescomplemented
23C22⋊#11717579C2212yescomplemented
24C23×C4×C101C2211yescharacteristic
25#117×C101—21no—
26#117×C101—21no—
27#117×C101C2212yes—
28#117×C101C2212yescomplemented
29C2×C4⋊C4×C101C2211yescharacteristic, complemented
30C2×C4⋊C4×C101C2211yescharacteristic, complemented
31C2×C22⋊C4×C101—21no—
32C2×C42×C101—21no—
33C23×D101—21no—
34C2×D404—24no—
35C2×D404—21no—
36C2×D404C2312yes—
37C2×C22⋊D101—22no—
38C2×C22⋊D101—22no—
39C2×C22⋊D101—44no—
40C22×Dic101—21no—
41#13108058—24no—
42C2×#11717579C2314yes—
43C4⋊Dic101—21no—
44C4⋊Dic101C2312yes—
45#13108051—22no—
46#13108051—22no—
47C24×C101D411yescharacteristic
48C22×C4×C101—24no—
49C22×C4×C101—21no—
50C22×C4×C101—21no—
51C22×C4×C101D411yescharacteristic
52C22×C4×C101D411yescharacteristic
53C22×C4×C101D411yescharacteristic
54C22×C4×C101C2311yescharacteristic
55C22×C4×C101C2311yescharacteristic
56C22×C4×C101C2311yescharacteristic
57C4⋊C4×C101—21no—
58C4⋊C4×C101—21no—
59C4⋊C4×C101C2312yes—
60C4⋊C4×C101C2312yes—
61C22⋊C4×C101—22no—
62C22⋊C4×C101—22no—
63C42×C101—22no—
64C42×C101—22no—
65C23×C101—22no—
66C23×C101—22no—
67C23×C101—41no—
68C23×C101C2×D411yescharacteristic
69C23×C101C2×D411yescharacteristic
70C23×C101C2×D411yescharacteristic
71C22×D101—22no—
72C22×D101—42no—
73C22×D101—42no—
74C22×D101—42no—
75C2×C4×C101—22no—
76C2×C4×C101—22no—
77C2×C4×C101—22no—
78C2×C4×C101—22no—
79C2×C4×C101—22no—
80C2×C4×C101—22no—
81C2×C4×C101—22no—
82C2×C4×C101—42no—
83C2×C4×C101—42no—
84C2×C4×C101C2×D411yescharacteristic
85C2×C4×C101C2×D411yescharacteristic
86C2×C4×C101C2×D411yescharacteristic
87C2×C4×C101#4114yes—
88C2×C4×C101C2411yescharacteristic
89C22⋊D101—44no—
90C22⋊D101—44no—
91C2×Dic101—22no—
92C2×Dic101—42no—
93D404—24no—
94D404—42no—
95#11717579—24no—
96C22×C101—21no—
97C22×C101—21no—
98C22×C101—21no—
99C22×C101—21no—
100C22×C101—21no—
101C22×C101—21no—
102C22×C101—21no—
103C22×C101—44no—
104C22×C101#12312yes—
105C22×C101AGL(1,ℤ/8)12yes—
106C22×C101C22×D411yescharacteristic
107C2×D101—42no—
108C2×D101—42no—
109C2×D101—42no—
110C2×D101—84no—
111C4×C101—21no—
112C4×C101—21no—
113C4×C101—42no—
114C4×C101—42no—
115C4×C101#12312yes—
116C4×C101C2×#4112yes—
117Dic101—42no—
118C2×C101—22no—
119C2×C101—22no—
120C2×C101—41no—
121C2×C101C2×#12311yescharacteristic
122C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
123C2×C101#57511yescharacteristic
124D101—82no—
125#3094—1011nomaximal, Sylow
126C101#309411yescharacteristic, complemented, Sylow
127C2×#121—1011no—
128C2×#117D10111yescharacteristic, complemented
129#482—1014no—
130#481—1012no—
131#480—1012no—
132#428—1012no—
133#418—1012no—
134#407—1011no—
135C22×D4—1011no—
136C22×D4—2021no—
137C23×C4C2×D10111yescharacteristic
138C2×M16—1012no—
139#121—1012no—
140#121—1012no—
141#121—2021no—
142#121—2021no—
143#117—21no—
144#117—21no—
145#117C2×D10112yes—
146#117C2×D10112yescomplemented
147C2×C4⋊C4—1011no—
148C2×C4⋊C4C2×D10111yescharacteristic, complemented
149C2×C4⋊C4C2×D10111yescharacteristic, complemented
150C2×C22⋊C4—21no—
151C2×C22⋊C4—2021no—
152C2×C42—21no—
153#107—1014no—
154#106—1014no—
155#102—1014no—
156#102—1014no—
157C22⋊C8—1012no—
158C22⋊C8—1012no—
159C24—2021no—
160C24C22⋊D10111yescharacteristic
161C2×D4—1012no—
162C2×D4—4044no—
163C2×D4—2022no—
164C2×D4—2022no—
165C2×D4—2021no—
166C2×D4—2024no—
167C22×C4—24no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—2021no—
171C22×C4C22⋊D10111yescharacteristic
172C22×C4C22⋊D10111yescharacteristic
173C22×C4C22⋊D10111yescharacteristic
174C22×C4C22×D10111yescharacteristic
175C22×C4C22×D10111yescharacteristic
176C22×C4C22×D10111yescharacteristic
177M16—2024no—
178C2×C8—1014no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—1012no—
182C4⋊C4—2021no—
183C4⋊C4C22×D10112yes—
184C4⋊C4C22×D10112yes—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—2022no—
188C22⋊C4—2022no—
189C42—22no—
190C42—22no—
191C23—22no—
192C23—22no—
193C23—41no—
194C23—4042no—
195C23—4042no—
196C23—4042no—
197C23—2022no—
198C23C2×C22⋊D10111yescharacteristic
199C23C2×C22⋊D10111yescharacteristic
200C23C2×C22⋊D10111yescharacteristic
201D4—4044no—
202D4—4044no—
203D4—4042no—
204D4—2024no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—4042no—
215C2×C4—2022no—
216C2×C4C2×C22⋊D10111yescharacteristic
217C2×C4C2×C22⋊D10111yescharacteristic
218C2×C4C2×C22⋊D10111yescharacteristic
219C2×C4#1310807414yes—
220C2×C4C23×D10111yescharacteristic
221C8—2024no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—8084no—
231C22—4042no—
232C22—4042no—
233C22—4042no—
234C22#1569223812yes—
235C22#1569228012yes—
236C22C22×C22⋊D10111yescharacteristic
237C4—21no—
238C4—21no—
239C4—42no—
240C4—42no—
241C4—4042no—
242C4#1569223812yes—
243C4C2×#1310807412yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—8082no—
248C2C2×#1569223811yescharacteristic
249C2C2×#1569228011yescharacteristic
250C2#2026597511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#117×C101C2
socleC22×C101C22×D4
prime radicalC2×#15692229C2
hypercenterC2×#117D101
hypocenterC101#3094
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20265501C2semidirect
3. #20265495C2semidirect
4. #20265490C2semidirect
5. #20265220C2semidirect
6. #20265181C2semidirect
7. #20265141C2semidirect
8. C2×#117×C101C2semidirect
18. #15692188C22semidirect
19. #15692186C22semidirect
20. #15692175C22semidirect
21. #15692175C22semidirect
22. C22⋊#11717579C22semidirect
23. C22⋊#11717579C22semidirect
28. #117×C101C22semidirect
29. C2×C4⋊C4×C101C22semidirect
30. C2×C4⋊C4×C101C22semidirect
126. C101#3094semidirect
128. C2×#117D101semidirect
146. #117C2×D101semidirect
148. C2×C4⋊C4C2×D101semidirect
149. C2×C4⋊C4C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922292
#202655012
#202654952
#202654902
#202652202
#202651812
#202651412
C2×#117×C1012
#3094101