Finite Group Databasebeta

← #27181068

Subgroups — #27181068

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×#15692214C211yesmaximal, characteristic, complemented
2#20265498C214yesmaximal, complemented
3#20265493C212yesmaximal, complemented
4#20265488C212yesmaximal, complemented
5#20265217C212yesmaximal, complemented
6#20265182C212yesmaximal, complemented
7#20265141C211yesmaximal, characteristic, complemented
8C2×#121×C101C211yesmaximal, characteristic
9C2×#13108058C2212yes—
10#15692214—21no—
11#15692214—21no—
12#15692214C2212yes—
13#15692214C2212yescomplemented
14C2×C4⋊Dic101C2211yescharacteristic, complemented
15C2×C4⋊Dic101C2211yescharacteristic, complemented
16C2×C22⋊Dic101—21no—
17C2×C4×Dic101—21no—
18#15692188C2214yescomplemented
19#15692186C2214yescomplemented
20#15692176C2214yescomplemented
21#15692176C2214yescomplemented
22C22⋊#11717579C2212yescomplemented
23C22⋊#11717579C2212yescomplemented
24C22×D4×C101—21no—
25C22×D4×C101C2211yescharacteristic
26C23×C4×C101C2211yescharacteristic
27#121×C101—21no—
28#121×C101—21no—
29#121×C101C2212yes—
30#121×C101C2212yes—
31C2×C4⋊C4×C101C2211yescharacteristic
32C2×C22⋊C4×C101—21no—
33C22×Dic101—21no—
34C22×Dic101—21no—
35#13108058—24no—
36C2×#11717579C2314yes—
37C4⋊Dic101—21no—
38C4⋊Dic101—21no—
39C4⋊Dic101C2312yes—
40C4⋊Dic101C2312yes—
41C22⋊Dic101—22no—
42C22⋊Dic101—22no—
43C4×Dic101—22no—
44C4×Dic101—22no—
45C24×C101—21no—
46C24×C101D411yescharacteristic
47C2×D4×C101—22no—
48C2×D4×C101—22no—
49C2×D4×C101—21no—
50C2×D4×C101—24no—
51C2×D4×C101—44no—
52C2×D4×C101C2312yes—
53C22×C4×C101—24no—
54C22×C4×C101—21no—
55C22×C4×C101D411yescharacteristic
56C22×C4×C101D411yescharacteristic
57C22×C4×C101D411yescharacteristic
58C22×C4×C101C2311yescharacteristic
59C22×C4×C101C2311yescharacteristic
60C22×C4×C101C2311yescharacteristic
61C4⋊C4×C101—21no—
62C4⋊C4×C101C2312yes—
63C22⋊C4×C101—22no—
64C22⋊C4×C101—22no—
65C23×C101—22no—
66C23×C101—22no—
67C23×C101—22no—
68C23×C101—41no—
69C23×C101—42no—
70C23×C101—42no—
71C23×C101—42no—
72C23×C101C2×D411yescharacteristic
73C23×C101C2×D411yescharacteristic
74C23×C101C2×D411yescharacteristic
75D4×C101—24no—
76D4×C101—44no—
77D4×C101—44no—
78D4×C101—42no—
79C2×C4×C101—22no—
80C2×C4×C101—22no—
81C2×C4×C101—22no—
82C2×C4×C101—22no—
83C2×C4×C101—22no—
84C2×C4×C101—22no—
85C2×C4×C101—42no—
86C2×C4×C101C2×D411yescharacteristic
87C2×C4×C101C2×D411yescharacteristic
88C2×C4×C101C2×D411yescharacteristic
89C2×C4×C101#4114yes—
90C2×C4×C101C2411yescharacteristic
91C2×Dic101—22no—
92C2×Dic101—22no—
93C2×Dic101—42no—
94C2×Dic101—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—42no—
105C22×C101—42no—
106C22×C101—42no—
107C22×C101—84no—
108C22×C101#12312yes—
109C22×C101AGL(1,ℤ/8)12yes—
110C22×C101C22×D411yescharacteristic
111C4×C101—21no—
112C4×C101—21no—
113C4×C101—42no—
114C4×C101#12312yes—
115C4×C101C2×#4112yes—
116Dic101—42no—
117Dic101—42no—
118C2×C101—22no—
119C2×C101—22no—
120C2×C101—41no—
121C2×C101—82no—
122C2×C101C2×#12311yescharacteristic
123C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
124C2×C101#57511yescharacteristic
125#3094—1011nomaximal, Sylow
126C101#309411yescharacteristic, complemented, Sylow
127C2×#121D10111yescharacteristic
128C2×#117—1011no—
129#482—1014no—
130#481—1012no—
131#480—1012no—
132#428—1012no—
133#418—1012no—
134#407—1011no—
135C22×D4—21no—
136C22×D4C2×D10111yescharacteristic
137C23×C4C2×D10111yescharacteristic
138C2×M16—1012no—
139#121—21no—
140#121—21no—
141#121C2×D10112yes—
142#121C2×D10112yes—
143#117—1012no—
144#117—1012no—
145#117—2021no—
146#117—2021no—
147C2×C4⋊C4—1011no—
148C2×C4⋊C4—1011no—
149C2×C4⋊C4C2×D10111yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—2021no—
152C2×C42—2021no—
153#107—1014no—
154#106—1014no—
155#102—1014no—
156#102—1014no—
157C22⋊C8—1012no—
158C22⋊C8—1012no—
159C24—21no—
160C24C22⋊D10111yescharacteristic
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—21no—
164C2×D4—24no—
165C2×D4—44no—
166C2×D4C22×D10112yes—
167C22×C4—24no—
168C22×C4—21no—
169C22×C4—2021no—
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—1012no—
181C4⋊C4—1012no—
182C4⋊C4—2021no—
183C4⋊C4—2021no—
184C4⋊C4C22×D10112yes—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—2022no—
188C22⋊C4—2022no—
189C42—2022no—
190C42—2022no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—41no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23C2×C22⋊D10111yescharacteristic
199C23C2×C22⋊D10111yescharacteristic
200C23C2×C22⋊D10111yescharacteristic
201D4—24no—
202D4—44no—
203D4—44no—
204D4—42no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—4042no—
213C2×C4—4042no—
214C2×C4—2022no—
215C2×C4—2022no—
216C2×C4C2×C22⋊D10111yescharacteristic
217C2×C4C2×C22⋊D10111yescharacteristic
218C2×C4C2×C22⋊D10111yescharacteristic
219C2×C4#1310807314yes—
220C2×C4C23×D10111yescharacteristic
221C8—2024no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—84no—
234C22#1569223612yes—
235C22#1569228112yes—
236C22C22×C22⋊D10111yescharacteristic
237C4—21no—
238C4—21no—
239C4—42no—
240C4—4042no—
241C4—4042no—
242C4#1569223612yes—
243C4C2×#1310807312yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—82no—
248C2C2×#1569223611yescharacteristic
249C2C2×#1569228111yescharacteristic
250C2#2026597511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting, prime radicalC2×#121×C101C2
socleC22×C101C22×D4
hypercenterC2×#121D101
hypocenterC101#3094
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#15692214C2semidirect
2. #20265498C2semidirect
3. #20265493C2semidirect
4. #20265488C2semidirect
5. #20265217C2semidirect
6. #20265182C2semidirect
7. #20265141C2semidirect
13. #15692214C22semidirect
14. C2×C4⋊Dic101C22semidirect
15. C2×C4⋊Dic101C22semidirect
18. #15692188C22semidirect
19. #15692186C22semidirect
20. #15692176C22semidirect
21. #15692176C22semidirect
22. C22⋊#11717579C22semidirect
23. C22⋊#11717579C22semidirect
126. C101#3094semidirect

Maximal subgroups

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

subgroupindex
C2×#156922142
#202654982
#202654932
#202654882
#202652172
#202651822
#202651412
C2×#121×C1012
#3094101