Finite Group Databasebeta

← #27187608

Subgroups — #27187608

Order 12928.

672 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 162 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#20266127C218yesmaximal, complemented
2#20266119C212yesmaximal, complemented
3#20266118C212yesmaximal
4C4×C22⋊#87938C212yesmaximal, complemented
5C2×C22⋊C4×D101C211yesmaximal, characteristic
6C2×C4⋊#87938C2214yescomplemented
7C2×C22⋊#87938C2214yes—
8C2×C22⋊#87938C2214yes—
9C2×C22⋊#87938C2214yes—
10C2×C4×#87938C2214yescomplemented
11(C2×#87938)⋊C4C2214yescomplemented
12(C2×#87938)⋊C4C2214yescomplemented
13C24×D101C2211yescharacteristic
14C22×C4×D101C2211yescharacteristic
15C22×C4×D101C2211yescharacteristic
16C2×C22⋊Dic101C411yescharacteristic, complemented
17C2×#13108051C411yescharacteristic, complemented
18C2×#13108051C411yescharacteristic, complemented
19C22⋊C4×D101C414yescomplemented
20C22⋊C4×D101C2214yes—
21C2×C22⋊C4×C101C411yescharacteristic, complemented
22C22×#87938C2318yes—
23C4⋊#87938—44no—
24C22⋊#87938—28no—
25C22⋊#87938—44no—
26C22⋊#87938—44no—
27C4×#87938—44no—
28C23×D101—24no—
29C23×D101—42no—
30C23×D101—42no—
31C23×D101C2×C412yes—
32C23×D101C2×C411yescharacteristic
33C23×D101C2312yes—
34C23×D101C2311yescharacteristic
35C22×Dic101C2×C411yescharacteristic
36C22×Dic101C2×C411yescharacteristic
37C2×C4×D101—42no—
38C2×C4×D101—42no—
39C2×C4×D101C2×C412yes—
40C2×C4×D101C2×C412yes—
41C2×C4×D101C2312yes—
42C2×C4×D101C2312yes—
43C22⋊Dic101C2×C414yes—
44#13108051C2×C414yes—
45#13108051C2×C414yes—
46C24×C101C2×C411yescharacteristic
47C22×C4×C101C2×C411yescharacteristic
48C22×C4×C101C2×C411yescharacteristic
49C22⋊C4×C101C2×C414yes—
50C23×C101—22no—
51C23×C101—41no—
52C23×C101—41no—
53C23×C101C22×C412yes—
54C23×C101C22×C411yescharacteristic
55C22×D101—22no—
56C22×D101—22no—
57C22×D101—22no—
58C22×D101—24no—
59C22×D101—24no—
60C22×D101—24no—
61C22×D101—24no—
62C22×D101—42no—
63C22×D101—42no—
64C22×D101—41no—
65C22×D101—41no—
66C22×D101—416no—
67C22×D101C22×C412yes—
68C22×D101C22×C411yescharacteristic
69C22×D101#4114yes—
70C22×D101C2411yescharacteristic
71C2×#87938—28no—
72C2×#87938—28no—
73C2×#87938—28no—
74C2×C4×C101—41no—
75C2×C4×C101—41no—
76C2×C4×C101C22×C412yes—
77C2×C4×C101C22×C412yes—
78C2×Dic101—41no—
79C2×Dic101—41no—
80C2×Dic101C22×C412yes—
81C2×Dic101C22×C412yes—
82C4×D101—24no—
83C4×D101—24no—
84C4×D101—42no—
85C4×D101—42no—
86C22×C101—21no—
87C22×C101—21no—
88C22×C101—22no—
89C22×C101—22no—
90C22×C101—22no—
91C22×C101—44no—
92C22×C101#11712yes—
93C22×C101C23×C411yescharacteristic
94C2×D101—21no—
95C2×D101—21no—
96C2×D101—28no—
97C2×D101—22no—
98C2×D101—22no—
99C2×D101—22no—
100C2×D101—48no—
101C2×D101—44no—
102C2×D101—48no—
103C2×D101#11712yes—
104C2×D101C2×#4112yes—
105C2×D101H(5,𝔽2)12yes—
106C2×D101H(5,𝔽2)12yes—
107#87938—48no—
108C4×C101—22no—
109C4×C101—22no—
110Dic101—22no—
111Dic101—22no—
112C2×C101—22no—
113C2×C101—42no—
114C2×C101C2×#11711yescharacteristic
115C2×C101#51811yescharacteristic
116C2×C101#51811yescharacteristic
117D101—22no—
118D101—42no—
119D101#53814yes—
120#2312—1011nomaximal, Sylow
121C101#231211yescharacteristic, complemented, Sylow
122C22×C22⋊C4—1011no—
123#388—1018no—
124#380—1012no—
125#379—1012no—
126C4×C22⋊C4—1012no—
127C25—1011no—
128C23×C4—1011no—
129C23×C4—1011no—
130C2×C4⋊C4—1014no—
131C2×C22⋊C4—1011no—
132C2×C22⋊C4—1011no—
133C2×C22⋊C4—1011no—
134C2×C22⋊C4—1014no—
135C2×C22⋊C4—1014no—
136C2×C22⋊C4—1014no—
137C2×C22⋊C4—1014no—
138C2×C22⋊C4—1014no—
139C2×C22⋊C4#8793811yescharacteristic, complemented
140C2×C42—1014no—
141#95—1014no—
142#95—1014no—
143C24—1011no—
144C24—1012no—
145C24—1011no—
146C24—1012no—
147C24—4042no—
148C24—4042no—
149C24—2024no—
150C24C2×#8793811yescharacteristic
151C22×C4—1011no—
152C22×C4—1011no—
153C22×C4—1012no—
154C22×C4—1012no—
155C22×C4—1012no—
156C22×C4—1012no—
157C22×C4—1018no—
158C22×C4—4042no—
159C22×C4—4042no—
160C22×C4C2×#8793811yescharacteristic
161C22×C4C2×#8793811yescharacteristic
162C4⋊C4—4044no—
163C22⋊C4—1014no—
164C22⋊C4—1014no—
165C22⋊C4—1014no—
166C22⋊C4—4044no—
167C22⋊C4—4044no—
168C22⋊C4—2028no—
169C22⋊C4C2×#8793814yes—
170C42—4044no—
171C23—22no—
172C23—41no—
173C23—41no—
174C23—1011no—
175C23—1014no—
176C23—1011no—
177C23—1012no—
178C23—4041no—
179C23—4041no—
180C23—4042no—
181C23—4042no—
182C23—40416no—
183C23—2024no—
184C23—2024no—
185C23—2024no—
186C23—2024no—
187C23—2022no—
188C23—2022no—
189C23—2022no—
190C23C22×#8793812yes—
191C23C22×#8793811yescharacteristic
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—1012no—
195C2×C4—1012no—
196C2×C4—4042no—
197C2×C4—4042no—
198C2×C4—4041no—
199C2×C4—4041no—
200C2×C4—2024no—
201C2×C4—2024no—
202C2×C4—2028no—
203C2×C4—2028no—
204C2×C4—2028no—
205C2×C4C22×#8793812yes—
206C2×C4C22×#8793812yes—
207C22—21no—
208C22—21no—
209C22—22no—
210C22—22no—
211C22—22no—
212C22—44no—
213C22—1012no—
214C22—1012no—
215C22—1012no—
216C22—1012no—
217C22—4044no—
218C22—4048no—
219C22—4048no—
220C22—2028no—
221C22—2022no—
222C22—2022no—
223C22—2022no—
224C22—2021no—
225C22—2021no—
226C22#1569233412yes—
227C22C23×#8793811yescharacteristic
228C4—22no—
229C4—22no—
230C4—4048no—
231C4—2022no—
232C4—2022no—
233C2—22no—
234C2—42no—
235C2—1014no—
236C2—4042no—
237C2—2022no—
238C2C2×#1569233411yescharacteristic
239C2#2026624411yescharacteristic
240C2#2026624411yescharacteristic
subgroupG/H
center, FrattiniC22C23×#87938
commutator, socleC22×C101C23×C4
FittingC2×C22⋊C4×C101C4
prime radicalC24×D101C22
cosocleC22×D101C24
hypercenterC2×C22⋊C4#87938
hypocenterC101#2312
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #20266127C2semidirect
2. #20266119C2semidirect
4. C4×C22⋊#87938C2semidirect
6. C2×C4⋊#87938C22semidirect
10. C2×C4×#87938C22semidirect
11. (C2×#87938)⋊C4C22semidirect
12. (C2×#87938)⋊C4C22semidirect
16. C2×C22⋊Dic101C4semidirect
17. C2×#13108051C4semidirect
18. C2×#13108051C4semidirect
19. C22⋊C4×D101C4semidirect
21. C2×C22⋊C4×C101C4semidirect
121. C101#2312semidirect
139. C2×C22⋊C4#87938semidirect

Maximal subgroups

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

subgroupindex
#202661272
#202661192
#202661182
C4×C22⋊#879382
C2×C22⋊C4×D1012
#2312101