Finite Group Databasebeta

← #27186569

Subgroups — #27186569

Order 12928.

340 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 70 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
1D101⋊(C22⋊C8)C211yesmaximal, characteristic, complemented
2#20266119C211yesmaximal, characteristic, complemented
3#20266029C214yesmaximal, complemented
4C2×#15692230C211yesmaximal, characteristic
5C2×D101⋊C8—21no—
6C2×C4⋊#87938—21no—
7C2×C22⋊#87938—21no—
8#15692312C2212yescomplemented
9C22⋊#11717581—21no—
10(C2×#87938)⋊C4C2212yescomplemented
11C22×D404—21no—
12C22×D404C411yescharacteristic, complemented
13C22×C4×D101C2211yescharacteristic
14#15692230—21no—
15#15692230—21no—
16#15692230C412yescomplemented
17#15692230C2212yes—
18C2×#13108051—21no—
19C2×C4⋊C4×C101C411yescharacteristic, complemented
20C22×#87938—22no—
21D101⋊C8—42no—
22C2×#11717581—22no—
23C4⋊#87938—42no—
24C22⋊#87938—42no—
25C23×D101—21no—
26C23×D101D411yescharacteristic
27C2×D404—22no—
28C2×D404—22no—
29C2×D404—21no—
30C2×D404—42no—
31C2×D404—42no—
32C2×D404—42no—
33C2×D404C2×C412yes—
34C22×Dic101D411yescharacteristic
35C2×C4×D101—42no—
36C2×C4×D101C2×C411yescharacteristic
37C2×C4×D101D411yescharacteristic
38C2×C4×D101D411yescharacteristic
39C2×C4×D101C2311yescharacteristic
40#13108051—22no—
41#13108051—22no—
42C22×C4×C101—21no—
43C22×C4×C101C2×C411yescharacteristic
44C4⋊C4×C101—21no—
45C4⋊C4×C101C2×C412yes—
46C23×C101C22⋊C411yescharacteristic
47C22×D101—22no—
48C22×D101—41no—
49C22×D101—41no—
50C22×D101—42no—
51C22×D101—42no—
52C22×D101—42no—
53C22×D101C22⋊C411yescharacteristic
54C22×D101D812yes—
55C22×D101SD1612yes—
56C22×D101C2×D411yescharacteristic
57C2×#87938—42no—
58C2×#87938—42no—
59C2×#87938—42no—
60C2×C4×C101—22no—
61C2×C4×C101—22no—
62C2×C4×C101—42no—
63C2×C4×C101C22⋊C411yescharacteristic
64C2×C4×C101C22×C411yescharacteristic
65C2×Dic101—41no—
66C2×Dic101C22⋊C411yescharacteristic
67C2×Dic101C2×D411yescharacteristic
68D404—42no—
69D404—44no—
70D404—42no—
71D404—42no—
72D404—42no—
73C4×D101—22no—
74C4×D101—22no—
75C4×D101—41no—
76C4×D101—41no—
77#11717581—42no—
78C22×C101—21no—
79C22×C101—21no—
80C22×C101#10212yes—
81C22×C101C2×C22⋊C411yescharacteristic
82C2×D101—21no—
83C2×D101—21no—
84C2×D101—21no—
85C2×D101—21no—
86C2×D101—42no—
87C2×D101—42no—
88C2×D101—42no—
89C2×D101—42no—
90C2×D101—42no—
91C2×D101—84no—
92C2×D101#10212yes—
93C2×D101C2×D811yescharacteristic
94C2×D101C2×SD1611yescharacteristic
95#87938—82no—
96C4×C101—21no—
97C4×C101—42no—
98C4×C101#9912yes—
99Dic101—21no—
100Dic101—21no—
101C2×C101—22no—
102C2×C101C2×#9911yescharacteristic
103C2×C101C2×#10211yescharacteristic
104C2×C101#42111yescharacteristic
105D101—21no—
106D101—22no—
107D101—21no—
108D101—82no—
109#1519—1011nomaximal, Sylow
110C101#151911yescharacteristic, complemented, Sylow
111C2×#121—1011no—
112C2×C22⋊C8—1011no—
113#380—1011no—
114#327—1014no—
115C22×D4—1011no—
116C22×D4—2021no—
117C23×C4—1011no—
118C22×C8—2021no—
119#121—1012no—
120#121—1012no—
121#121—2021no—
122#121—2021no—
123C2×C4⋊C4—2021no—
124C2×C4⋊C4#8793811yescharacteristic, complemented
125C2×C22⋊C4—2021no—
126C2×C22⋊C4—2021no—
127C22⋊C8—1012no—
128C22⋊C8—2021no—
129#95—1012no—
130C24—1011no—
131C24—2021no—
132C2×D4—1012no—
133C2×D4—4042no—
134C2×D4—4042no—
135C2×D4—4042no—
136C2×D4—2022no—
137C2×D4—2022no—
138C2×D4—2021no—
139C22×C4—21no—
140C22×C4—1011no—
141C22×C4—1011no—
142C22×C4—1011no—
143C22×C4—1011no—
144C22×C4—1011no—
145C22×C4—4042no—
146C22×C4—2022no—
147C22×C4C2×#8793811yescharacteristic
148C2×C8—4042no—
149C2×C8—2022no—
150C4⋊C4—21no—
151C4⋊C4—4042no—
152C4⋊C4C2×#8793812yes—
153C22⋊C4—4042no—
154C22⋊C4—2022no—
155C22⋊C4—2022no—
156C23—1012no—
157C23—1012no—
158C23—1011no—
159C23—1011no—
160C23—4042no—
161C23—4042no—
162C23—4041no—
163C23—4041no—
164C23—4042no—
165C23—2022no—
166C23C22⋊#8793811yescharacteristic
167D4—4042no—
168D4—4042no—
169D4—4044no—
170D4—4042no—
171D4—4042no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—42no—
175C2×C4—1011no—
176C2×C4—1011no—
177C2×C4—4041no—
178C2×C4—4041no—
179C2×C4—4042no—
180C2×C4—4042no—
181C2×C4—4042no—
182C2×C4—4041no—
183C2×C4—2022no—
184C2×C4—2022no—
185C2×C4C22⋊#8793811yescharacteristic
186C2×C4C22×#8793811yescharacteristic
187C8—4042no—
188C22—21no—
189C22—21no—
190C22—8084no—
191C22—1011no—
192C22—1012no—
193C22—1011no—
194C22—4042no—
195C22—4042no—
196C22—4042no—
197C22—4042no—
198C22—4042no—
199C22—2021no—
200C22—2021no—
201C22—2021no—
202C22—2021no—
203C22#1569231712yes—
204C22C2×C22⋊#8793811yescharacteristic
205C4—21no—
206C4—42no—
207C4—8082no—
208C4—2021no—
209C4—2021no—
210C4#1569231412yes—
211C2—22no—
212C2—8082no—
213C2—2021no—
214C2—2022no—
215C2—2021no—
216C2C2×#1569231411yescharacteristic
217C2C2×#1569231711yescharacteristic
218C2#2026617611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC2×C4⋊C4×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC2×#15692230C2
cosocleC2×C4×D101C23
hypercenterC2×C4⋊C4#87938
hypocenterC101#1519
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. D101⋊(C22⋊C8)C2semidirect
2. #20266119C2semidirect
3. #20266029C2semidirect
8. #15692312C22semidirect
10. (C2×#87938)⋊C4C22semidirect
12. C22×D404C4semidirect
16. #15692230C4semidirect
19. C2×C4⋊C4×C101C4semidirect
110. C101#1519semidirect
124. C2×C4⋊C4#87938semidirect

Maximal subgroups

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

subgroupindex
D101⋊(C22⋊C8)2
#202661192
#202660292
C2×#156922302
#1519101