Finite Group Databasebeta

← #15671965

Subgroups — #15671965

Order 2944.

340 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 97 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×C22⋊(C23⋊C8)C211yesmaximal, characteristic, complemented
2#13092277C211yesmaximal, characteristic, complemented
3#13092151C214yesmaximal, complemented
4C2×C23×#121C211yesmaximal, characteristic
5C22×D4×C23—21no—
6C22×D4×C23C411yescharacteristic, complemented
7C23×C4×C23C2211yescharacteristic
8C23×#121—21no—
9C23×#121—21no—
10C23×#121C412yescomplemented
11C23×#121C2212yes—
12C2×C4⋊C4×C23C411yescharacteristic, complemented
13C2×C22⋊C4×C23—21no—
14C2×C22⋊Dic23—21no—
15C2×C4⋊Dic23—21no—
16C22×C23⋊C8—21no—
17C23⋊#95C2212yescomplemented
18C22⋊(C23⋊C8)—21no—
19C22⋊(C23⋊C8)C2212yescomplemented
20C24×C23—21no—
21C24×C23D411yescharacteristic
22C2×D4×C23—22no—
23C2×D4×C23—22no—
24C2×D4×C23—21no—
25C2×D4×C23—42no—
26C2×D4×C23—42no—
27C2×D4×C23—42no—
28C2×D4×C23C2×C412yes—
29C22×C4×C23—21no—
30C22×C4×C23—42no—
31C22×C4×C23C2×C411yescharacteristic
32C22×C4×C23C2×C411yescharacteristic
33C22×C4×C23D411yescharacteristic
34C22×C4×C23D411yescharacteristic
35C22×C4×C23D411yescharacteristic
36C22×C4×C23C2311yescharacteristic
37C22×Dic23—22no—
38C4⋊C4×C23—21no—
39C4⋊C4×C23C2×C412yes—
40C22⋊C4×C23—22no—
41C22⋊C4×C23—22no—
42C22⋊Dic23—42no—
43C4⋊Dic23—42no—
44C2×C23⋊C8—22no—
45C2×C23⋊C8—42no—
46C23×C23—22no—
47C23×C23—41no—
48C23×C23—41no—
49C23×C23—42no—
50C23×C23—42no—
51C23×C23—42no—
52C23×C23C22⋊C411yescharacteristic
53C23×C23C22⋊C411yescharacteristic
54C23×C23D812yes—
55C23×C23SD1612yes—
56C23×C23C2×D411yescharacteristic
57D4×C23—42no—
58D4×C23—44no—
59D4×C23—42no—
60D4×C23—42no—
61D4×C23—42no—
62C2×C4×C23—22no—
63C2×C4×C23—22no—
64C2×C4×C23—22no—
65C2×C4×C23—22no—
66C2×C4×C23—41no—
67C2×C4×C23—41no—
68C2×C4×C23—41no—
69C2×C4×C23—42no—
70C2×C4×C23C22⋊C411yescharacteristic
71C2×C4×C23C22⋊C411yescharacteristic
72C2×C4×C23C22×C411yescharacteristic
73C2×C4×C23C2×D411yescharacteristic
74C2×Dic23—42no—
75C2×Dic23—42no—
76C2×Dic23—42no—
77C23⋊C8—42no—
78#1519—231nomaximal, Sylow
79C22×C23—21no—
80C22×C23—21no—
81C22×C23—21no—
82C22×C23—21no—
83C22×C23—21no—
84C22×C23—21no—
85C22×C23—42no—
86C22×C23—42no—
87C22×C23—42no—
88C22×C23—42no—
89C22×C23—42no—
90C22×C23—84no—
91C22×C23#10212yes—
92C22×C23#10212yes—
93C22×C23C2×C22⋊C411yescharacteristic
94C22×C23C2×D811yescharacteristic
95C22×C23C2×SD1611yescharacteristic
96C4×C23—21no—
97C4×C23—21no—
98C4×C23—21no—
99C4×C23—42no—
100C4×C23#9912yes—
101Dic23—82no—
102C2×#121D2311yescharacteristic
103C2×C22⋊C8—231no—
104#380—231no—
105#327—234no—
106C2×C23—21no—
107C2×C23—22no—
108C2×C23—22no—
109C2×C23—21no—
110C2×C23—82no—
111C2×C23C2×#9911yescharacteristic
112C2×C23C2×#10211yescharacteristic
113C2×C23#42111yescharacteristic
114C22×D4—21no—
115C22×D4Dic2311yescharacteristic, complemented
116C23×C4C2×D2311yescharacteristic
117C22×C8—461no—
118#121—21no—
119#121—21no—
120#121Dic2312yescomplemented
121#121C2×D2312yes—
122C2×C4⋊C4—461no—
123C2×C4⋊C4Dic2311yescharacteristic, complemented
124C2×C22⋊C4—21no—
125C2×C22⋊C4—461no—
126C22⋊C8—232no—
127C22⋊C8—461no—
128#95—232no—
129C23#151911yescharacteristic, complemented, Sylow
130C24—21no—
131C24C22⋊D2311yescharacteristic
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—21no—
135C2×D4—42no—
136C2×D4—42no—
137C2×D4—42no—
138C2×D4C2×Dic2312yes—
139C22×C4—21no—
140C22×C4—42no—
141C22×C4—462no—
142C22×C4C2×Dic2311yescharacteristic
143C22×C4C2×Dic2311yescharacteristic
144C22×C4C22⋊D2311yescharacteristic
145C22×C4C22⋊D2311yescharacteristic
146C22×C4C22⋊D2311yescharacteristic
147C22×C4C22×D2311yescharacteristic
148C2×C8—462no—
149C2×C8—922no—
150C4⋊C4—21no—
151C4⋊C4—922no—
152C4⋊C4C2×Dic2312yes—
153C22⋊C4—22no—
154C22⋊C4—22no—
155C22⋊C4—922no—
156C23—22no—
157C23—41no—
158C23—41no—
159C23—42no—
160C23—42no—
161C23—42no—
162C23#6727212yes—
163C23#6727312yes—
164C23C22⋊Dic2311yescharacteristic
165C23C22⋊Dic2311yescharacteristic
166C23C2×C22⋊D2311yescharacteristic
167D4—42no—
168D4—42no—
169D4—42no—
170D4—44no—
171D4—42no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—922no—
181C2×C4—922no—
182C2×C4—922no—
183C2×C4C22⋊Dic2311yescharacteristic
184C2×C4C22⋊Dic2311yescharacteristic
185C2×C4C22×Dic2311yescharacteristic
186C2×C4C2×C22⋊D2311yescharacteristic
187C8—922no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—42no—
198C22—42no—
199C22—84no—
200C22#1062501412yes—
201C22#1062501412yes—
202C22C2×#6727211yescharacteristic
203C22C2×#6727311yescharacteristic
204C22C2×C22⋊Dic2311yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—42no—
209C4—1842no—
210C4#1062501612yes—
211C2—21no—
212C2—22no—
213C2—21no—
214C2—22no—
215C2—82no—
216C2C2×#1062501611yescharacteristic
217C2C2×#1062501411yescharacteristic
218C2#1309243411yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic23
commutatorC2×C4×C23C22×C4
FrattiniC22×C4C22×D23
Fitting, prime radicalC2×C23×#121C2
socleC22×C23C2×C22⋊C4
cosocleC22×C4×C23C23
hypercenterC2×#121D23
hypocenterC23#1519
LCSG ▷ C2×C4×C23 ▷ C22×C23 ▷ C23
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊(C23⋊C8)C2semidirect
2. #13092277C2semidirect
3. #13092151C2semidirect
6. C22×D4×C23C4semidirect
10. C23×#121C4semidirect
12. C2×C4⋊C4×C23C4semidirect
17. C23⋊#95C22semidirect
19. C22⋊(C23⋊C8)C22semidirect
115. C22×D4Dic23semidirect
120. #121Dic23semidirect
123. C2×C4⋊C4Dic23semidirect
129. C23#1519semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊(C23⋊C8)2
#130922772
#130921512
C2×C23×#1212
#151923