Finite Group Databasebeta

← C22×#10625055 #15686618

Subgroups — C22×#10625055 #15686618

Order 2944.

2528 conjugacy classes of subgroups; only the 645 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 645 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
1C23×D92C211yesmaximal, characteristic, complemented
2C23×C22⋊D23C211yesmaximal, characteristic, complemented
3C23×C22⋊D23C211yesmaximal, characteristic, complemented
4C23×C4×D23C211yesmaximal, characteristic, complemented
5C2×#10625055C2124yesmaximal, direct product factor
6C22×Dic23⋊C4C211yesmaximal, characteristic, complemented
7C22×(C2×D23)⋊C4C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C23C211yesmaximal, characteristic, complemented
9C25×C23C2211yescharacteristic
10C24×D23C2211yescharacteristic
11C24×D23C2211yescharacteristic
12C23×C4×C23C2211yescharacteristic
13C23×C4×C23C2211yescharacteristic
14C22×C22⋊D23—28no—
15C22×C22⋊D23—28no—
16C22×C22⋊D23C22112yescomplemented
17C22×C22⋊D23C22112yescomplemented
18C23×Dic23C2211yescharacteristic
19C23×Dic23C2211yescharacteristic
20C22×D92—28no—
21C22×D92C22112yescomplemented
22C22×C4×D23—28no—
23C22×C4×D23C22112yescomplemented
24C2×C22⋊C4×C23C22112yescomplemented
25C2×(C2×D23)⋊C4C22112yescomplemented
26C2×Dic23⋊C4C22112yescomplemented
27#10625055C22164yesdirect product factor
28C24×C23—24no—
29C24×C23—24no—
30C24×C23—24no—
31C24×C23C2316yes—
32C24×C23C2311yescharacteristic
33C23×D23—24no—
34C23×D23—24no—
35C23×D23—24no—
36C23×D23—24no—
37C23×D23—24no—
38C23×D23D418yescomplemented
39C23×D23C2316yes—
40C23×D23C2316yes—
41C22×C4×C23—24no—
42C22×C4×C23—24no—
43C22×C4×C23C2316yes—
44C22×C4×C23C2316yes—
45C2×C22⋊D23—248no—
46C2×C22⋊D23—248no—
47C2×C22⋊D23C23116yescomplemented
48C2×C22⋊D23C23116yescomplemented
49C22×Dic23—24no—
50C22×Dic23D418yescomplemented
51C22×Dic23C2316yes—
52C22×Dic23C2316yes—
53C2×D92—248no—
54C2×D92C23116yescomplemented
55C2×C4×D23—248no—
56C2×C4×D23C23116yescomplemented
57C22⋊C4×C23C23116yescomplemented
58(C2×D23)⋊C4C23116yescomplemented
59Dic23⋊C4C23116yescomplemented
60C23×C23—212no—
61C23×C23—212no—
62C23×C23—212no—
63C23×C23—416no—
64C23×C23C2×D414yes—
65C23×C23C2×D414yes—
66C23×C23#4114yes—
67C23×C23C2414yes—
68C23×C23C2413yes—
69C22×D23—232no—
70C22×D23—212no—
71C22×D23—212no—
72C22×D23—212no—
73C22×D23—212no—
74C22×D23—212no—
75C22×D23—416no—
76C22×D23C2×D4124yescomplemented
77C22×D23C2414yes—
78C22×D23C2414yes—
79C2×C4×C23—212no—
80C2×C4×C23—212no—
81C2×C4×C23C2414yes—
82C2×C4×C23C2414yes—
83C22⋊D23—232no—
84C22⋊D23—232no—
85C2×Dic23—212no—
86C2×Dic23C2×D4124yescomplemented
87C2×Dic23C2414yes—
88C2×Dic23C2414yes—
89D92—232no—
90C4×D23—232no—
91C22×#121—231nomaximal, Sylow
92C22×C23—24no—
93C22×C23—24no—
94C22×C23—24no—
95C22×C23—424no—
96C22×C23#121116yescomplemented
97C22×C23C22×D416yes—
98C22×C23C22×D416yes—
99C22×C23C2×#4116yes—
100C22×C23C2511yescharacteristic
101C2×D23—248no—
102C2×D23—24no—
103C2×D23—24no—
104C2×D23—24no—
105C2×D23—24no—
106C2×D23—24no—
107C2×D23—424no—
108C2×D23C22×D418yescomplemented
109C4×C23—24no—
110C4×C23—24no—
111Dic23—24no—
112Dic23C22×D418yescomplemented
113C23×D4—231no—
114C23×D4—231no—
115C23×D4—231no—
116C24×C4—231no—
117C2×#121—2324no—
118C22×C4⋊C4—231no—
119C22×C22⋊C4—231no—
120C22×C22⋊C4D2311yescharacteristic, complemented
121C2×C23—44no—
122C2×C23C2×#121112yescomplemented
123C2×C23C23×D411yescharacteristic
124C2×C23C23×D411yescharacteristic
125C2×C23C22×#4111yescharacteristic
126D23—28no—
127D23—44no—
128C25—231no—
129C25—231no—
130C25C2×D2311yescharacteristic
131C22×D4—2312no—
132C22×D4—2312no—
133C22×D4—2312no—
134C22×D4—468no—
135C22×D4—468no—
136C22×D4—468no—
137C23×C4—231no—
138C23×C4—231no—
139C23×C4—2312no—
140C23×C4—468no—
141C23×C4C2×D2311yescharacteristic
142C23×C4C2×D2311yescharacteristic
143#121—2364no—
144C2×C4⋊C4—2312no—
145C2×C22⋊C4—2312no—
146C2×C22⋊C4C2×D23112yescomplemented
147C23C22×#12111yescharacteristic, complemented, Sylow
148C24—24no—
149C24—24no—
150C24—24no—
151C24—238no—
152C24—236no—
153C24—236no—
154C24—464no—
155C24—464no—
156C24—464no—
157C24—464no—
158C24—464no—
159C24C22×D2316yes—
160C24C22×D2311yescharacteristic
161C2×D4—2316no—
162C2×D4—2316no—
163C2×D4—2316no—
164C2×D4—4648no—
165C2×D4—4648no—
166C2×D4—4648no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—238no—
170C22×C4—236no—
171C22×C4—236no—
172C22×C4—2316no—
173C22×C4—4648no—
174C22×C4—464no—
175C22×C4C22×D2316yes—
176C22×C4C22×D2316yes—
177C4⋊C4—2316no—
178C22⋊C4—2316no—
179C22⋊C4C22×D23116yescomplemented
180C23—212no—
181C23—212no—
182C23—212no—
183C23—416no—
184C23—2324no—
185C23—234no—
186C23—234no—
187C23—4632no—
188C23—4612no—
189C23—4612no—
190C23—4612no—
191C23—4612no—
192C23—4612no—
193C23—9216no—
194C23#6728814yes—
195C23D4×D2314yes—
196C23D4×D2314yes—
197C23C23×D2314yes—
198C23C23×D2313yes—
199D4—4632no—
200D4—4632no—
201D4—4632no—
202C2×C4—212no—
203C2×C4—212no—
204C2×C4—2324no—
205C2×C4—234no—
206C2×C4—234no—
207C2×C4—4632no—
208C2×C4—4612no—
209C2×C4C23×D2314yes—
210C2×C4C23×D2314yes—
211C22—24no—
212C22—24no—
213C22—24no—
214C22—424no—
215C22—238no—
216C22—4648no—
217C22—464no—
218C22—464no—
219C22—464no—
220C22—464no—
221C22—464no—
222C22—9224no—
223C22#10625055116yesdirect product factor
224C22C2×#6728816yes—
225C22C2×D4×D2316yes—
226C22C2×D4×D2316yes—
227C22C24×D2311yescharacteristic
228C4—24no—
229C4—24no—
230C4—238no—
231C4—464no—
232C2—44no—
233C2—468no—
234C2—924no—
235C2C2×#10625055112yesdirect product factor
236C2C22×D4×D2311yescharacteristic
237C2C22×D4×D2311yescharacteristic
238C2C22×#6728811yescharacteristic
subgroupG/H
centerC24C22×D23
commutator, cosocleC22×C23C25
FrattiniC22C24×D23
FittingC22×C22⋊C4×C23C2
socleC24×C23C23
hypercenterC22×C22⋊C4D23
hypocenterC23C22×#121
LCSG ▷ C22×C23 ▷ C23
UCSC1 ◁ C24 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×D92C2semidirect
2. C23×C22⋊D23C2semidirect
3. C23×C22⋊D23C2semidirect
4. C23×C4×D23C2semidirect
5. C2×#10625055C2direct (N × H)
6. C22×Dic23⋊C4C2semidirect
7. C22×(C2×D23)⋊C4C2semidirect
8. C22×C22⋊C4×C23C2semidirect
16. C22×C22⋊D23C22semidirect
17. C22×C22⋊D23C22semidirect
21. C22×D92C22semidirect
23. C22×C4×D23C22semidirect
24. C2×C22⋊C4×C23C22semidirect
25. C2×(C2×D23)⋊C4C22semidirect
26. C2×Dic23⋊C4C22semidirect
27. #10625055C22direct (N × H)
38. C23×D23D4semidirect
47. C2×C22⋊D23C23semidirect
48. C2×C22⋊D23C23semidirect
50. C22×Dic23D4semidirect
54. C2×D92C23semidirect
56. C2×C4×D23C23semidirect
57. C22⋊C4×C23C23semidirect
58. (C2×D23)⋊C4C23semidirect
59. Dic23⋊C4C23semidirect
76. C22×D23C2×D4semidirect
86. C2×Dic23C2×D4semidirect
96. C22×C23#121semidirect
108. C2×D23C22×D4semidirect
112. Dic23C22×D4semidirect
120. C22×C22⋊C4D23semidirect
122. C2×C23C2×#121semidirect
146. C2×C22⋊C4C2×D23semidirect
147. C23C22×#121semidirect
179. C22⋊C4C22×D23semidirect
223. C22#10625055direct (N × H)
235. C2C2×#10625055direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D922
C23×C22⋊D232
C23×C4×D232
C2×#106250552
C22×Dic23⋊C42
C22×(C2×D23)⋊C42
C22×C22⋊C4×C232
C22×#12123