Finite Group Databasebeta

← D23×#488 #15683710

Subgroups — D23×#488 #15683710

Order 2944.

412 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 125 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#125×D23C211yesmaximal, characteristic, complemented
2#124×D23C211yesmaximal, characteristic
3#13092774C211yesmaximal, characteristic, complemented
4#13092773C212yesmaximal, complemented
5#13092772C211yesmaximal, characteristic, complemented
6#13092771C211yesmaximal, characteristic, complemented
7#13092770C212yesmaximal, complemented
8#103×D23C212yesmaximal, complemented
9#102×D23C212yesmaximal
10C4×C8×D23C211yesmaximal, characteristic, complemented
11C23×#488C211yesmaximal, characteristic, complemented
12C2×Q8×D23C2211yescharacteristic
13C2×D4×D23C2211yescharacteristic
14C23×#125C2211yescharacteristic, complemented
15C23×#124C2211yescharacteristic
16#10625118C2211yescharacteristic
17#10625107C2211yescharacteristic
18C2×C8×D23C2212yes—
19C4⋊C4×D23—22no—
20C4⋊C4×D23C2212yes—
21#10625061C2212yescomplemented
22#10625060—22no—
23#10625056—22no—
24C22⋊C4×D23—22no—
25#10625046C2211yescharacteristic
26C42×D23C2211yescharacteristic
27#10625041C2211yescharacteristic, complemented
28C23×#103C2212yescomplemented
29C23×#102C2212yes—
30C4×C8×C23C2211yescharacteristic, complemented
31#10625017C2212yescomplemented
32#10625014C2212yes—
33#10625003C2212yes—
34#10624998C2212yescomplemented
35C8×Dic23C2212yescomplemented
36#10624992C2212yescomplemented
37#10624991C2212yes—
38C4×C23⋊C8C2211yescharacteristic, complemented
39C23×D23—21no—
40C2×Q8×C23C2311yescharacteristic
41C2×D4×C23C2311yescharacteristic
42C2×C22⋊D23—21no—
43Q8×D23—24no—
44D4×D23—24no—
45C2×D92C2311yescharacteristic
46C2×C4×D23—21no—
47C2×C4×D23—22no—
48C2×C4×D23D412yes—
49C2×C4×D23C2311yescharacteristic
50C2×Dic46—21no—
51C2×Dic46C2311yescharacteristic
52C2×C8×C23C2312yes—
53C4⋊C4×C23—22no—
54C4⋊C4×C23C2312yes—
55C22⋊C4×C23—22no—
56C42×C23C2311yescharacteristic
57C22⋊Dic23—22no—
58(C2×D23)⋊C4—22no—
59(C2×D23)⋊C4—22no—
60C4⋊Dic23—22no—
61C4⋊Dic23C2312yes—
62Dic23⋊C4—22no—
63Dic23⋊C4—22no—
64Dic23⋊C4—22no—
65C4×Dic23D412yes—
66C4×Dic23C2311yescharacteristic
67C2×C23⋊C8C2312yes—
68C8×D23—24no—
69C23×C23—21no—
70C22×D23—21no—
71C22×D23—42no—
72C22×D23—44no—
73C22×D23C2×D411yescharacteristic
74Q8×C23—22no—
75D4×C23—22no—
76C2×C4×C23—22no—
77C2×C4×C23—21no—
78C2×C4×C23C2×D412yes—
79C2×C4×C23C2411yescharacteristic
80C22⋊D23—42no—
81C2×Dic23—22no—
82C2×Dic23—21no—
83C2×Dic23C2×D412yes—
84C2×Dic23C2×D411yescharacteristic
85D92—22no—
86C4×D23—24no—
87C4×D23—42no—
88C4×D23—44no—
89C4×D23#4114yes—
90Dic46—22no—
91Dic46—42no—
92C8×C23—22no—
93C23⋊C8—22no—
94C2×#488—231nomaximal, Sylow
95C22×C23—41no—
96C22×C23—42no—
97C22×C23C22×D411yescharacteristic
98C2×D23—41no—
99C2×D23—42no—
100C2×D23—84no—
101C2×D23#12412yes—
102C2×D23#13514yes—
103C4×C23—22no—
104C4×C23—41no—
105C4×C23—42no—
106C4×C23C2×#4112yes—
107Dic23—22no—
108Dic23—41no—
109Dic23—42no—
110Dic23#12412yes—
111C2×#125—231no—
112C2×#124—231no—
113#488—231no—
114#488—231no—
115#488—232no—
116#488—232no—
117#488—231no—
118#488D2311yescharacteristic, direct product factor
119C2×#103—232no—
120C2×#102—232no—
121C2×C4×C8—231no—
122C2×C23—81no—
123C2×C23C2×#12411yescharacteristic
124C2×C23C2×#13512yes—
125D23—81no—
126D23#48814yesdirect product factor
127C22×Q8—231no—
128C22×D4—231no—
129C22×C8—232no—
130#125—232no—
131#125—231no—
132#125—462no—
133#125C2×D2311yescharacteristic, complemented
134#124—231no—
135#124—231no—
136#124—231no—
137#124—462no—
138#124C2×D2311yescharacteristic
139C2×C4⋊C4—232no—
140C2×C4⋊C4—462no—
141C2×C22⋊C4—462no—
142C2×C42—231no—
143#103—232no—
144#103—232no—
145#103—232no—
146#103C2×D2312yescomplemented
147#102—232no—
148#102—232no—
149#102—232no—
150#102C2×D2312yes—
151C4×C8—232no—
152C4×C8—231no—
153C4×C8C2×D2311yescharacteristic, complemented
154C23C2×#48811yescharacteristic, complemented, Sylow
155C24—461no—
156C2×Q8—231no—
157C2×Q8—461no—
158C2×Q8—464no—
159C2×Q8C22×D2311yescharacteristic
160C2×D4—231no—
161C2×D4—464no—
162C2×D4—461no—
163C2×D4C22×D2311yescharacteristic
164C22×C4—231no—
165C22×C4—232no—
166C22×C4—461no—
167C22×C4—462no—
168C2×C8—232no—
169C2×C8—464no—
170C2×C8C22×D2312yes—
171C4⋊C4—22no—
172C4⋊C4—232no—
173C4⋊C4—462no—
174C4⋊C4—462no—
175C4⋊C4—462no—
176C4⋊C4—462no—
177C4⋊C4C22×D2312yes—
178C22⋊C4—22no—
179C22⋊C4—462no—
180C22⋊C4—462no—
181C22⋊C4—462no—
182C42—232no—
183C42—231no—
184C42C22×D2311yescharacteristic
185C23—21no—
186C23—231no—
187C23—461no—
188C23—922no—
189C23—924no—
190Q8—22no—
191Q8—462no—
192Q8—922no—
193D4—22no—
194D4—462no—
195D4—922no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—231no—
199C2×C4—234no—
200C2×C4—232no—
201C2×C4—464no—
202C2×C4—461no—
203C2×C4—462no—
204C2×C4—922no—
205C2×C4—924no—
206C2×C4D4×D2312yes—
207C2×C4C23×D2311yescharacteristic
208C8—22no—
209C8—462no—
210C22—41no—
211C22—42no—
212C22—232no—
213C22—234no—
214C22—921no—
215C22—922no—
216C22—1844no—
217C22C2×D4×D2311yescharacteristic
218C4—22no—
219C4—41no—
220C4—42no—
221C4—232no—
222C4—462no—
223C4—921no—
224C4—922no—
225C4#41×D2312yes—
226C2—81no—
227C2—234no—
228C2—1841no—
229C2#124×D2311yescharacteristic
230C2#135×D2312yes—
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#488C2
socleC22×C23C22×D4
prime radicalC2×D4×D23C22
hypercenter#488D23
hypocenterC23C2×#488
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #125×D23C2semidirect
3. #13092774C2semidirect
4. #13092773C2semidirect
5. #13092772C2semidirect
6. #13092771C2semidirect
7. #13092770C2semidirect
8. #103×D23C2semidirect
10. C4×C8×D23C2semidirect
11. C23×#488C2semidirect
14. C23×#125C22semidirect
21. #10625061C22semidirect
27. #10625041C22semidirect
28. C23×#103C22semidirect
30. C4×C8×C23C22semidirect
31. #10625017C22semidirect
34. #10624998C22semidirect
35. C8×Dic23C22semidirect
36. #10624992C22semidirect
38. C4×C23⋊C8C22semidirect
118. #488D23direct (N × H)
126. D23#488direct (N × H)
133. #125C2×D23semidirect
146. #103C2×D23semidirect
153. C4×C8C2×D23semidirect
154. C23C2×#488semidirect

Maximal subgroups

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

subgroupindex
#125×D232
#124×D232
#130927742
#130927732
#130927722
#130927712
#130927702
#103×D232
#102×D232
C4×C8×D232
C23×#4882
C2×#48823