Finite Group Databasebeta

← D23×#483 #15683271

Subgroups — D23×#483 #15683271

Order 2944.

460 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C22⋊Q8×D23C211yesmaximal, characteristic
2C2×C4⋊C4×D23C211yesmaximal, characteristic
3#13092749C212yesmaximal, complemented
4#13092748C211yesmaximal, characteristic, complemented
5#13092747C211yesmaximal, characteristic, complemented
6#13092746C212yesmaximal, complemented
7#13092745C211yesmaximal, characteristic, complemented
8#106×D23C212yesmaximal, complemented
9#103×D23C212yesmaximal, complemented
10C22⋊C8×D23C211yesmaximal, characteristic
11C23×#483C211yesmaximal, characteristic, complemented
12C2×Q8×D23C2211yescharacteristic
13C22×C4×D23—21no—
14C22×C4×D23C2211yescharacteristic
15C23×C22⋊Q8C2211yescharacteristic
16C2×C4⋊C4×C23C2211yescharacteristic
17#10625117C2211yescharacteristic
18C2×C4⋊Dic23C2211yescharacteristic
19#10625095C2211yescharacteristic
20C2×Dic23⋊C4—21no—
21C2×C8×D23C2212yes—
22#10625068C2211yescharacteristic
23#10625067—21no—
24C4⋊C4×D23—21no—
25C4⋊C4×D23—21no—
26C4⋊C4×D23—21no—
27C4⋊C4×D23C2212yes—
28C4⋊C4×D23C2212yes—
29C4⋊C4×D23C2211yescharacteristic
30C22⋊C4×D23—21no—
31#10625049—21no—
32C23×#106C2212yescomplemented
33C23×#103C2212yescomplemented
34C23×C22⋊C8C2211yescharacteristic
35#10625017C2212yescomplemented
36C22⋊(C23⋊C8)C2211yescharacteristic
37(C2×D23)⋊C8C2212yes—
38#10624999C2212yescomplemented
39#10624998C2212yescomplemented
40#10624992C2212yescomplemented
41#10624990C2212yescomplemented
42#10624990C2212yescomplemented
43C23×D23D411yescharacteristic
44C2×Q8×C23C2311yescharacteristic
45C22×C4×C23—21no—
46C22×C4×C23C2311yescharacteristic
47C22×Dic23—21no—
48C22×Dic23D411yescharacteristic
49Q8×D23—24no—
50C2×C4×D23—24no—
51C2×C4×D23—21no—
52C2×C4×D23—21no—
53C2×C4×D23—22no—
54C2×C4×D23—22no—
55C2×C4×D23—44no—
56C2×C4×D23D411yescharacteristic
57C2×C4×D23D411yescharacteristic
58C2×C4×D23C2311yescharacteristic
59C2×C4×D23C2311yescharacteristic
60C2×Dic46—21no—
61C2×Dic46C2311yescharacteristic
62C2×C8×C23C2312yes—
63C4⋊C4×C23—21no—
64C4⋊C4×C23—21no—
65C4⋊C4×C23C2312yes—
66C4⋊C4×C23C2311yescharacteristic
67C22⋊C4×C23—21no—
68C22⋊Dic23—21no—
69(C2×D23)⋊C4—21no—
70(C2×D23)⋊C4—21no—
71C4⋊Dic23—21no—
72C4⋊Dic23—21no—
73C4⋊Dic23C2312yes—
74C4⋊Dic23C2311yescharacteristic
75Dic23⋊C4—21no—
76Dic23⋊C4—21no—
77Dic23⋊C4—21no—
78Dic23⋊C4—22no—
79Dic23⋊C4—22no—
80C2×C23⋊C8C2312yes—
81C8×D23—24no—
82C23×C23C2×D411yescharacteristic
83C22×D23—22no—
84C22×D23—22no—
85C22×D23SD1614yes—
86C22×D23C2×D411yescharacteristic
87C22×D23C2×D411yescharacteristic
88Q8×C23—22no—
89C2×C4×C23—22no—
90C2×C4×C23—22no—
91C2×C4×C23—21no—
92C2×C4×C23—21no—
93C2×C4×C23—42no—
94C2×C4×C23C2×D411yescharacteristic
95C2×C4×C23C2411yescharacteristic
96C2×Dic23—22no—
97C2×Dic23—21no—
98C2×Dic23—22no—
99C2×Dic23—21no—
100C2×Dic23—42no—
101C2×Dic23C2×D411yescharacteristic
102C2×Dic23C2×D411yescharacteristic
103C4×D23—22no—
104C4×D23—22no—
105C4×D23—22no—
106C4×D23—42no—
107C4×D23—44no—
108C4×D23—44no—
109C4×D23—42no—
110C4×D23#4114yes—
111Dic46—22no—
112Dic46—42no—
113C8×C23—22no—
114C23⋊C8—22no—
115C2×#483—231nomaximal, Sylow
116C22×C23—21no—
117C22×C23—21no—
118C22×C23C2×SD1612yes—
119C22×C23C22×D411yescharacteristic
120C2×D23—21no—
121C2×D23—21no—
122C2×D23—28no—
123C2×D23#12312yes—
124C2×D23C2×SD1612yes—
125C2×D23C2×SD1612yes—
126C2×D23#13712yes—
127C4×C23—21no—
128C4×C23—41no—
129C4×C23—42no—
130C4×C23—41no—
131C4×C23C2×#4112yes—
132Dic23—21no—
133Dic23—41no—
134Dic23—42no—
135Dic23—41no—
136Dic23#12312yes—
137C2×C22⋊Q8—231no—
138C22×C4⋊C4—231no—
139#483—231no—
140#483—231no—
141#483—232no—
142#483—232no—
143#483—231no—
144#483D2311yescharacteristic, direct product factor
145C2×#106—232no—
146C2×#103—232no—
147C2×C22⋊C8—231no—
148C2×C23—22no—
149C2×C23C2×#12311yescharacteristic
150C2×C23C22×SD1611yescharacteristic
151C2×C23C2×#13711yescharacteristic
152D23—22no—
153D23#48314yesdirect product factor
154C22×Q8—231no—
155C23×C4—231no—
156C23×C4—461no—
157C22×C8—232no—
158C22⋊Q8—231no—
159C22⋊Q8—231no—
160C22⋊Q8—231no—
161C22⋊Q8—461no—
162C22⋊Q8—461no—
163C22⋊Q8C2×D2311yescharacteristic
164C2×C4⋊C4—231no—
165C2×C4⋊C4—232no—
166C2×C4⋊C4—232no—
167C2×C4⋊C4—231no—
168C2×C4⋊C4—461no—
169C2×C4⋊C4—461no—
170C2×C4⋊C4—461no—
171C2×C4⋊C4—461no—
172C2×C4⋊C4C2×D2311yescharacteristic
173C2×C22⋊C4—461no—
174#106—232no—
175#106—232no—
176#106—232no—
177#106C2×D2312yescomplemented
178#103—232no—
179#103—232no—
180#103—232no—
181#103C2×D2312yescomplemented
182C22⋊C8—231no—
183C22⋊C8—232no—
184C22⋊C8C2×D2311yescharacteristic
185C23C2×#48311yescharacteristic, complemented, Sylow
186C24—231no—
187C2×Q8—231no—
188C2×Q8—461no—
189C2×Q8—464no—
190C2×Q8C22×D2311yescharacteristic
191C22×C4—21no—
192C22×C4—231no—
193C22×C4—231no—
194C22×C4—231no—
195C22×C4—231no—
196C22×C4—231no—
197C22×C4—464no—
198C22×C4—461no—
199C22×C4—461no—
200C22×C4—462no—
201C22×C4—462no—
202C22×C4—461no—
203C22×C4—924no—
204C22×C4C22×D2311yescharacteristic
205C2×C8—232no—
206C2×C8—464no—
207C2×C8C22×D2312yes—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—232no—
211C4⋊C4—231no—
212C4⋊C4—462no—
213C4⋊C4—462no—
214C4⋊C4—461no—
215C4⋊C4—461no—
216C4⋊C4—461no—
217C4⋊C4—461no—
218C4⋊C4—461no—
219C4⋊C4C22×D2312yes—
220C4⋊C4C22×D2311yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—461no—
223C22⋊C4—461no—
224C22⋊C4—461no—
225C23—231no—
226C23—234no—
227C23—231no—
228C23—462no—
229C23—462no—
230C23D4×D2311yescharacteristic
231Q8—22no—
232Q8—462no—
233Q8—922no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—231no—
240C2×C4—234no—
241C2×C4—231no—
242C2×C4—462no—
243C2×C4—462no—
244C2×C4—462no—
245C2×C4—462no—
246C2×C4—461no—
247C2×C4—461no—
248C2×C4—462no—
249C2×C4—924no—
250C2×C4—924no—
251C2×C4—922no—
252C2×C4—922no—
253C2×C4—922no—
254C2×C4D4×D2311yescharacteristic
255C2×C4C23×D2311yescharacteristic
256C8—22no—
257C8—462no—
258C22—21no—
259C22—21no—
260C22—232no—
261C22—232no—
262C22—232no—
263C22—232no—
264C22—468no—
265C22—461no—
266C22—461no—
267C22SD16×D2312yes—
268C22C2×D4×D2311yescharacteristic
269C4—21no—
270C4—41no—
271C4—42no—
272C4—41no—
273C4—232no—
274C4—461no—
275C4—921no—
276C4—922no—
277C4—921no—
278C4#41×D2312yes—
279C2—22no—
280C2—234no—
281C2—462no—
282C2#123×D2311yescharacteristic
283C2C2×SD16×D2311yescharacteristic
284C2#137×D2311yescharacteristic
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#483C2
socleC22×C23C22×D4
prime radicalC23×D23D4
hypercenter#483D23
hypocenterC23C2×#483
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #13092749C2semidirect
4. #13092748C2semidirect
5. #13092747C2semidirect
6. #13092746C2semidirect
7. #13092745C2semidirect
8. #106×D23C2semidirect
9. #103×D23C2semidirect
11. C23×#483C2semidirect
32. C23×#106C22semidirect
33. C23×#103C22semidirect
35. #10625017C22semidirect
38. #10624999C22semidirect
39. #10624998C22semidirect
40. #10624992C22semidirect
41. #10624990C22semidirect
42. #10624990C22semidirect
144. #483D23direct (N × H)
153. D23#483direct (N × H)
177. #106C2×D23semidirect
181. #103C2×D23semidirect
185. C23C2×#483semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D232
C2×C4⋊C4×D232
#130927492
#130927482
#130927472
#130927462
#130927452
#106×D232
#103×D232
C22⋊C8×D232
C23×#4832
C2×#48323