Finite Group Databasebeta

← D23×#489 #15683723

Subgroups — D23×#489 #15683723

Order 2944.

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

normal subgroups only

This group has 436 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
1C4⋊Q8×D23C211yesmaximal, characteristic, complemented
2#124×D23C211yesmaximal, characteristic
3#13092779C211yesmaximal, characteristic, complemented
4#13092778C212yesmaximal, complemented
5#13092777C211yesmaximal, characteristic, complemented
6#13092776C211yesmaximal, characteristic, complemented
7#13092775C212yesmaximal, complemented
8#103×D23C212yesmaximal, complemented
9#102×D23C212yesmaximal
10#97×D23C211yesmaximal, characteristic, complemented
11C23×#489C211yesmaximal, characteristic, complemented
12C2×Q8×D23—21no—
13C2×Q8×D23C2211yescharacteristic
14C2×D4×D23C2211yescharacteristic
15C23×C4⋊Q8C2211yescharacteristic, complemented
16C23×#124C2211yescharacteristic
17#10625118C2211yescharacteristic
18#10625115—21no—
19#10625107C2211yescharacteristic
20C2×C8×D23C2212yes—
21C4⋊C4×D23—21no—
22C4⋊C4×D23C2212yes—
23#10625059—21no—
24#10625059C2212yescomplemented
25#10625056—21no—
26#10625056—21no—
27C22⋊C4×D23—21no—
28C22⋊C4×D23—21no—
29#10625046C2211yescharacteristic
30C42×D23C2211yescharacteristic
31#10625040C2211yescharacteristic, complemented
32C23×#103C2212yescomplemented
33C23×#102C2212yes—
34C23×#97C2211yescharacteristic, complemented
35#10625017C2212yescomplemented
36#10625014C2212yes—
37#10625003C2212yes—
38#10624998C2212yescomplemented
39#10624997C2212yescomplemented
40#10624992C2212yescomplemented
41#10624991C2212yes—
42#10624985C2211yescharacteristic, complemented
43C23×D23—21no—
44C2×Q8×C23—21no—
45C2×Q8×C23C2311yescharacteristic
46C2×D4×C23C2311yescharacteristic
47C2×C22⋊D23—21no—
48Q8×D23—24no—
49Q8×D23—44no—
50D4×D23—24no—
51C2×D92C2311yescharacteristic
52C2×C4×D23—21no—
53C2×C4×D23—22no—
54C2×C4×D23D411yescharacteristic
55C2×C4×D23D411yescharacteristic
56C2×C4×D23C2311yescharacteristic
57C2×Dic46—21no—
58C2×Dic46—21no—
59C2×Dic46—22no—
60C2×Dic46C2311yescharacteristic
61C2×C8×C23C2312yes—
62C4⋊C4×C23—21no—
63C4⋊C4×C23C2312yes—
64C22⋊C4×C23—21no—
65C22⋊C4×C23—21no—
66C42×C23C2311yescharacteristic
67C22⋊Dic23—21no—
68C22⋊Dic23—21no—
69(C2×D23)⋊C4—21no—
70(C2×D23)⋊C4—21no—
71(C2×D23)⋊C4—21no—
72(C2×D23)⋊C4—21no—
73C4⋊Dic23—21no—
74C4⋊Dic23C2312yes—
75Dic23⋊C4—22no—
76Dic23⋊C4—22no—
77C4×Dic23D411yescharacteristic
78C4×Dic23D411yescharacteristic
79C4×Dic23C2311yescharacteristic
80C2×C23⋊C8C2312yes—
81C8×D23—24no—
82C23×C23—21no—
83C22×D23—21no—
84C22×D23—42no—
85C22×D23—42no—
86C22×D23—42no—
87C22×D23C2×D411yescharacteristic
88Q8×C23—22no—
89Q8×C23—42no—
90D4×C23—22no—
91C2×C4×C23—22no—
92C2×C4×C23—21no—
93C2×C4×C23C2×D411yescharacteristic
94C2×C4×C23C2×D411yescharacteristic
95C2×C4×C23C2411yescharacteristic
96C22⋊D23—42no—
97C2×Dic23—22no—
98C2×Dic23—21no—
99C2×Dic23C2×D411yescharacteristic
100C2×Dic23C2×D411yescharacteristic
101C2×Dic23C2×D411yescharacteristic
102D92—22no—
103C4×D23—22no—
104C4×D23—22no—
105C4×D23—42no—
106C4×D23—44no—
107C4×D23#4114yes—
108Dic46—22no—
109Dic46—44no—
110Dic46—42no—
111Dic46—42no—
112C8×C23—22no—
113C23⋊C8—22no—
114C2×#489—231nomaximal, Sylow
115C22×C23—41no—
116C22×C23—41no—
117C22×C23—41no—
118C22×C23C22×D411yescharacteristic
119C2×D23—41no—
120C2×D23—41no—
121C2×D23—41no—
122C2×D23—84no—
123C2×D23#12412yes—
124C2×D23AGL(1,ℤ/8)12yes—
125C2×D23#13712yes—
126C4×C23—21no—
127C4×C23—21no—
128C4×C23—41no—
129C4×C23—42no—
130C4×C23C2×#4112yes—
131Dic23—21no—
132Dic23—21no—
133Dic23—41no—
134Dic23—42no—
135Dic23#12412yes—
136C2×C4⋊Q8—231no—
137C2×#124—231no—
138#489—231no—
139#489—231no—
140#489—232no—
141#489—232no—
142#489—231no—
143#489D2311yescharacteristic, direct product factor
144C2×#103—232no—
145C2×#102—232no—
146C2×#97—231no—
147C2×C23—81no—
148C2×C23C2×#12411yescharacteristic
149C2×C23C2×AGL(1,ℤ/8)11yescharacteristic
150C2×C23C2×#13711yescharacteristic
151D23—81no—
152D23#48914yesdirect product factor
153C22×Q8—231no—
154C22×Q8—461no—
155C22×D4—231no—
156C22×C8—232no—
157C4⋊Q8—232no—
158C4⋊Q8—231no—
159C4⋊Q8—461no—
160C4⋊Q8—461no—
161C4⋊Q8C2×D2311yescharacteristic, complemented
162#124—231no—
163#124—231no—
164#124—231no—
165#124—461no—
166#124—461no—
167#124C2×D2311yescharacteristic
168C2×C4⋊C4—232no—
169C2×C4⋊C4—461no—
170C2×C22⋊C4—461no—
171C2×C22⋊C4—461no—
172C2×C42—231no—
173#103—232no—
174#103—232no—
175#103—232no—
176#103C2×D2312yescomplemented
177#102—232no—
178#102—232no—
179#102—232no—
180#102C2×D2312yes—
181#97—232no—
182#97—231no—
183#97C2×D2311yescharacteristic, complemented
184C23C2×#48911yescharacteristic, complemented, Sylow
185C24—461no—
186C2×Q8—21no—
187C2×Q8—231no—
188C2×Q8—461no—
189C2×Q8—462no—
190C2×Q8—464no—
191C2×Q8—461no—
192C2×Q8—924no—
193C2×Q8C22×D2311yescharacteristic
194C2×D4—231no—
195C2×D4—464no—
196C2×D4—461no—
197C2×D4C22×D2311yescharacteristic
198C22×C4—231no—
199C22×C4—231no—
200C22×C4—231no—
201C22×C4—461no—
202C22×C4—462no—
203C2×C8—232no—
204C2×C8—464no—
205C2×C8C22×D2312yes—
206C4⋊C4—21no—
207C4⋊C4—232no—
208C4⋊C4—462no—
209C4⋊C4—462no—
210C4⋊C4—461no—
211C4⋊C4C22×D2312yes—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—461no—
215C22⋊C4—461no—
216C22⋊C4—461no—
217C22⋊C4—461no—
218C22⋊C4—461no—
219C22⋊C4—461no—
220C42—231no—
221C42—231no—
222C42—231no—
223C42C22×D2311yescharacteristic
224C23—21no—
225C23—231no—
226C23—461no—
227C23—922no—
228C23—922no—
229C23—922no—
230Q8—22no—
231Q8—42no—
232Q8—462no—
233Q8—922no—
234Q8—922no—
235Q8—924no—
236D4—22no—
237D4—462no—
238D4—922no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—231no—
242C2×C4—234no—
243C2×C4—231no—
244C2×C4—231no—
245C2×C4—462no—
246C2×C4—462no—
247C2×C4—461no—
248C2×C4—462no—
249C2×C4—922no—
250C2×C4—924no—
251C2×C4D4×D2311yescharacteristic
252C2×C4D4×D2311yescharacteristic
253C2×C4C23×D2311yescharacteristic
254C8—22no—
255C8—462no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—232no—
260C22—232no—
261C22—232no—
262C22—921no—
263C22—921no—
264C22—921no—
265C22—1844no—
266C22C2×D4×D2311yescharacteristic
267C4—21no—
268C4—21no—
269C4—41no—
270C4—42no—
271C4—232no—
272C4—461no—
273C4—461no—
274C4—921no—
275C4—922no—
276C4#41×D2312yes—
277C2—81no—
278C2—234no—
279C2—1841no—
280C2#124×D2311yescharacteristic
281C2AGL(1,ℤ/8)×D2311yescharacteristic
282C2#137×D2311yescharacteristic
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#489C2
socleC22×C23C22×D4
prime radicalC2×D4×D23C22
hypercenter#489D23
hypocenterC23C2×#489
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D23C2semidirect
3. #13092779C2semidirect
4. #13092778C2semidirect
5. #13092777C2semidirect
6. #13092776C2semidirect
7. #13092775C2semidirect
8. #103×D23C2semidirect
10. #97×D23C2semidirect
11. C23×#489C2semidirect
15. C23×C4⋊Q8C22semidirect
24. #10625059C22semidirect
31. #10625040C22semidirect
32. C23×#103C22semidirect
34. C23×#97C22semidirect
35. #10625017C22semidirect
38. #10624998C22semidirect
39. #10624997C22semidirect
40. #10624992C22semidirect
42. #10624985C22semidirect
143. #489D23direct (N × H)
152. D23#489direct (N × H)
161. C4⋊Q8C2×D23semidirect
176. #103C2×D23semidirect
183. #97C2×D23semidirect
184. C23C2×#489semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D232
#124×D232
#130927792
#130927782
#130927772
#130927762
#130927752
#103×D232
#102×D232
#97×D232
C23×#4892
C2×#48923