Finite Group Databasebeta

← D23×#496 #15683827

Subgroups — D23×#496 #15683827

Order 2944.

668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 145 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×SD16×D23C212yesmaximal, complemented
2C2×D8×D23C212yesmaximal, complemented
3#127×D23C211yesmaximal, characteristic, complemented
4#124×D23C211yesmaximal, characteristic, complemented
5#13092807C211yesmaximal, characteristic, complemented
6#13092806C211yesmaximal, characteristic, complemented
7#13092805C212yesmaximal, complemented
8#13092804C211yesmaximal, characteristic, complemented
9#13092803C212yesmaximal, complemented
10#97×D23C211yesmaximal, characteristic, complemented
11C23×#496C211yesmaximal, characteristic, complemented
12C2×Q8×D23C2211yescharacteristic
13C2×D4×D23—22no—
14C2×D4×D23C2211yescharacteristic
15C2×D4×D23C2212yes—
16C2×SD16×C23C2212yescomplemented
17C2×D8×C23C2212yescomplemented
18C23×#127C2211yescharacteristic, complemented
19C23×#124C2211yescharacteristic, complemented
20#10625118C2211yescharacteristic, complemented
21C2×#67274C2212yescomplemented
22#10625111—22no—
23#10625111C2212yescomplemented
24#10625107C2211yescharacteristic, complemented
25C2×#67273C2212yescomplemented
26C2×#67272C2212yescomplemented
27C2×#67272C2212yescomplemented
28SD16×D23—28no—
29D8×D23—28no—
30C2×D184C2212yescomplemented
31C2×#67263C2212yescomplemented
32C2×C8×D23C2212yes—
33#10625056—22no—
34C22⋊C4×D23—22no—
35#10625046C2211yescharacteristic, complemented
36#10625045C2211yescharacteristic, complemented
37C42×D23C2211yescharacteristic
38C23×#97C2211yescharacteristic, complemented
39#10624997C2212yescomplemented
40#10624985C2211yescharacteristic, complemented
41C23×D23—21no—
42C23×D23—22no—
43C2×Q8×C23C2311yescharacteristic
44C2×D4×C23—22no—
45C2×D4×C23C2311yescharacteristic
46C2×D4×C23C2312yes—
47C2×C22⋊D23—21no—
48C2×C22⋊D23—22no—
49C2×C22⋊D23—24no—
50Q8×D23—24no—
51D4×D23—24no—
52D4×D23—24no—
53D4×D23—24no—
54D4×D23—48no—
55C2×D92—22no—
56C2×D92C2312yes—
57C2×D92C2311yescharacteristic
58C2×C4×D23—21no—
59C2×C4×D23D412yes—
60C2×C4×D23C2311yescharacteristic
61C2×Dic46—21no—
62C2×Dic46C2311yescharacteristic
63SD16×C23—24no—
64D8×C23—24no—
65C2×C8×C23C2312yes—
66C22⋊C4×C23—22no—
67C42×C23C2311yescharacteristic
68C22⋊Dic23—22no—
69#67274—24no—
70#67273—24no—
71#67272—24no—
72#67272—24no—
73(C2×D23)⋊C4—22no—
74(C2×D23)⋊C4—22no—
75C4×Dic23D412yes—
76C4×Dic23C2311yescharacteristic
77C2×C23⋊C8C2312yes—
78D184—24no—
79#67263—24no—
80C8×D23D418yescomplemented
81C23×C23—21no—
82C23×C23—22no—
83C22×D23—21no—
84C22×D23—22no—
85C22×D23—42no—
86C22×D23—44no—
87C22×D23—48no—
88C22×D23—44no—
89C22×D23C2×D411yescharacteristic
90Q8×C23—22no—
91D4×C23—22no—
92D4×C23—22no—
93D4×C23—22no—
94D4×C23—44no—
95C2×C4×C23—21no—
96C2×C4×C23C2×D412yes—
97C2×C4×C23C2411yescharacteristic
98C22⋊D23—48no—
99C22⋊D23—42no—
100C22⋊D23—44no—
101C2×Dic23—21no—
102C2×Dic23C2×D412yes—
103C2×Dic23C2×D411yescharacteristic
104D92—22no—
105D92—22no—
106D92—22no—
107D92—44no—
108C4×D23—24no—
109C4×D23—42no—
110C4×D23C2×D414yes—
111Dic46—22no—
112Dic46—42no—
113C8×C23C2×D414yescomplemented
114C23⋊C8C2×D414yescomplemented
115C2×#496—231nomaximal, Sylow
116C22×C23—41no—
117C22×C23—42no—
118C22×C23—44no—
119C22×C23—42no—
120C22×C23C22×D411yescharacteristic
121C2×D23—42no—
122C2×D23—41no—
123C2×D23—44no—
124C2×D23—42no—
125C2×D23—88no—
126C2×D23—84no—
127C2×D23#12712yes—
128C2×D23AGL(1,ℤ/8)14yes—
129C4×C23—22no—
130C4×C23—41no—
131C4×C23C22×D412yes—
132Dic23—22no—
133Dic23—41no—
134Dic23#12712yes—
135C22×SD16—232no—
136C22×D8—232no—
137C2×#127—231no—
138C2×#124—231no—
139#496—232no—
140#496—232no—
141#496—231no—
142#496—231no—
143#496—231no—
144#496D2311yescharacteristic, direct product factor
145C2×#97—231no—
146C2×C23—82no—
147C2×C23—81no—
148C2×C23C2×#12711yescharacteristic
149C2×C23C2×AGL(1,ℤ/8)12yes—
150D23—82no—
151D23—81no—
152D23#49614yesdirect product factor
153C22×Q8—231no—
154C22×D4—231no—
155C22×D4—232no—
156C22×D4—462no—
157C2×SD16—232no—
158C2×SD16—232no—
159C2×SD16—232no—
160C2×SD16—468no—
161C2×SD16C2×D2312yescomplemented
162C2×D8—232no—
163C2×D8—232no—
164C2×D8—232no—
165C2×D8—468no—
166C2×D8C2×D2312yescomplemented
167C22×C8—232no—
168#127—232no—
169#127—231no—
170#127—462no—
171#127C2×D2311yescharacteristic, complemented
172#124—231no—
173#124—231no—
174#124—231no—
175#124—462no—
176#124C2×D2311yescharacteristic, complemented
177C2×C22⋊C4—462no—
178C2×C42—231no—
179#97—232no—
180#97—231no—
181#97C2×D2311yescharacteristic, complemented
182C23C2×#49611yescharacteristic, complemented, Sylow
183C24—461no—
184C24—462no—
185C2×Q8—231no—
186C2×Q8—461no—
187C2×Q8—464no—
188C2×Q8C22×D2311yescharacteristic
189C2×D4—22no—
190C2×D4—231no—
191C2×D4—232no—
192C2×D4—464no—
193C2×D4—464no—
194C2×D4—464no—
195C2×D4—461no—
196C2×D4—464no—
197C2×D4—462no—
198C2×D4—462no—
199C2×D4—928no—
200C2×D4C22×D2311yescharacteristic
201C2×D4C22×D2312yes—
202C22×C4—231no—
203C22×C4—232no—
204C22×C4—461no—
205SD16—24no—
206SD16—464no—
207SD16—464no—
208SD16—464no—
209D8—24no—
210D8—464no—
211D8—464no—
212D8—464no—
213C2×C8—232no—
214C2×C8—238no—
215C2×C8C22×D2312yes—
216C22⋊C4—22no—
217C22⋊C4—462no—
218C22⋊C4—462no—
219C22⋊C4—462no—
220C42—232no—
221C42—231no—
222C42C22×D2311yescharacteristic
223C23—21no—
224C23—22no—
225C23—231no—
226C23—461no—
227C23—462no—
228C23—922no—
229C23—924no—
230C23—928no—
231C23—924no—
232Q8—22no—
233Q8—462no—
234Q8—922no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—462no—
240D4—462no—
241D4—462no—
242D4—924no—
243D4—922no—
244D4—924no—
245D4—928no—
246C2×C4—21no—
247C2×C4—231no—
248C2×C4—234no—
249C2×C4—232no—
250C2×C4—464no—
251C2×C4—461no—
252C2×C4—922no—
253C2×C4D4×D2312yes—
254C2×C4C23×D2311yescharacteristic
255C8—234no—
256C8D4×D2314yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—234no—
262C22—232no—
263C22—924no—
264C22—922no—
265C22—921no—
266C22—922no—
267C22—1848no—
268C22—1844no—
269C22C2×D4×D2311yescharacteristic
270C4—22no—
271C4—41no—
272C4—232no—
273C4—462no—
274C4—921no—
275C4C2×D4×D2312yes—
276C2—82no—
277C2—81no—
278C2—234no—
279C2—1842no—
280C2—1841no—
281C2#127×D2311yescharacteristic
282C2AGL(1,ℤ/8)×D2312yes—
subgroupG/H
centerC22C2×D4×D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#496C2
socleC22×C23C22×D4
hypercenter#496D23
hypocenterC23C2×#496
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D23C2semidirect
2. C2×D8×D23C2semidirect
3. #127×D23C2semidirect
4. #124×D23C2semidirect
5. #13092807C2semidirect
6. #13092806C2semidirect
7. #13092805C2semidirect
8. #13092804C2semidirect
9. #13092803C2semidirect
10. #97×D23C2semidirect
11. C23×#496C2semidirect
16. C2×SD16×C23C22semidirect
17. C2×D8×C23C22semidirect
18. C23×#127C22semidirect
19. C23×#124C22semidirect
20. #10625118C22semidirect
21. C2×#67274C22semidirect
23. #10625111C22semidirect
24. #10625107C22semidirect
25. C2×#67273C22semidirect
26. C2×#67272C22semidirect
27. C2×#67272C22semidirect
30. C2×D184C22semidirect
31. C2×#67263C22semidirect
35. #10625046C22semidirect
36. #10625045C22semidirect
38. C23×#97C22semidirect
39. #10624997C22semidirect
40. #10624985C22semidirect
80. C8×D23D4semidirect
113. C8×C23C2×D4semidirect
114. C23⋊C8C2×D4semidirect
144. #496D23direct (N × H)
152. D23#496direct (N × H)
161. C2×SD16C2×D23semidirect
166. C2×D8C2×D23semidirect
171. #127C2×D23semidirect
176. #124C2×D23semidirect
181. #97C2×D23semidirect
182. C23C2×#496semidirect
256. C8D4×D23semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D232
C2×D8×D232
#127×D232
#124×D232
#130928072
#130928062
#130928052
#130928042
#130928032
#97×D232
C23×#4962
C2×#49623