Finite Group Databasebeta

← D23×#509 #15686517

Subgroups — D23×#509 #15686517

Order 2944.

400 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 115 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#135×D23C211yesmaximal, characteristic, complemented
2C2×D8×D23C211yesmaximal, characteristic, complemented
3#13092855C211yesmaximal, characteristic, complemented
4#13092854C212yesmaximal, complemented
5#13092853C211yesmaximal, characteristic, complemented
6#13092852C211yesmaximal, characteristic, complemented
7#13092851C212yesmaximal, complemented
8SD32×D23C212yesmaximal, complemented
9D16×D23C212yesmaximal, complemented
10M32×D23C211yesmaximal, characteristic, complemented
11C23×#509C211yesmaximal, characteristic, complemented
12#41×D23—21no—
13C2×D4×D23—21no—
14C23×#135C2211yescharacteristic, complemented
15C2×D8×C23C2211yescharacteristic, complemented
16#10625121—21no—
17C2×#67272—21no—
18#10625090C2211yescharacteristic, complemented
19Q16×D23C2211yescharacteristic, complemented
20#10625087—21no—
21SD16×D23—21no—
22#10625083C2211yescharacteristic, complemented
23D8×D23—21no—
24D8×D23—21no—
25D8×D23C2212yes—
26D8×D23C2212yescomplemented
27D8×D23C2211yescharacteristic, complemented
28#10625075C2211yescharacteristic, complemented
29C2×D184C2211yescharacteristic, complemented
30C2×C8×D23C2211yescharacteristic
31C23×SD32C2212yescomplemented
32C23×D16C2212yescomplemented
33C23×M32C2211yescharacteristic, complemented
34#10625010C2212yescomplemented
35#10625009C2212yescomplemented
36#10625008C2212yescomplemented
37#10625008C2212yescomplemented
38#10624994C2211yescharacteristic, complemented
39#10624982C2212yescomplemented
40D368C2212yescomplemented
41#10624980C2212yescomplemented
42C16×D23C2212yescomplemented
43C23×D23—41no—
44#41×C23—21no—
45C2×D4×C23—21no—
46C2×C22⋊D23—41no—
47#67292—21no—
48Q8×D23—21no—
49#67290—21no—
50#67290—41no—
51D4×D23—22no—
52D4×D23—22no—
53D4×D23—21no—
54D4×D23—41no—
55D4×D23—41no—
56D4×D23—41no—
57#67288—21no—
58#67288—41no—
59C2×D92—21no—
60C2×C4×D23—41no—
61C2×C4×D23D411yescharacteristic
62Q16×C23C2311yescharacteristic, complemented
63SD16×C23—21no—
64D8×C23—21no—
65D8×C23C2312yes—
66D8×C23C2311yescharacteristic, complemented
67C2×C8×C23C2311yescharacteristic
68#67275—21no—
69#67274—21no—
70#67273—21no—
71#67272—22no—
72#67272—22no—
73#67272—21no—
74C2×C23⋊C8D411yescharacteristic
75Dic92C2311yescharacteristic, complemented
76D184—21no—
77D184C2312yes—
78D184C2311yescharacteristic, complemented
79#67263—21no—
80C8×D23D411yescharacteristic
81C8×D23D411yescharacteristic
82C8×D23C2311yescharacteristic
83C8×D23C2311yescharacteristic
84C16×C23C2312yescomplemented
85C23⋊C16C2312yescomplemented
86C23×C23—41no—
87C22×D23—41no—
88C22×D23—41no—
89C22×D23—42no—
90C22×D23—42no—
91C22×D23—84no—
92C22×D23D811yescharacteristic
93Q8×C23—21no—
94D4×C23—22no—
95D4×C23—21no—
96D4×C23—41no—
97D4×C23—41no—
98C2×C4×C23—41no—
99C2×C4×C23C2×D411yescharacteristic
100C22⋊D23—41no—
101C22⋊D23—41no—
102C22⋊D23—41no—
103C22⋊D23—42no—
104C22⋊D23—42no—
105C2×Dic23—41no—
106C2×Dic23D811yescharacteristic
107D92—21no—
108D92—22no—
109D92—41no—
110D92—41no—
111C4×D23—41no—
112C4×D23—41no—
113C4×D23—41no—
114C4×D23D811yescharacteristic
115C4×D23D811yescharacteristic
116C4×D23C2×D411yescharacteristic
117C4×D23C2×D411yescharacteristic
118Dic46—21no—
119Dic46—41no—
120C8×C23C2×D411yescharacteristic
121C8×C23C2411yescharacteristic
122C23⋊C8C2×D411yescharacteristic
123C23⋊C8C2×D411yescharacteristic
124C2×#509—231nomaximal, Sylow
125C22×C23—41no—
126C22×C23—42no—
127C22×C23—82no—
128C22×C23C2×D811yescharacteristic
129C2×D23—22no—
130C2×D23—41no—
131C2×D23—42no—
132C2×D23—84no—
133C2×D23—82no—
134C2×D23—82no—
135C2×D23—84no—
136C2×D23C2×D811yescharacteristic
137C2×D23C2×D811yescharacteristic
138C4×C23—41no—
139C4×C23C2×D811yescharacteristic
140C4×C23C22×D411yescharacteristic
141Dic23—41no—
142Dic23C2×D811yescharacteristic
143Dic23C2×D811yescharacteristic
144C2×#135—231no—
145C22×D8—231no—
146#509—231no—
147#509—231no—
148#509—232no—
149#509—232no—
150#509—231no—
151#509D2311yescharacteristic, direct product factor
152C2×SD32—232no—
153C2×D16—232no—
154C2×M32—231no—
155C2×C23—21no—
156C2×C23—82no—
157C2×C23—81no—
158C2×C23C22×D811yescharacteristic
159D23—21no—
160D23—81no—
161D23—82no—
162D23#50912yesdirect product factor
163C2×#41—461no—
164C22×D4—461no—
165#135—231no—
166#135—231no—
167#135—231no—
168#135—461no—
169#135—461no—
170#135C2×D2311yescharacteristic, complemented
171C2×Q16—231no—
172C2×SD16—461no—
173C2×D8—232no—
174C2×D8—232no—
175C2×D8—231no—
176C2×D8—231no—
177C2×D8—461no—
178C2×D8—461no—
179C2×D8—461no—
180C2×D8C2×D2311yescharacteristic, complemented
181C22×C8—231no—
182SD32—232no—
183SD32—232no—
184SD32—232no—
185SD32C2×D2312yescomplemented
186D16—232no—
187D16—232no—
188D16—232no—
189D16C2×D2312yescomplemented
190M32—231no—
191M32—232no—
192M32C2×D2311yescharacteristic, complemented
193C2×C16—232no—
194C23C2×#50911yescharacteristic, complemented, Sylow
195C24—921no—
196#41—21no—
197#41—461no—
198#41—461no—
199#41—461no—
200#41—921no—
201#41—921no—
202C2×Q8—461no—
203C2×D4—21no—
204C2×D4—462no—
205C2×D4—462no—
206C2×D4—461no—
207C2×D4—461no—
208C2×D4—921no—
209C2×D4—921no—
210C2×D4—921no—
211C2×D4—921no—
212C22×C4—231no—
213C22×C4—921no—
214Q16—231no—
215Q16—461no—
216Q16C22×D2311yescharacteristic, complemented
217SD16—21no—
218SD16—461no—
219SD16—461no—
220SD16—461no—
221D8—21no—
222D8—231no—
223D8—232no—
224D8—462no—
225D8—462no—
226D8—461no—
227D8—461no—
228D8C22×D2312yes—
229D8C22×D2311yescharacteristic, complemented
230C2×C8—231no—
231C2×C8—231no—
232C2×C8—231no—
233C2×C8—231no—
234C2×C8—231no—
235C2×C8C22×D2311yescharacteristic
236C16—232no—
237C16C22×D2312yescomplemented
238C23—41no—
239C23—231no—
240C23—921no—
241C23—922no—
242C23—922no—
243C23—921no—
244C23—1844no—
245Q8—21no—
246Q8—461no—
247Q8—921no—
248D4—22no—
249D4—21no—
250D4—41no—
251D4—41no—
252D4—462no—
253D4—461no—
254D4—921no—
255D4—922no—
256D4—922no—
257D4—921no—
258D4—921no—
259D4—921no—
260D4—921no—
261C2×C4—41no—
262C2×C4—231no—
263C2×C4—231no—
264C2×C4—231no—
265C2×C4—231no—
266C2×C4—231no—
267C2×C4—921no—
268C2×C4—921no—
269C2×C4—921no—
270C2×C4—921no—
271C2×C4D4×D2311yescharacteristic
272C8—231no—
273C8—231no—
274C8D4×D2311yescharacteristic
275C8C23×D2311yescharacteristic
276C22—41no—
277C22—42no—
278C22—82no—
279C22—231no—
280C22—231no—
281C22—462no—
282C22—921no—
283C22—922no—
284C22—1842no—
285C22—1844no—
286C22—1842no—
287C22—1844no—
288C22D8×D2311yescharacteristic
289C4—41no—
290C4—231no—
291C4—231no—
292C4—921no—
293C4D8×D2311yescharacteristic
294C4C2×D4×D2311yescharacteristic
295C2—21no—
296C2—82no—
297C2—81no—
298C2—232no—
299C2—461no—
300C2—1842no—
301C2—1841no—
302C2C2×D8×D2311yescharacteristic
subgroupG/H
centerC2C2×D8×D23
commutator, cosocleC8×C23C24
FrattiniC8C23×D23
FittingC23×#509C2
socleC2×C23C22×D8
hypercenter#509D23
hypocenterC23C2×#509
LCSG ▷ C8×C23 ▷ C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #135×D23C2semidirect
2. C2×D8×D23C2semidirect
3. #13092855C2semidirect
4. #13092854C2semidirect
5. #13092853C2semidirect
6. #13092852C2semidirect
7. #13092851C2semidirect
8. SD32×D23C2semidirect
9. D16×D23C2semidirect
10. M32×D23C2semidirect
11. C23×#509C2semidirect
14. C23×#135C22semidirect
15. C2×D8×C23C22semidirect
18. #10625090C22semidirect
19. Q16×D23C22semidirect
22. #10625083C22semidirect
26. D8×D23C22semidirect
27. D8×D23C22semidirect
28. #10625075C22semidirect
29. C2×D184C22semidirect
31. C23×SD32C22semidirect
32. C23×D16C22semidirect
33. C23×M32C22semidirect
34. #10625010C22semidirect
35. #10625009C22semidirect
36. #10625008C22semidirect
37. #10625008C22semidirect
38. #10624994C22semidirect
39. #10624982C22semidirect
40. D368C22semidirect
41. #10624980C22semidirect
42. C16×D23C22semidirect
62. Q16×C23C23semidirect
66. D8×C23C23semidirect
75. Dic92C23semidirect
78. D184C23semidirect
84. C16×C23C23semidirect
85. C23⋊C16C23semidirect
151. #509D23direct (N × H)
162. D23#509direct (N × H)
170. #135C2×D23semidirect
180. C2×D8C2×D23semidirect
185. SD32C2×D23semidirect
189. D16C2×D23semidirect
192. M32C2×D23semidirect
194. C23C2×#509semidirect
216. Q16C22×D23semidirect
229. D8C22×D23semidirect
237. C16C22×D23semidirect

Maximal subgroups

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

subgroupindex
#135×D232
C2×D8×D232
#130928552
#130928542
#130928532
#130928522
#130928512
SD32×D232
D16×D232
M32×D232
C23×#5092
C2×#50923