Finite Group Databasebeta

← D23×#508 #15686511

Subgroups — D23×#508 #15686511

Order 2944.

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

normal subgroups only

This group has 368 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×D23C212yesmaximal, complemented
2#13092850C211yesmaximal, characteristic, complemented
3#13092849C211yesmaximal, characteristic, complemented
4#13092848C212yesmaximal, complemented
5#13092847C212yesmaximal, complemented
6#13092846C211yesmaximal, characteristic, complemented
7Q32×D23C211yesmaximal, characteristic, complemented
8SD32×D23C212yesmaximal, complemented
9D16×D23C211yesmaximal, characteristic, complemented
10C2×C16×D23C211yesmaximal, characteristic, complemented
11C23×#508C211yesmaximal, characteristic, complemented
12#41×D23—22no—
13C23×#135C2212yescomplemented
14#10625121—22no—
15#10625090C2212yescomplemented
16Q16×D23C2212yes—
17#10625087—22no—
18SD16×D23—22no—
19#10625083C2212yescomplemented
20D8×D23C2212yes—
21#10625075C2212yescomplemented
22C2×C8×D23C2211yescharacteristic
23C23×Q32C2211yescharacteristic, complemented
24C23×SD32C2212yescomplemented
25C23×D16C2211yescharacteristic, complemented
26C2×C16×C23C2211yescharacteristic, complemented
27#10625011C2212yescomplemented
28#10625010C2212yescomplemented
29#10625009C2212yescomplemented
30#10625008C2212yescomplemented
31C2×C23⋊C16C2211yescharacteristic, complemented
32Dic184C2211yescharacteristic, complemented
33#10624982C2212yescomplemented
34D368C2211yescharacteristic, complemented
35C16×D23C2211yescharacteristic
36C16×D23C2211yescharacteristic
37C16×D23C2211yescharacteristic, complemented
38C16×D23C2211yescharacteristic, complemented
39#41×C23—22no—
40#67292—22no—
41Q8×D23—22no—
42#67290—22no—
43#67290—42no—
44D4×D23—22no—
45D4×D23—42no—
46#67288—22no—
47#67288—42no—
48C2×C4×D23—42no—
49C2×C4×D23D411yescharacteristic
50Q16×C23C2312yes—
51SD16×C23—22no—
52D8×C23C2312yes—
53C2×C8×C23C2311yescharacteristic
54#67275—22no—
55#67274—22no—
56#67273—22no—
57#67272—22no—
58C2×C23⋊C8D411yescharacteristic
59Dic92C2312yes—
60D184C2312yes—
61#67263—22no—
62C8×D23D411yescharacteristic
63C8×D23D411yescharacteristic
64C8×D23C2311yescharacteristic
65C8×D23C2311yescharacteristic
66C16×C23C2311yescharacteristic
67C16×C23C2311yescharacteristic
68C23⋊C16C2311yescharacteristic
69C23⋊C16C2311yescharacteristic
70C22×D23—42no—
71C22×D23D811yescharacteristic
72Q8×C23—22no—
73D4×C23—22no—
74D4×C23—42no—
75C2×C4×C23—42no—
76C2×C4×C23C2×D411yescharacteristic
77C22⋊D23—42no—
78C22⋊D23—42no—
79C22⋊D23—42no—
80C2×Dic23—42no—
81C2×Dic23D811yescharacteristic
82D92—22no—
83D92—42no—
84C4×D23—42no—
85C4×D23—42no—
86C4×D23—42no—
87C4×D23D811yescharacteristic
88C4×D23D811yescharacteristic
89C4×D23C2×D411yescharacteristic
90C4×D23C2×D411yescharacteristic
91Dic46—22no—
92Dic46—42no—
93C8×C23C2×D411yescharacteristic
94C8×C23C2411yescharacteristic
95C23⋊C8C2×D411yescharacteristic
96C23⋊C8C2×D411yescharacteristic
97C2×#508—231nomaximal, Sylow
98C22×C23—42no—
99C22×C23C2×D811yescharacteristic
100C2×D23—22no—
101C2×D23—42no—
102C2×D23—84no—
103C2×D23C2×D811yescharacteristic
104C2×D23C2×D811yescharacteristic
105C4×C23—42no—
106C4×C23C2×D811yescharacteristic
107C4×C23C22×D411yescharacteristic
108Dic23—42no—
109Dic23C2×D811yescharacteristic
110Dic23C2×D811yescharacteristic
111C2×#135—232no—
112#508—232no—
113#508—231no—
114#508—232no—
115#508—231no—
116#508—231no—
117#508D2311yescharacteristic, direct product factor
118C2×Q32—231no—
119C2×SD32—232no—
120C2×D16—231no—
121C22×C16—231no—
122C2×C23—21no—
123C2×C23—82no—
124C2×C23C22×D811yescharacteristic
125D23—21no—
126D23—82no—
127D23#50812yesdirect product factor
128C2×#41—462no—
129#135—232no—
130#135—232no—
131#135—232no—
132#135—462no—
133#135—462no—
134#135C2×D2312yescomplemented
135C2×Q16—232no—
136C2×SD16—462no—
137C2×D8—232no—
138C22×C8—231no—
139Q32—232no—
140Q32—231no—
141Q32C2×D2311yescharacteristic, complemented
142SD32—232no—
143SD32—232no—
144SD32—232no—
145SD32C2×D2312yescomplemented
146D16—232no—
147D16—231no—
148D16C2×D2311yescharacteristic, complemented
149C2×C16—231no—
150C2×C16—231no—
151C2×C16—231no—
152C2×C16—231no—
153C2×C16—231no—
154C2×C16C2×D2311yescharacteristic, complemented
155C23C2×#50811yescharacteristic, complemented, Sylow
156#41—22no—
157#41—462no—
158#41—462no—
159#41—462no—
160#41—922no—
161#41—922no—
162C2×Q8—462no—
163C2×D4—462no—
164C2×D4—922no—
165C22×C4—231no—
166C22×C4—922no—
167Q16—232no—
168Q16—462no—
169Q16C22×D2312yes—
170SD16—22no—
171SD16—462no—
172SD16—462no—
173SD16—462no—
174D8—232no—
175D8—462no—
176D8C22×D2312yes—
177C2×C8—231no—
178C2×C8—231no—
179C2×C8—231no—
180C2×C8—231no—
181C2×C8—231no—
182C2×C8C22×D2311yescharacteristic
183C16—231no—
184C16—231no—
185C16C22×D2311yescharacteristic
186C16C22×D2311yescharacteristic
187C23—231no—
188C23—922no—
189Q8—22no—
190Q8—462no—
191Q8—922no—
192D4—22no—
193D4—42no—
194D4—462no—
195D4—922no—
196D4—922no—
197D4—922no—
198D4—922no—
199C2×C4—42no—
200C2×C4—231no—
201C2×C4—231no—
202C2×C4—231no—
203C2×C4—231no—
204C2×C4—231no—
205C2×C4—922no—
206C2×C4—922no—
207C2×C4—922no—
208C2×C4—922no—
209C2×C4D4×D2311yescharacteristic
210C8—231no—
211C8—231no—
212C8D4×D2311yescharacteristic
213C8C23×D2311yescharacteristic
214C22—42no—
215C22—231no—
216C22—231no—
217C22—462no—
218C22—922no—
219C22—1844no—
220C22D8×D2311yescharacteristic
221C4—42no—
222C4—231no—
223C4—231no—
224C4—922no—
225C4D8×D2311yescharacteristic
226C4C2×D4×D2311yescharacteristic
227C2—21no—
228C2—82no—
229C2—232no—
230C2—461no—
231C2—1842no—
232C2C2×D8×D2311yescharacteristic
subgroupG/H
centerC4D8×D23
commutator, cosocleC8×C23C24
FrattiniC8C23×D23
FittingC23×#508C2
socleC2×C23C22×D8
hypercenter#508D23
hypocenterC23C2×#508
LCSG ▷ C8×C23 ▷ C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #135×D23C2semidirect
2. #13092850C2semidirect
3. #13092849C2semidirect
4. #13092848C2semidirect
5. #13092847C2semidirect
6. #13092846C2semidirect
7. Q32×D23C2semidirect
8. SD32×D23C2semidirect
9. D16×D23C2semidirect
10. C2×C16×D23C2semidirect
11. C23×#508C2semidirect
13. C23×#135C22semidirect
15. #10625090C22semidirect
19. #10625083C22semidirect
21. #10625075C22semidirect
23. C23×Q32C22semidirect
24. C23×SD32C22semidirect
25. C23×D16C22semidirect
26. C2×C16×C23C22semidirect
27. #10625011C22semidirect
28. #10625010C22semidirect
29. #10625009C22semidirect
30. #10625008C22semidirect
31. C2×C23⋊C16C22semidirect
32. Dic184C22semidirect
33. #10624982C22semidirect
34. D368C22semidirect
37. C16×D23C22semidirect
38. C16×D23C22semidirect
117. #508D23direct (N × H)
127. D23#508direct (N × H)
134. #135C2×D23semidirect
141. Q32C2×D23semidirect
145. SD32C2×D23semidirect
148. D16C2×D23semidirect
154. C2×C16C2×D23semidirect
155. C23C2×#508semidirect

Maximal subgroups

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

subgroupindex
#135×D232
#130928502
#130928492
#130928482
#130928472
#130928462
Q32×D232
SD32×D232
D16×D232
C2×C16×D232
C23×#5082
C2×#50823