Finite Group Databasebeta

← #15685165

Subgroups — #15685165

Order 2944.

464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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#13093225C212yesmaximal, complemented
2#13093221C212yesmaximal, complemented
3#13092807C211yesmaximal, characteristic, complemented
4#13092685C212yesmaximal, complemented
5#13092683C212yesmaximal, complemented
6#13092594C211yesmaximal, characteristic, complemented
7#13092586C212yesmaximal, complemented
8#13092536C212yesmaximal, complemented
9C23×#454C211yesmaximal, characteristic, complemented
10C23×2−1+4C2211yescharacteristic, complemented
11#10625160C2212yescomplemented
12#10625155C2211yescharacteristic
13C23×AGL(1,ℤ/8)C2212yescomplemented
14C23×#124C2211yescharacteristic, complemented
15#10625121—22no—
16#10625121C2212yes—
17#10625120—22no—
18#10625120C2212yescomplemented
19#10625119C2212yescomplemented
20#10625113—22no—
21#10625113C2212yescomplemented
22C2×#67274—22no—
23C2×#67274C2212yescomplemented
24C2×#67272C2212yescomplemented
25#10625087—22no—
26#10625085—22no—
27#10625082—22no—
28D8×D23—22no—
29#10625079C2212yes—
30#10625078C2212yescomplemented
31#10625045C2211yescharacteristic, complemented
32C23×#104C2212yescomplemented
33C23×#101C2211yescharacteristic, complemented
34#10625015C2211yescharacteristic, complemented
35#10625005C2212yescomplemented
36#10625004C2212yescomplemented
37#10624987C2212yescomplemented
38#10624985C2211yescharacteristic, complemented
39#41×C23—22no—
40#41×C23—41no—
41#41×C23C2312yes—
42C2×Q8×C23—22no—
43C2×Q8×C23C2311yescharacteristic
44C2×D4×C23C2311yescharacteristic
45C2×C22⋊D23—41no—
46#67292—22no—
47#67290—22no—
48D4×D23—22no—
49D4×D23—22no—
50D4×D23—42no—
51#67288—22no—
52#67288C2312yes—
53C2×D92—22no—
54C2×D92—41no—
55C2×D92C2312yes—
56SD16×C23—22no—
57D8×C23—22no—
58M16×C23C2312yes—
59C22⋊C4×C23—41no—
60C42×C23C2311yescharacteristic
61#67275—22no—
62#67275D412yescomplemented
63#67274—22no—
64#67274—22no—
65#67274—22no—
66#67274—22no—
67#67274D412yescomplemented
68#67273D412yescomplemented
69#67272—22no—
70#67272—22no—
71#67272—22no—
72#67272D412yescomplemented
73#67267—22no—
74#67267C2312yes—
75C2×C23⋊C8—22no—
76C2×C23⋊C8C2312yes—
77D184—22no—
78#67263—22no—
79#67262—22no—
80C8×D23—22no—
81C23×C23—21no—
82C22×D23—22no—
83C22×D23—41no—
84C22×D23—42no—
85Q8×C23—22no—
86Q8×C23—41no—
87Q8×C23C2×D412yes—
88D4×C23—22no—
89D4×C23—22no—
90D4×C23—41no—
91D4×C23C2×D412yes—
92C2×C4×C23—22no—
93C2×C4×C23—21no—
94C2×C4×C23—21no—
95C2×C4×C23—42no—
96C2×C4×C23C2411yescharacteristic
97C22⋊D23—22no—
98C22⋊D23—42no—
99C22⋊D23—42no—
100C2×Dic23—41no—
101D92—22no—
102D92—22no—
103D92—42no—
104D92—82no—
105D92C2×D412yes—
106C4×D23—22no—
107C4×D23—42no—
108Dic46C2×D412yes—
109C8×C23—22no—
110C23⋊C8—22no—
111C23⋊C8C2×D412yes—
112C23⋊C8C2×D412yes—
113#3288—231nomaximal, Sylow
114C22×C23—22no—
115C22×C23—41no—
116C22×C23—81no—
117C22×C23H(5,𝔽2)11yescharacteristic
118C2×D23—22no—
119C2×D23—42no—
120C2×D23—82no—
121C2×D23—161no—
122C2×D23—162no—
123C4×C23—22no—
124C4×C23—41no—
125C4×C23—41no—
126C4×C23C22×D412yes—
127Dic23—22no—
128#577—232no—
129#576—232no—
130#496—231no—
131#472—232no—
132#471—232no—
133#455—231no—
134#454D2311yescharacteristic, complemented
135#453—232no—
136#444—232no—
137C2×C23—21no—
138C2×C23—42no—
139C2×C23—81no—
140C2×C23D4211yescharacteristic
141D23—42no—
142D23—82no—
1432−1+4C2×D2311yescharacteristic, complemented
144H(5,𝔽2)—232no—
145H(5,𝔽2)—231no—
146#137—232no—
147#137—462no—
148AGL(1,ℤ/8)—232no—
149AGL(1,ℤ/8)—232no—
150AGL(1,ℤ/8)—462no—
151AGL(1,ℤ/8)—462no—
152AGL(1,ℤ/8)—462no—
153AGL(1,ℤ/8)C2×D2312yescomplemented
154#135—232no—
155#135—462no—
156#135—462no—
157C2×SD16—232no—
158C2×SD16—462no—
159C2×D8—232no—
160C2×D8—462no—
161#131—232no—
162#131—232no—
163#127—231no—
164#124C2×D2311yescharacteristic, complemented
165#108—232no—
166#104—232no—
167#104C2×D2312yescomplemented
168#101C2×D2311yescharacteristic, complemented
169#100—231no—
170#100—232no—
171#97—231no—
172C23#328811yescharacteristic, complemented, Sylow
173#41—22no—
174#41—41no—
175#41—232no—
176#41—462no—
177#41—462no—
178#41—462no—
179#41C22×D2312yes—
180C2×Q8—22no—
181C2×Q8C22×D2311yescharacteristic
182C2×D4—232no—
183C2×D4—462no—
184C2×D4—462no—
185C2×D4—462no—
186C2×D4—921no—
187C2×D4—922no—
188C2×D4—921no—
189C2×D4C22×D2311yescharacteristic
190Q16—232no—
191Q16—462no—
192SD16—22no—
193SD16—232no—
194SD16—232no—
195SD16—462no—
196SD16—462no—
197SD16—462no—
198SD16—462no—
199SD16—462no—
200D8—22no—
201D8—232no—
202D8—462no—
203D8—462no—
204D8—462no—
205D8—462no—
206M16—232no—
207M16—462no—
208M16—462no—
209M16C22×D2312yes—
210C2×C8—232no—
211C2×C8—462no—
212C2×C8—462no—
213C22⋊C4—41no—
214C42C22×D2311yescharacteristic
215C23—21no—
216C23—462no—
217C23—922no—
218C23—921no—
219Q8—22no—
220Q8—41no—
221Q8—232no—
222Q8D4×D2312yes—
223D4—22no—
224D4—22no—
225D4—41no—
226D4—232no—
227D4—462no—
228D4—462no—
229D4—462no—
230D4—922no—
231D4—922no—
232D4—922no—
233D4—1842no—
234D4D4×D2312yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—462no—
240C2×C4—922no—
241C2×C4—921no—
242C2×C4C23×D2311yescharacteristic
243C8—22no—
244C8—232no—
245C8—232no—
246C8—462no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—462no—
251C22—922no—
252C22—3682no—
253C22—3681no—
254C22—1842no—
255C22#1062515511yescharacteristic
256C4—22no—
257C4—41no—
258C4—41no—
259C4—462no—
260C4C2×D4×D2312yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—922no—
265C2—1842no—
266C2#1309304911yescharacteristic
subgroupG/H
centerC2#13093049
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#454C2
socleC2×C23D42
hypercenter#454D23
hypocenterC23#3288
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #13093225C2semidirect
2. #13093221C2semidirect
3. #13092807C2semidirect
4. #13092685C2semidirect
5. #13092683C2semidirect
6. #13092594C2semidirect
7. #13092586C2semidirect
8. #13092536C2semidirect
9. C23×#454C2semidirect
10. C23×2−1+4C22semidirect
11. #10625160C22semidirect
13. C23×AGL(1,ℤ/8)C22semidirect
14. C23×#124C22semidirect
18. #10625120C22semidirect
19. #10625119C22semidirect
21. #10625113C22semidirect
23. C2×#67274C22semidirect
24. C2×#67272C22semidirect
30. #10625078C22semidirect
31. #10625045C22semidirect
32. C23×#104C22semidirect
33. C23×#101C22semidirect
34. #10625015C22semidirect
35. #10625005C22semidirect
36. #10625004C22semidirect
37. #10624987C22semidirect
38. #10624985C22semidirect
62. #67275D4semidirect
67. #67274D4semidirect
68. #67273D4semidirect
72. #67272D4semidirect
134. #454D23semidirect
143. 2−1+4C2×D23semidirect
153. AGL(1,ℤ/8)C2×D23semidirect
164. #124C2×D23semidirect
167. #104C2×D23semidirect
168. #101C2×D23semidirect
172. C23#3288semidirect

Maximal subgroups

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

subgroupindex
#130932252
#130932212
#130928072
#130926852
#130926832
#130925942
#130925862
#130925362
C23×#4542
#328823