Finite Group Databasebeta

← #15685194

Subgroups — #15685194

Order 2944.

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

normal subgroups only

This group has 476 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
1#13093220C212yesmaximal, complemented
2#13093219C211yesmaximal, characteristic, complemented
3#13092792C211yesmaximal, characteristic, complemented
4#13092687C211yesmaximal, characteristic, complemented
5#13092685C212yesmaximal, complemented
6#13092684C211yesmaximal, characteristic, complemented
7#13092585C212yesmaximal, complemented
8#13092583C212yesmaximal, complemented
9#13092534C211yesmaximal, characteristic, complemented
10#13092532C211yesmaximal, characteristic, complemented
11C23×#576C211yesmaximal, characteristic, complemented
12C23×H(5,𝔽2)C2212yescomplemented
13#10625160C2212yescomplemented
14C23×AGL(1,ℤ/8)—22no—
15C23×AGL(1,ℤ/8)C2212yescomplemented
16C23×#135—21no—
17C23×#135C2211yescharacteristic
18C2×D8×C23—21no—
19C2×D8×C23C2211yescharacteristic, complemented
20C23×#131C2211yescharacteristic, complemented
21#10625121—22no—
22#10625121C2211yescharacteristic
23#10625120—22no—
24#10625120C2212yescomplemented
25#10625120C2212yescomplemented
26#10625119C2212yescomplemented
27#10625111C2212yescomplemented
28#10625103—22no—
29#10625103C2212yescomplemented
30C2×#67272—22no—
31C2×#67272C2212yescomplemented
32#10625090—21no—
33#10625085—22no—
34D8×D23—21no—
35C2×D184C2211yescharacteristic, complemented
36#10625072C2211yescharacteristic, complemented
37#10625019C2211yescharacteristic, complemented
38#10625019C2212yescomplemented
39#10625015C2212yescomplemented
40#10625007C2211yescharacteristic, complemented
41#10625005C2212yescomplemented
42#10625004C2211yescharacteristic, complemented
43#10625001C2211yescharacteristic, complemented
44C8×Dic23C2211yescharacteristic, complemented
45#41×C23—22no—
46#41×C23—22no—
47#41×C23C2312yes—
48#41×C23C2311yescharacteristic
49C2×D4×C23—22no—
50C2×D4×C23—22no—
51C2×D4×C23—42no—
52C2×D4×C23C2312yes—
53C2×C22⋊D23—42no—
54#67292—22no—
55D4×D23—22no—
56D4×D23—42no—
57#67288—22no—
58#67288C2311yescharacteristic
59C2×D92—22no—
60C2×D92C2312yes—
61Q16×C23D411yescharacteristic, complemented
62SD16×C23—22no—
63SD16×C23D412yescomplemented
64D8×C23—22no—
65D8×C23—21no—
66D8×C23—21no—
67D8×C23D411yescharacteristic, complemented
68M16×C23—21no—
69M16×C23C2311yescharacteristic
70C2×C8×C23—21no—
71C2×C8×C23C2311yescharacteristic
72#67275D411yescharacteristic, complemented
73#67274—22no—
74#67274—22no—
75#67274D411yescharacteristic, complemented
76#67273—22no—
77#67273D411yescharacteristic, complemented
78#67272—22no—
79#67272—22no—
80#67272—22no—
81#67272—22no—
82#67272—22no—
83#67272D411yescharacteristic, complemented
84C4×Dic23C2311yescharacteristic
85#67267—22no—
86#67267C2311yescharacteristic
87#67267C2312yes—
88C2×C23⋊C8—22no—
89C2×C23⋊C8C2311yescharacteristic
90D184—21no—
91D184—21no—
92#67262—21no—
93C8×D23—21no—
94C23×C23—22no—
95C23×C23—42no—
96C22×D23—22no—
97C22×D23—42no—
98Q8×C23C2×D411yescharacteristic
99Q8×C23C2×D412yes—
100D4×C23—22no—
101D4×C23—21no—
102D4×C23—22no—
103D4×C23—22no—
104D4×C23—42no—
105D4×C23—42no—
106D4×C23C2×D411yescharacteristic
107D4×C23C2×D412yes—
108C2×C4×C23—22no—
109C2×C4×C23—21no—
110C2×C4×C23—42no—
111C2×C4×C23C2411yescharacteristic
112C22⋊D23—21no—
113C22⋊D23—42no—
114C22⋊D23—84no—
115C2×Dic23—21no—
116D92—22no—
117D92—22no—
118D92—42no—
119D92C2×D411yescharacteristic
120C4×D23—21no—
121C4×D23—42no—
122Dic46C2×D411yescharacteristic
123C8×C23—21no—
124C8×C23C2×D411yescharacteristic
125C8×C23C2×D411yescharacteristic
126C23⋊C8—21no—
127C23⋊C8—22no—
128C23⋊C8C2×D411yescharacteristic
129C23⋊C8C2×D411yescharacteristic
130#3292—231nomaximal, Sylow
131C22×C23—22no—
132C22×C23—21no—
133C22×C23—42no—
134C22×C23—82no—
135C22×C23—162no—
136C22×C23H(5,𝔽2)11yescharacteristic
137C2×D23—21no—
138C2×D23—42no—
139C2×D23—82no—
140C2×D23—162no—
141C4×C23—22no—
142C4×C23—21no—
143C4×C23C22×D411yescharacteristic
144C4×C23C22×D411yescharacteristic
145Dic23—22no—
146Dic23—21no—
147#576—232no—
148#576—231no—
149#576D2311yescharacteristic, complemented
150#493—231no—
151#472—232no—
152#472—231no—
153#472—231no—
154#453—232no—
155#453—232no—
156#443—231no—
157#443—231no—
158C2×C23—21no—
159C2×C23—42no—
160C2×C23—41no—
161C2×C23—82no—
162C2×C23D4211yescharacteristic
163D23—41no—
164D23—82no—
165H(5,𝔽2)—232no—
166H(5,𝔽2)C2×D2312yescomplemented
167AGL(1,ℤ/8)—22no—
168AGL(1,ℤ/8)—232no—
169AGL(1,ℤ/8)—232no—
170AGL(1,ℤ/8)—232no—
171AGL(1,ℤ/8)—462no—
172AGL(1,ℤ/8)—462no—
173AGL(1,ℤ/8)—462no—
174AGL(1,ℤ/8)C2×D2312yescomplemented
175#135—21no—
176#135—231no—
177#135—462no—
178#135—461no—
179#135C2×D2311yescharacteristic
180C2×D8—21no—
181C2×D8—232no—
182C2×D8—231no—
183C2×D8—462no—
184C2×D8—461no—
185C2×D8C2×D2311yescharacteristic, complemented
186#131—232no—
187#131—231no—
188#131C2×D2311yescharacteristic, complemented
189#127—232no—
190#108—231no—
191#108—231no—
192#104—231no—
193#104—232no—
194#104—231no—
195#100—232no—
196#100—232no—
197C4×C8—231no—
198C23#329211yescharacteristic, complemented, Sylow
199#41—22no—
200#41—22no—
201#41—231no—
202#41—462no—
203#41—462no—
204#41C22×D2311yescharacteristic
205#41C22×D2312yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—232no—
210C2×D4—462no—
211C2×D4—462no—
212C2×D4—922no—
213C2×D4—922no—
214C2×D4C22×D2312yes—
215Q16—231no—
216Q16C22⋊D2311yescharacteristic, complemented
217SD16—22no—
218SD16—231no—
219SD16—231no—
220SD16—462no—
221SD16—462no—
222SD16—462no—
223SD16C22⋊D2312yescomplemented
224D8—22no—
225D8—21no—
226D8—21no—
227D8—231no—
228D8—462no—
229D8—462no—
230D8—462no—
231D8—462no—
232D8—462no—
233D8—461no—
234D8—461no—
235D8C22⋊D2311yescharacteristic, complemented
236M16—21no—
237M16—232no—
238M16—231no—
239M16—462no—
240M16—461no—
241M16C22×D2311yescharacteristic
242C2×C8—21no—
243C2×C8—231no—
244C2×C8—462no—
245C2×C8—461no—
246C2×C8C22×D2311yescharacteristic
247C42—231no—
248C23—22no—
249C23—42no—
250C23—462no—
251C23—922no—
252Q8—231no—
253Q8D4×D2311yescharacteristic
254Q8C2×C22⋊D2312yes—
255D4—21no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—231no—
262D4—461no—
263D4—462no—
264D4—462no—
265D4—922no—
266D4—922no—
267D4—1844no—
268D4D4×D2311yescharacteristic
269D4C2×C22⋊D2312yes—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—461no—
274C2×C4—461no—
275C2×C4—922no—
276C2×C4C23×D2311yescharacteristic
277C8—21no—
278C8—231no—
279C8—231no—
280C8—462no—
281C8—461no—
282C8C2×C22⋊D2311yescharacteristic
283C8C2×C22⋊D2311yescharacteristic
284C22—21no—
285C22—22no—
286C22—42no—
287C22—82no—
288C22—162no—
289C22—461no—
290C22—922no—
291C22—3682no—
292C22—1842no—
293C22#1062515511yescharacteristic
294C4—21no—
295C4—22no—
296C4—462no—
297C4—461no—
298C4C2×D4×D2311yescharacteristic
299C4C22×C22⋊D2311yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—921no—
305C2—1842no—
306C2D4×C22⋊D2311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D23
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#576C2
socleC2×C23D42
hypercenter#576D23
hypocenterC23#3292
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #13093220C2semidirect
2. #13093219C2semidirect
3. #13092792C2semidirect
4. #13092687C2semidirect
5. #13092685C2semidirect
6. #13092684C2semidirect
7. #13092585C2semidirect
8. #13092583C2semidirect
9. #13092534C2semidirect
10. #13092532C2semidirect
11. C23×#576C2semidirect
12. C23×H(5,𝔽2)C22semidirect
13. #10625160C22semidirect
15. C23×AGL(1,ℤ/8)C22semidirect
19. C2×D8×C23C22semidirect
20. C23×#131C22semidirect
24. #10625120C22semidirect
25. #10625120C22semidirect
26. #10625119C22semidirect
27. #10625111C22semidirect
29. #10625103C22semidirect
31. C2×#67272C22semidirect
35. C2×D184C22semidirect
36. #10625072C22semidirect
37. #10625019C22semidirect
38. #10625019C22semidirect
39. #10625015C22semidirect
40. #10625007C22semidirect
41. #10625005C22semidirect
42. #10625004C22semidirect
43. #10625001C22semidirect
44. C8×Dic23C22semidirect
61. Q16×C23D4semidirect
63. SD16×C23D4semidirect
67. D8×C23D4semidirect
72. #67275D4semidirect
75. #67274D4semidirect
77. #67273D4semidirect
83. #67272D4semidirect
149. #576D23semidirect
166. H(5,𝔽2)C2×D23semidirect
174. AGL(1,ℤ/8)C2×D23semidirect
185. C2×D8C2×D23semidirect
188. #131C2×D23semidirect
198. C23#3292semidirect
216. Q16C22⋊D23semidirect
223. SD16C22⋊D23semidirect
235. D8C22⋊D23semidirect

Maximal subgroups

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

subgroupindex
#130932202
#130932192
#130927922
#130926872
#130926852
#130926842
#130925852
#130925832
#130925342
#130925322
C23×#5762
#329223