Finite Group Databasebeta

← #15685195

Subgroups — #15685195

Order 2944.

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

normal subgroups only

This group has 476 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#13093222C211yesmaximal, characteristic, complemented
2#13093221C212yesmaximal, complemented
3#13093218C211yesmaximal, characteristic, complemented
4#13092792C211yesmaximal, characteristic, complemented
5#13092688C211yesmaximal, characteristic, complemented
6#13092686C212yesmaximal, complemented
7#13092587C212yesmaximal, complemented
8#13092583C212yesmaximal, complemented
9#13092533C211yesmaximal, characteristic, complemented
10#13092532C211yesmaximal, characteristic, complemented
11C23×#472C211yesmaximal, characteristic, complemented
12#10625160C2212yescomplemented
13#10625155C2212yescomplemented
14C23×AGL(1,ℤ/8)C2212yescomplemented
15C2×D8×C23C2211yescharacteristic, complemented
16C23×#131C2211yescharacteristic, complemented
17#10625121C2211yescharacteristic
18#10625111C2212yescomplemented
19#10625103C2212yescomplemented
20C2×#67272C2212yescomplemented
21#10625087—22no—
22#10625085—22no—
23#10625083—21no—
24#10625082—22no—
25#10625082—22no—
26D8×D23—22no—
27D8×D23—21no—
28#10625079—22no—
29#10625079C2212yescomplemented
30#10625079C2212yescomplemented
31#10625078C2212yescomplemented
32#10625075—21no—
33#10625075C2211yescharacteristic
34C2×D184—21no—
35C2×D184C2211yescharacteristic, complemented
36#10625072C2211yescharacteristic, complemented
37C23×#108C2211yescharacteristic, complemented
38C23×#100C2212yescomplemented
39#10625019C2211yescharacteristic, complemented
40#10625007C2211yescharacteristic, complemented
41#10625007C2212yescomplemented
42#10625005C2212yescomplemented
43#10625004C2211yescharacteristic, complemented
44C8×Dic23C2211yescharacteristic, complemented
45#41×C23C2311yescharacteristic
46C2×D4×C23C2312yes—
47C2×C22⋊D23—42no—
48C2×C22⋊D23—42no—
49#67292—22no—
50#67290—22no—
51#67290—22no—
52D4×D23—22no—
53D4×D23—22no—
54D4×D23—22no—
55D4×D23—42no—
56#67288—22no—
57#67288C2311yescharacteristic
58#67288C2312yes—
59C2×D92—22no—
60C2×D92C2312yes—
61SD16×C23—22no—
62D8×C23—22no—
63D8×C23—21no—
64D8×C23—21no—
65M16×C23—21no—
66M16×C23C2311yescharacteristic
67M16×C23C2312yes—
68C2×C8×C23—21no—
69C2×C8×C23C2311yescharacteristic
70#67275D411yescharacteristic, complemented
71#67274D411yescharacteristic, complemented
72#67273—22no—
73#67273D411yescharacteristic, complemented
74#67272—22no—
75#67272—22no—
76#67272—22no—
77#67272D411yescharacteristic, complemented
78C4×Dic23C2311yescharacteristic
79#67267C2311yescharacteristic
80C2×C23⋊C8C2311yescharacteristic
81Dic92D411yescharacteristic, complemented
82D184—22no—
83D184—22no—
84D184—21no—
85D184—21no—
86D184D411yescharacteristic, complemented
87#67263—22no—
88#67263—22no—
89#67263D412yescomplemented
90#67262—21no—
91#67262—22no—
92C8×D23—21no—
93C8×D23—22no—
94C23×C23—22no—
95C22×D23—22no—
96C22×D23—42no—
97C22×D23—42no—
98Q8×C23C2×D411yescharacteristic
99D4×C23—21no—
100D4×C23—22no—
101D4×C23—22no—
102D4×C23C2×D411yescharacteristic
103C2×C4×C23—21no—
104C2×C4×C23C2411yescharacteristic
105C22⋊D23—22no—
106C22⋊D23—21no—
107C22⋊D23—42no—
108C22⋊D23—42no—
109C22⋊D23—42no—
110C22⋊D23—84no—
111C2×Dic23—21no—
112C2×Dic23—42no—
113D92—22no—
114D92—22no—
115D92—42no—
116D92C2×D412yes—
117D92C2×D411yescharacteristic
118C4×D23—21no—
119C4×D23—22no—
120C4×D23—42no—
121Dic46C2×D411yescharacteristic
122Dic46C2×D412yes—
123C8×C23—22no—
124C8×C23—21no—
125C8×C23C2×D411yescharacteristic
126C8×C23C2×D411yescharacteristic
127C23⋊C8—21no—
128C23⋊C8C2×D411yescharacteristic
129C23⋊C8C2×D411yescharacteristic
130#3292—231nomaximal, Sylow
131C22×C23—21no—
132C22×C23—42no—
133C22×C23—82no—
134C22×C23H(5,𝔽2)11yescharacteristic
135C2×D23—22no—
136C2×D23—21no—
137C2×D23—42no—
138C2×D23—82no—
139C2×D23—162no—
140C2×D23—162no—
141C4×C23—21no—
142C4×C23C22×D411yescharacteristic
143C4×C23C22×D411yescharacteristic
144Dic23—22no—
145Dic23—22no—
146Dic23—21no—
147#576—232no—
148#576—231no—
149#576—231no—
150#493—231no—
151#472—231no—
152#472—232no—
153#472D2311yescharacteristic, complemented
154#453—232no—
155#453—232no—
156#443—231no—
157#443—231no—
158C2×C23—21no—
159C2×C23—41no—
160C2×C23—82no—
161C2×C23D4211yescharacteristic
162D23—42no—
163D23—41no—
164D23—82no—
165H(5,𝔽2)—232no—
166H(5,𝔽2)—232no—
167AGL(1,ℤ/8)—232no—
168AGL(1,ℤ/8)—232no—
169AGL(1,ℤ/8)—232no—
170AGL(1,ℤ/8)—462no—
171AGL(1,ℤ/8)—462no—
172AGL(1,ℤ/8)—462no—
173AGL(1,ℤ/8)—462no—
174AGL(1,ℤ/8)C2×D2312yescomplemented
175#135—231no—
176#135—231no—
177#135—461no—
178#135—461no—
179#135—462no—
180C2×D8—232no—
181C2×D8—231no—
182C2×D8—461no—
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#108C2×D2311yescharacteristic, complemented
192#104—231no—
193#104—232no—
194#104—231no—
195#100—232no—
196#100C2×D2312yescomplemented
197C4×C8—231no—
198C23#329211yescharacteristic, complemented, Sylow
199#41—232no—
200#41—231no—
201#41—462no—
202#41—462no—
203#41—462no—
204#41—462no—
205#41C22×D2311yescharacteristic
206C2×D4—232no—
207C2×D4—462no—
208C2×D4—462no—
209C2×D4—462no—
210C2×D4—462no—
211C2×D4—922no—
212C2×D4—922no—
213C2×D4—922no—
214C2×D4C22×D2312yes—
215Q16—231no—
216Q16—231no—
217SD16—22no—
218SD16—231no—
219SD16—231no—
220SD16—232no—
221SD16—462no—
222SD16—462no—
223SD16—462no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—231no—
228D8—231no—
229D8—462no—
230D8—462no—
231D8—462no—
232D8—462no—
233D8—462no—
234D8—461no—
235D8—461no—
236M16—21no—
237M16—231no—
238M16—462no—
239M16—461no—
240M16C22×D2312yes—
241M16C22×D2311yescharacteristic
242C2×C8—21no—
243C2×C8—231no—
244C2×C8—462no—
245C2×C8—461no—
246C2×C8C22×D2311yescharacteristic
247C42—231no—
248C23—22no—
249C23—462no—
250C23—922no—
251C23—922no—
252Q8—231no—
253Q8—232no—
254Q8D4×D2311yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—231no—
259D4—232no—
260D4—462no—
261D4—461no—
262D4—462no—
263D4—462no—
264D4—922no—
265D4—922no—
266D4—922no—
267D4—922no—
268D4—1844no—
269D4D4×D2311yescharacteristic
270C2×C4—21no—
271C2×C4—461no—
272C2×C4—462no—
273C2×C4—461no—
274C2×C4—922no—
275C2×C4—922no—
276C2×C4C23×D2311yescharacteristic
277C8—21no—
278C8—22no—
279C8—231no—
280C8—231no—
281C8—461no—
282C8D4×D2311yescharacteristic
283C8D4×D2311yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—462no—
288C22—461no—
289C22—922no—
290C22—3682no—
291C22—3682no—
292C22—1842no—
293C22#1062516011yescharacteristic
294C4—21no—
295C4—462no—
296C4—462no—
297C4—461no—
298C4C2×D4×D2311yescharacteristic
299C4C2×D4×D2311yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—922no—
304C2—921no—
305C2—1842no—
306C2#1309305111yescharacteristic
subgroupG/H
centerC2#13093051
commutator, cosocleC2×C4×C23C24
FrattiniC2×C4C23×D23
FittingC23×#472C2
socleC2×C23D42
hypercenter#472D23
hypocenterC23#3292
LCSG ▷ C2×C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #13093222C2semidirect
2. #13093221C2semidirect
3. #13093218C2semidirect
4. #13092792C2semidirect
5. #13092688C2semidirect
6. #13092686C2semidirect
7. #13092587C2semidirect
8. #13092583C2semidirect
9. #13092533C2semidirect
10. #13092532C2semidirect
11. C23×#472C2semidirect
12. #10625160C22semidirect
13. #10625155C22semidirect
14. C23×AGL(1,ℤ/8)C22semidirect
15. C2×D8×C23C22semidirect
16. C23×#131C22semidirect
18. #10625111C22semidirect
19. #10625103C22semidirect
20. C2×#67272C22semidirect
29. #10625079C22semidirect
30. #10625079C22semidirect
31. #10625078C22semidirect
35. C2×D184C22semidirect
36. #10625072C22semidirect
37. C23×#108C22semidirect
38. C23×#100C22semidirect
39. #10625019C22semidirect
40. #10625007C22semidirect
41. #10625007C22semidirect
42. #10625005C22semidirect
43. #10625004C22semidirect
44. C8×Dic23C22semidirect
70. #67275D4semidirect
71. #67274D4semidirect
73. #67273D4semidirect
77. #67272D4semidirect
81. Dic92D4semidirect
86. D184D4semidirect
89. #67263D4semidirect
153. #472D23semidirect
174. AGL(1,ℤ/8)C2×D23semidirect
185. C2×D8C2×D23semidirect
188. #131C2×D23semidirect
191. #108C2×D23semidirect
196. #100C2×D23semidirect
198. C23#3292semidirect

Maximal subgroups

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

subgroupindex
#130932222
#130932212
#130932182
#130927922
#130926882
#130926862
#130925872
#130925832
#130925332
#130925322
C23×#4722
#329223