Finite Group Databasebeta

← #49133058

Subgroups — #49133058

Order 36224.

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#31827268C212yesmaximal, complemented
2#31827267C211yesmaximal, characteristic, complemented
3#31826840C211yesmaximal, characteristic, complemented
4#31826735C211yesmaximal, characteristic, complemented
5#31826733C212yesmaximal, complemented
6#31826732C211yesmaximal, characteristic, complemented
7#31826633C212yesmaximal, complemented
8#31826631C212yesmaximal, complemented
9#31826582C211yesmaximal, characteristic, complemented
10#31826580C211yesmaximal, characteristic, complemented
11#576×C283C211yesmaximal, characteristic, complemented
12#22583656C2212yescomplemented
13#22583637—22no—
14#22583636—22no—
15#22583636C2212yescomplemented
16#22583635—22no—
17#22583635C2212yescomplemented
18#22583635C2212yescomplemented
19#22583633—21no—
20#22583630—22no—
21#22583630C2211yescharacteristic
22C2×D2264C2211yescharacteristic, complemented
23C2×#17985377—22no—
24C2×#17985377C2212yescomplemented
25D8×D283—21no—
26#22583617C2212yescomplemented
27#22583616C2211yescharacteristic, complemented
28#22583606C2212yescomplemented
29#22583546C2211yescharacteristic, complemented
30#22583545C2211yescharacteristic, complemented
31#22583537C2211yescharacteristic, complemented
32#22583537C2212yescomplemented
33#22583536C2211yescharacteristic, complemented
34#22583527C2212yescomplemented
35#22583526C2212yescomplemented
36C8×Dic283C2211yescharacteristic, complemented
37H(5,𝔽2)×C283C2212yescomplemented
38AGL(1,ℤ/8)×C283—22no—
39AGL(1,ℤ/8)×C283C2212yescomplemented
40#135×C283—21no—
41#135×C283C2211yescharacteristic
42C2×D8×C283—21no—
43C2×D8×C283C2211yescharacteristic, complemented
44#131×C283C2211yescharacteristic, complemented
45#17985393—22no—
46#17985392—22no—
47#17985392C2311yescharacteristic
48C2×D1132—22no—
49C2×D1132C2312yes—
50C2×C22⋊D283—42no—
51D4×D283—22no—
52D4×D283—42no—
53#17985382D411yescharacteristic, complemented
54#17985380—22no—
55#17985380D411yescharacteristic, complemented
56#17985379—22no—
57#17985379—22no—
58#17985379D411yescharacteristic, complemented
59D2264—21no—
60D2264—21no—
61#17985377—22no—
62#17985377—22no—
63#17985377—22no—
64#17985377—22no—
65#17985377—22no—
66#17985377D411yescharacteristic, complemented
67#17985376—22no—
68#17985376C2311yescharacteristic
69#17985376C2312yes—
70#17985375—21no—
71C2×C283⋊C8—22no—
72C2×C283⋊C8C2311yescharacteristic
73C8×D283—21no—
74C4×Dic283C2311yescharacteristic
75#41×C283—22no—
76#41×C283—22no—
77#41×C283C2312yes—
78#41×C283C2311yescharacteristic
79C2×D4×C283—22no—
80C2×D4×C283—22no—
81C2×D4×C283—42no—
82C2×D4×C283C2312yes—
83Q16×C283D411yescharacteristic, complemented
84SD16×C283—22no—
85SD16×C283D412yescomplemented
86D8×C283—22no—
87D8×C283—21no—
88D8×C283—21no—
89D8×C283D411yescharacteristic, complemented
90M16×C283—21no—
91M16×C283C2311yescharacteristic
92C2×C8×C283—21no—
93C2×C8×C283C2311yescharacteristic
94C22×D283—22no—
95C22×D283—42no—
96Dic566C2×D411yescharacteristic
97D1132—22no—
98D1132—22no—
99D1132—42no—
100D1132C2×D411yescharacteristic
101C22⋊D283—21no—
102C22⋊D283—42no—
103C22⋊D283—84no—
104C2×Dic283—21no—
105C4×D283—21no—
106C4×D283—42no—
107C283⋊C8—21no—
108C283⋊C8—22no—
109C283⋊C8C2×D411yescharacteristic
110C283⋊C8C2×D411yescharacteristic
111C23×C283—22no—
112C23×C283—42no—
113Q8×C283C2×D411yescharacteristic
114Q8×C283C2×D412yes—
115D4×C283—21no—
116D4×C283—22no—
117D4×C283—22no—
118D4×C283—22no—
119D4×C283—42no—
120D4×C283—42no—
121D4×C283C2×D411yescharacteristic
122D4×C283C2×D412yes—
123C2×C4×C283—21no—
124C2×C4×C283—22no—
125C2×C4×C283—42no—
126C2×C4×C283C2411yescharacteristic
127C8×C283—21no—
128C8×C283C2×D411yescharacteristic
129C8×C283C2×D411yescharacteristic
130C22×C283—21no—
131C22×C283—22no—
132C22×C283—42no—
133C22×C283—82no—
134C22×C283—162no—
135C22×C283H(5,𝔽2)11yescharacteristic
136C2×D283—21no—
137C2×D283—42no—
138C2×D283—82no—
139C2×D283—162no—
140C4×C283—21no—
141C4×C283—22no—
142C4×C283C22×D411yescharacteristic
143C4×C283C22×D411yescharacteristic
144Dic283—22no—
145Dic283—21no—
146C2×C283—21no—
147C2×C283—42no—
148C2×C283—41no—
149C2×C283—82no—
150C2×C283D4211yescharacteristic
151D283—41no—
152D283—82no—
153C283#329211yescharacteristic, complemented, Sylow
154#3292—2831nomaximal, Sylow
155#576—2832no—
156#576—2831no—
157#576D28311yescharacteristic, complemented
158#493—2831no—
159#472—2832no—
160#472—2831no—
161#472—2831no—
162#453—2832no—
163#453—2832no—
164#443—2831no—
165#443—2831no—
166H(5,𝔽2)—2832no—
167H(5,𝔽2)C2×D28312yescomplemented
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—2832no—
170AGL(1,ℤ/8)—2832no—
171AGL(1,ℤ/8)—2832no—
172AGL(1,ℤ/8)—5662no—
173AGL(1,ℤ/8)—5662no—
174AGL(1,ℤ/8)—5662no—
175AGL(1,ℤ/8)C2×D28312yescomplemented
176#135—21no—
177#135—2831no—
178#135—5662no—
179#135—5661no—
180#135C2×D28311yescharacteristic
181C2×D8—21no—
182C2×D8—2832no—
183C2×D8—2831no—
184C2×D8—5662no—
185C2×D8—5661no—
186C2×D8C2×D28311yescharacteristic, complemented
187#131—2832no—
188#131—2831no—
189#131C2×D28311yescharacteristic, complemented
190#127—2832no—
191#108—2831no—
192#108—2831no—
193#104—2831no—
194#104—2832no—
195#104—2831no—
196#100—2832no—
197#100—2832no—
198C4×C8—2831no—
199#41—22no—
200#41—22no—
201#41—2831no—
202#41—5662no—
203#41—5662no—
204#41C22×D28311yescharacteristic
205#41C22×D28312yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—2832no—
210C2×D4—5662no—
211C2×D4—5662no—
212C2×D4—11322no—
213C2×D4—11322no—
214C2×D4C22×D28312yes—
215Q16—2831no—
216Q16C22⋊D28311yescharacteristic, complemented
217SD16—22no—
218SD16—2831no—
219SD16—2831no—
220SD16—5662no—
221SD16—5662no—
222SD16—5662no—
223SD16C22⋊D28312yescomplemented
224D8—22no—
225D8—21no—
226D8—21no—
227D8—2831no—
228D8—5662no—
229D8—5662no—
230D8—5662no—
231D8—5662no—
232D8—5662no—
233D8—5661no—
234D8—5661no—
235D8C22⋊D28311yescharacteristic, complemented
236M16—21no—
237M16—2832no—
238M16—2831no—
239M16—5662no—
240M16—5661no—
241M16C22×D28311yescharacteristic
242C2×C8—21no—
243C2×C8—2831no—
244C2×C8—5662no—
245C2×C8—5661no—
246C2×C8C22×D28311yescharacteristic
247C42—2831no—
248C23—22no—
249C23—42no—
250C23—5662no—
251C23—11322no—
252Q8—2831no—
253Q8D4×D28311yescharacteristic
254Q8C2×C22⋊D28312yes—
255D4—21no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—2831no—
262D4—5661no—
263D4—5662no—
264D4—5662no—
265D4—11322no—
266D4—11322no—
267D4—22644no—
268D4D4×D28311yescharacteristic
269D4C2×C22⋊D28312yes—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—5661no—
274C2×C4—5661no—
275C2×C4—11322no—
276C2×C4C23×D28311yescharacteristic
277C8—21no—
278C8—2831no—
279C8—2831no—
280C8—5662no—
281C8—5661no—
282C8C2×C22⋊D28311yescharacteristic
283C8C2×C22⋊D28311yescharacteristic
284C22—21no—
285C22—22no—
286C22—42no—
287C22—82no—
288C22—162no—
289C22—5661no—
290C22—11322no—
291C22—45282no—
292C22—22642no—
293C22#2258365511yescharacteristic
294C4—21no—
295C4—22no—
296C4—5662no—
297C4—5661no—
298C4C2×D4×D28311yescharacteristic
299C4C22×C22⋊D28311yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—11321no—
305C2—22642no—
306C2D4×C22⋊D28311yescharacteristic
subgroupG/H
centerC2D4×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#576×C283C2
socleC2×C283D42
hypercenter#576D283
hypocenterC283#3292
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #31827268C2semidirect
2. #31827267C2semidirect
3. #31826840C2semidirect
4. #31826735C2semidirect
5. #31826733C2semidirect
6. #31826732C2semidirect
7. #31826633C2semidirect
8. #31826631C2semidirect
9. #31826582C2semidirect
10. #31826580C2semidirect
11. #576×C283C2semidirect
12. #22583656C22semidirect
15. #22583636C22semidirect
17. #22583635C22semidirect
18. #22583635C22semidirect
22. C2×D2264C22semidirect
24. C2×#17985377C22semidirect
26. #22583617C22semidirect
27. #22583616C22semidirect
28. #22583606C22semidirect
29. #22583546C22semidirect
30. #22583545C22semidirect
31. #22583537C22semidirect
32. #22583537C22semidirect
33. #22583536C22semidirect
34. #22583527C22semidirect
35. #22583526C22semidirect
36. C8×Dic283C22semidirect
37. H(5,𝔽2)×C283C22semidirect
39. AGL(1,ℤ/8)×C283C22semidirect
43. C2×D8×C283C22semidirect
44. #131×C283C22semidirect
53. #17985382D4semidirect
55. #17985380D4semidirect
58. #17985379D4semidirect
66. #17985377D4semidirect
83. Q16×C283D4semidirect
85. SD16×C283D4semidirect
89. D8×C283D4semidirect
153. C283#3292semidirect
157. #576D283semidirect
167. H(5,𝔽2)C2×D283semidirect
175. AGL(1,ℤ/8)C2×D283semidirect
186. C2×D8C2×D283semidirect
189. #131C2×D283semidirect
216. Q16C22⋊D283semidirect
223. SD16C22⋊D283semidirect
235. D8C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
#318272682
#318272672
#318268402
#318267352
#318267332
#318267322
#318266332
#318266312
#318265822
#318265802
#576×C2832
#3292283