Finite Group Databasebeta

← #49133059

Subgroups — #49133059

Order 36224.

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#31827270C211yesmaximal, characteristic, complemented
2#31827269C212yesmaximal, complemented
3#31827266C211yesmaximal, characteristic, complemented
4#31826840C211yesmaximal, characteristic, complemented
5#31826736C211yesmaximal, characteristic, complemented
6#31826734C212yesmaximal, complemented
7#31826635C212yesmaximal, complemented
8#31826631C212yesmaximal, complemented
9#31826581C211yesmaximal, characteristic, complemented
10#31826580C211yesmaximal, characteristic, complemented
11#472×C283C211yesmaximal, characteristic, complemented
12#22583656C2212yescomplemented
13#22583655C2212yescomplemented
14#22583638—22no—
15#22583638C2212yescomplemented
16#22583638C2212yescomplemented
17#22583637—22no—
18#22583636C2212yescomplemented
19#22583634—22no—
20#22583634—22no—
21#22583632—21no—
22#22583632C2211yescharacteristic
23#22583631—22no—
24#22583630C2211yescharacteristic
25#22583629—21no—
26C2×D2264—21no—
27C2×D2264C2211yescharacteristic, complemented
28C2×#17985377C2212yescomplemented
29D8×D283—21no—
30D8×D283—22no—
31#22583618C2212yescomplemented
32#22583616C2211yescharacteristic, complemented
33#22583606C2212yescomplemented
34#22583545C2211yescharacteristic, complemented
35#22583537C2211yescharacteristic, complemented
36#22583536C2211yescharacteristic, complemented
37#22583536C2212yescomplemented
38#22583526C2212yescomplemented
39C8×Dic283C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C283C2212yescomplemented
41C2×D8×C283C2211yescharacteristic, complemented
42#131×C283C2211yescharacteristic, complemented
43#108×C283C2211yescharacteristic, complemented
44#100×C283C2212yescomplemented
45#17985393—22no—
46#17985392—22no—
47#17985392C2311yescharacteristic
48#17985392C2312yes—
49#17985391—22no—
50#17985391—22no—
51C2×D1132—22no—
52C2×D1132C2312yes—
53C2×C22⋊D283—42no—
54C2×C22⋊D283—42no—
55D4×D283—22no—
56D4×D283—22no—
57D4×D283—22no—
58D4×D283—42no—
59Dic1132D411yescharacteristic, complemented
60#17985382D411yescharacteristic, complemented
61#17985381—22no—
62#17985381—22no—
63#17985381D412yescomplemented
64#17985380—22no—
65#17985380D411yescharacteristic, complemented
66#17985379D411yescharacteristic, complemented
67D2264—22no—
68D2264—22no—
69D2264—21no—
70D2264—21no—
71D2264D411yescharacteristic, complemented
72#17985377—22no—
73#17985377—22no—
74#17985377—22no—
75#17985377D411yescharacteristic, complemented
76#17985376C2311yescharacteristic
77#17985375—22no—
78#17985375—21no—
79C2×C283⋊C8C2311yescharacteristic
80C8×D283—22no—
81C8×D283—21no—
82C4×Dic283C2311yescharacteristic
83#41×C283C2311yescharacteristic
84C2×D4×C283C2312yes—
85SD16×C283—22no—
86D8×C283—22no—
87D8×C283—21no—
88D8×C283—21no—
89M16×C283—21no—
90M16×C283C2312yes—
91M16×C283C2311yescharacteristic
92C2×C8×C283—21no—
93C2×C8×C283C2311yescharacteristic
94C22×D283—22no—
95C22×D283—42no—
96C22×D283—42no—
97Dic566C2×D412yes—
98Dic566C2×D411yescharacteristic
99D1132—22no—
100D1132—22no—
101D1132—42no—
102D1132C2×D412yes—
103D1132C2×D411yescharacteristic
104C22⋊D283—22no—
105C22⋊D283—21no—
106C22⋊D283—42no—
107C22⋊D283—42no—
108C22⋊D283—42no—
109C22⋊D283—84no—
110C2×Dic283—21no—
111C2×Dic283—42no—
112C4×D283—21no—
113C4×D283—22no—
114C4×D283—42no—
115C283⋊C8—21no—
116C283⋊C8C2×D411yescharacteristic
117C283⋊C8C2×D411yescharacteristic
118C23×C283—22no—
119Q8×C283C2×D411yescharacteristic
120D4×C283—21no—
121D4×C283—22no—
122D4×C283—22no—
123D4×C283C2×D411yescharacteristic
124C2×C4×C283—21no—
125C2×C4×C283C2411yescharacteristic
126C8×C283—21no—
127C8×C283—22no—
128C8×C283C2×D411yescharacteristic
129C8×C283C2×D411yescharacteristic
130C22×C283—21no—
131C22×C283—42no—
132C22×C283—82no—
133C22×C283H(5,𝔽2)11yescharacteristic
134C2×D283—21no—
135C2×D283—22no—
136C2×D283—42no—
137C2×D283—82no—
138C2×D283—162no—
139C2×D283—162no—
140C4×C283—21no—
141C4×C283C22×D411yescharacteristic
142C4×C283C22×D411yescharacteristic
143Dic283—22no—
144Dic283—22no—
145Dic283—21no—
146C2×C283—21no—
147C2×C283—41no—
148C2×C283—82no—
149C2×C283D4211yescharacteristic
150D283—41no—
151D283—42no—
152D283—82no—
153C283#329211yescharacteristic, complemented, Sylow
154#3292—2831nomaximal, Sylow
155#576—2832no—
156#576—2831no—
157#576—2831no—
158#493—2831no—
159#472—2831no—
160#472—2832no—
161#472D28311yescharacteristic, complemented
162#453—2832no—
163#453—2832no—
164#443—2831no—
165#443—2831no—
166H(5,𝔽2)—2832no—
167H(5,𝔽2)—2832no—
168AGL(1,ℤ/8)—2832no—
169AGL(1,ℤ/8)—2832no—
170AGL(1,ℤ/8)—2832no—
171AGL(1,ℤ/8)—5662no—
172AGL(1,ℤ/8)—5662no—
173AGL(1,ℤ/8)—5662no—
174AGL(1,ℤ/8)—5662no—
175AGL(1,ℤ/8)C2×D28312yescomplemented
176#135—2831no—
177#135—2831no—
178#135—5661no—
179#135—5661no—
180#135—5662no—
181C2×D8—2832no—
182C2×D8—2831no—
183C2×D8—5661no—
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#108C2×D28311yescharacteristic, complemented
193#104—2831no—
194#104—2832no—
195#104—2831no—
196#100—2832no—
197#100C2×D28312yescomplemented
198C4×C8—2831no—
199#41—2831no—
200#41—2832no—
201#41—5662no—
202#41—5662no—
203#41—5662no—
204#41—5662no—
205#41C22×D28311yescharacteristic
206C2×D4—2832no—
207C2×D4—5662no—
208C2×D4—5662no—
209C2×D4—5662no—
210C2×D4—5662no—
211C2×D4—11322no—
212C2×D4—11322no—
213C2×D4—11322no—
214C2×D4C22×D28312yes—
215Q16—2831no—
216Q16—2831no—
217SD16—22no—
218SD16—2831no—
219SD16—2831no—
220SD16—2832no—
221SD16—5662no—
222SD16—5662no—
223SD16—5662no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—2831no—
228D8—2831no—
229D8—5662no—
230D8—5662no—
231D8—5662no—
232D8—5662no—
233D8—5662no—
234D8—5661no—
235D8—5661no—
236M16—21no—
237M16—2831no—
238M16—5662no—
239M16—5661no—
240M16C22×D28312yes—
241M16C22×D28311yescharacteristic
242C2×C8—21no—
243C2×C8—2831no—
244C2×C8—5662no—
245C2×C8—5661no—
246C2×C8C22×D28311yescharacteristic
247C42—2831no—
248C23—22no—
249C23—5662no—
250C23—11322no—
251C23—11322no—
252Q8—2831no—
253Q8—2832no—
254Q8D4×D28311yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—2831no—
259D4—2832no—
260D4—5662no—
261D4—5661no—
262D4—5662no—
263D4—5662no—
264D4—11322no—
265D4—11322no—
266D4—11322no—
267D4—11322no—
268D4—22644no—
269D4D4×D28311yescharacteristic
270C2×C4—21no—
271C2×C4—5661no—
272C2×C4—5662no—
273C2×C4—5661no—
274C2×C4—11322no—
275C2×C4—11322no—
276C2×C4C23×D28311yescharacteristic
277C8—21no—
278C8—22no—
279C8—2831no—
280C8—2831no—
281C8—5661no—
282C8D4×D28311yescharacteristic
283C8D4×D28311yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—5662no—
288C22—5661no—
289C22—11322no—
290C22—45282no—
291C22—45282no—
292C22—22642no—
293C22#2258365611yescharacteristic
294C4—21no—
295C4—5662no—
296C4—5662no—
297C4—5661no—
298C4C2×D4×D28311yescharacteristic
299C4C2×D4×D28311yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—11322no—
304C2—11321no—
305C2—22642no—
306C2#3182709911yescharacteristic
subgroupG/H
centerC2#31827099
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#472×C283C2
socleC2×C283D42
hypercenter#472D283
hypocenterC283#3292
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #31827270C2semidirect
2. #31827269C2semidirect
3. #31827266C2semidirect
4. #31826840C2semidirect
5. #31826736C2semidirect
6. #31826734C2semidirect
7. #31826635C2semidirect
8. #31826631C2semidirect
9. #31826581C2semidirect
10. #31826580C2semidirect
11. #472×C283C2semidirect
12. #22583656C22semidirect
13. #22583655C22semidirect
15. #22583638C22semidirect
16. #22583638C22semidirect
18. #22583636C22semidirect
27. C2×D2264C22semidirect
28. C2×#17985377C22semidirect
31. #22583618C22semidirect
32. #22583616C22semidirect
33. #22583606C22semidirect
34. #22583545C22semidirect
35. #22583537C22semidirect
36. #22583536C22semidirect
37. #22583536C22semidirect
38. #22583526C22semidirect
39. C8×Dic283C22semidirect
40. AGL(1,ℤ/8)×C283C22semidirect
41. C2×D8×C283C22semidirect
42. #131×C283C22semidirect
43. #108×C283C22semidirect
44. #100×C283C22semidirect
59. Dic1132D4semidirect
60. #17985382D4semidirect
63. #17985381D4semidirect
65. #17985380D4semidirect
66. #17985379D4semidirect
71. D2264D4semidirect
75. #17985377D4semidirect
153. C283#3292semidirect
161. #472D283semidirect
175. AGL(1,ℤ/8)C2×D283semidirect
186. C2×D8C2×D283semidirect
189. #131C2×D283semidirect
192. #108C2×D283semidirect
197. #100C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272702
#318272692
#318272662
#318268402
#318267362
#318267342
#318266352
#318266312
#318265812
#318265802
#472×C2832
#3292283