Finite Group Databasebeta

← #49133027

Subgroups — #49133027

Order 36224.

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#31827271C212yesmaximal, complemented
2#31827266C212yesmaximal, complemented
3#31826732C212yesmaximal, complemented
4#31826725C212yesmaximal, complemented
5#31826642C211yesmaximal, characteristic, complemented
6#31826637C211yesmaximal, characteristic, complemented
7#31826632C212yesmaximal, complemented
8#31826588C212yesmaximal, complemented
9#496×C283C211yesmaximal, characteristic, complemented
10#22583658C2211yescharacteristic, complemented
11#22583655C2211yescharacteristic
12#22583655C2212yescomplemented
13#22583641C2212yescomplemented
14#22583639—22no—
15#22583637—22no—
16#22583636C2212yes—
17#22583636C2212yescomplemented
18#22583636C2212yescomplemented
19#22583634—22no—
20#22583634—22no—
21#22583632C2212yes—
22#22583631—22no—
23#22583629—22no—
24SD16×D283—22no—
25D8×D283—22no—
26#22583616C2212yescomplemented
27#22583616C2212yescomplemented
28#22583546C2212yescomplemented
29#22583538C2212yescomplemented
30#22583538C2212yescomplemented
31#22583529C2211yescharacteristic, complemented
32#22583527C2212yescomplemented
33#22583527C2211yescharacteristic, complemented
34C2×SD16×C283C2212yescomplemented
35C2×D8×C283C2212yescomplemented
36#127×C283C2211yescharacteristic, complemented
37#124×C283C2211yescharacteristic, complemented
38#97×C283C2211yescharacteristic, complemented
39#17985393—22no—
40#17985392—41no—
41#17985392C2312yes—
42#17985392C2312yes—
43#17985391—22no—
44#17985391—22no—
45#17985391—22no—
46Q8×D283—22no—
47C2×C22⋊D283—41no—
48C2×C22⋊D283—42no—
49D4×D283—22no—
50D4×D283—22no—
51D4×D283—22no—
52Dic1132D412yescomplemented
53#17985382—22no—
54#17985381D412yescomplemented
55#17985381D412yescomplemented
56#17985380—22no—
57#17985380—22no—
58#17985380—22no—
59#17985379—22no—
60D2264D412yescomplemented
61#17985377—22no—
62#17985377—22no—
63#17985377—22no—
64#17985376C2312yes—
65#17985376C2312yes—
66#17985375—22no—
67#17985375—22no—
68C8×D283—22no—
69C8×D283—22no—
70C2×Q8×C283C2311yescharacteristic
71C2×D4×C283—41no—
72C2×D4×C283C2312yes—
73C2×D4×C283C2311yescharacteristic
74SD16×C283—22no—
75SD16×C283—22no—
76D8×C283—22no—
77D8×C283—22no—
78C2×C8×C283C2312yes—
79C22⋊C4×C283—41no—
80C42×C283C2311yescharacteristic
81C22×D283—42no—
82C22×D283—41no—
83Dic566—41no—
84Dic566C2×D412yes—
85Dic566C2×D412yes—
86D1132—41no—
87D1132C2×D412yes—
88D1132C2×D412yes—
89C22⋊D283—22no—
90C22⋊D283—22no—
91C22⋊D283—42no—
92C22⋊D283—42no—
93C22⋊D283—42no—
94C2×Dic283—42no—
95C2×Dic283—41no—
96C4×D283—22no—
97C4×D283—22no—
98C4×D283—42no—
99C283⋊C8—22no—
100C283⋊C8—22no—
101C23×C283—21no—
102C23×C283—22no—
103Q8×C283—22no—
104D4×C283—22no—
105D4×C283—22no—
106D4×C283—22no—
107D4×C283—82no—
108C2×C4×C283—21no—
109C2×C4×C283—21no—
110C2×C4×C283C2411yescharacteristic
111C8×C283C2×D412yes—
112C8×C283C2×D412yes—
113C22×C283—41no—
114C22×C283—42no—
115C22×C283—82no—
116C22×C283—81no—
117C22×C283H(5,𝔽2)11yescharacteristic
118C2×D283—22no—
119C2×D283—22no—
120C2×D283—162no—
121C2×D283—161no—
122C4×C283—41no—
123C4×C283—41no—
124C4×C283C22×D412yes—
125Dic283—22no—
126Dic283—22no—
127C2×C283—21no—
128C2×C283—82no—
129C2×C283—81no—
130C2×C283D4211yescharacteristic
131D283—42no—
132D283—42no—
133C283#328811yescharacteristic, complemented, Sylow
134#3288—2831nomaximal, Sylow
135#577—2832no—
136#576—2832no—
137#496D28311yescharacteristic, complemented
138#472—2832no—
139#471—2832no—
140#455—2831no—
141#454—2831no—
142#453—2832no—
143#444—2832no—
1442−1+4—2831no—
145H(5,𝔽2)—2831no—
146H(5,𝔽2)—2832no—
147#137—2832no—
148#137—5662no—
149AGL(1,ℤ/8)—2832no—
150AGL(1,ℤ/8)—2832no—
151AGL(1,ℤ/8)—2832no—
152AGL(1,ℤ/8)—5662no—
153AGL(1,ℤ/8)—5662no—
154AGL(1,ℤ/8)—5662no—
155#135—2832no—
156#135—5662no—
157#135—5662no—
158C2×SD16—5662no—
159C2×SD16C2×D28312yescomplemented
160C2×D8—5662no—
161C2×D8C2×D28312yescomplemented
162#131—2832no—
163#131—2832no—
164#127C2×D28311yescharacteristic, complemented
165#124C2×D28311yescharacteristic, complemented
166#108—2832no—
167#104—2832no—
168#104—2832no—
169#101—2831no—
170#100—2832no—
171#100—2831no—
172#97C2×D28311yescharacteristic, complemented
173#41—2832no—
174#41—2832no—
175#41—5662no—
176#41—5662no—
177#41—5662no—
178#41—5662no—
179#41—11321no—
180C2×Q8—5662no—
181C2×Q8C22×D28311yescharacteristic
182C2×D4—41no—
183C2×D4—5662no—
184C2×D4—5662no—
185C2×D4—5662no—
186C2×D4—11321no—
187C2×D4—11322no—
188C2×D4C22×D28311yescharacteristic
189C2×D4C22×D28312yes—
190Q16—2832no—
191Q16—5662no—
192SD16—22no—
193SD16—22no—
194SD16—2832no—
195SD16—2832no—
196SD16—5662no—
197SD16—5662no—
198SD16—5662no—
199SD16—5662no—
200D8—22no—
201D8—22no—
202D8—2832no—
203D8—5662no—
204D8—5662no—
205D8—5662no—
206M16—2832no—
207M16—2832no—
208M16—5662no—
209M16—5662no—
210C2×C8—5662no—
211C2×C8—5662no—
212C2×C8C22×D28312yes—
213C22⋊C4—41no—
214C42C22×D28311yescharacteristic
215C23—21no—
216C23—22no—
217C23—11322no—
218C23—11321no—
219Q8—22no—
220Q8—2832no—
221Q8—2832no—
222Q8—11321no—
223D4—22no—
224D4—22no—
225D4—22no—
226D4—82no—
227D4—2832no—
228D4—2832no—
229D4—5662no—
230D4—5662no—
231D4—11321no—
232D4—11322no—
233D4—11322no—
234D4—11322no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—5662no—
238C2×C4—5662no—
239C2×C4—11322no—
240C2×C4—11321no—
241C2×C4—11322no—
242C2×C4C23×D28311yescharacteristic
243C8—5662no—
244C8—5662no—
245C8D4×D28312yes—
246C8D4×D28312yes—
247C22—41no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—5662no—
252C22—5662no—
253C22—45281no—
254C22—45282no—
255C22#2258365511yescharacteristic
256C4—41no—
257C4—41no—
258C4—5662no—
259C4—5662no—
260C4C2×D4×D28312yes—
261C2—21no—
262C2—82no—
263C2—81no—
264C2—11322no—
265C2—11322no—
266C2#3182710011yescharacteristic
subgroupG/H
centerC2#31827100
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#496×C283C2
socleC2×C283D42
hypercenter#496D283
hypocenterC283#3288
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #31827271C2semidirect
2. #31827266C2semidirect
3. #31826732C2semidirect
4. #31826725C2semidirect
5. #31826642C2semidirect
6. #31826637C2semidirect
7. #31826632C2semidirect
8. #31826588C2semidirect
9. #496×C283C2semidirect
10. #22583658C22semidirect
12. #22583655C22semidirect
13. #22583641C22semidirect
17. #22583636C22semidirect
18. #22583636C22semidirect
26. #22583616C22semidirect
27. #22583616C22semidirect
28. #22583546C22semidirect
29. #22583538C22semidirect
30. #22583538C22semidirect
31. #22583529C22semidirect
32. #22583527C22semidirect
33. #22583527C22semidirect
34. C2×SD16×C283C22semidirect
35. C2×D8×C283C22semidirect
36. #127×C283C22semidirect
37. #124×C283C22semidirect
38. #97×C283C22semidirect
52. Dic1132D4semidirect
54. #17985381D4semidirect
55. #17985381D4semidirect
60. D2264D4semidirect
133. C283#3288semidirect
137. #496D283semidirect
159. C2×SD16C2×D283semidirect
161. C2×D8C2×D283semidirect
164. #127C2×D283semidirect
165. #124C2×D283semidirect
172. #97C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272712
#318272662
#318267322
#318267252
#318266422
#318266372
#318266322
#318265882
#496×C2832
#3288283