Finite Group Databasebeta

← #49133061

Subgroups — #49133061

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#31827270C212yesmaximal, complemented
2#31827267C212yesmaximal, complemented
3#31826842C211yesmaximal, characteristic, complemented
4#31826736C212yesmaximal, complemented
5#31826735C212yesmaximal, complemented
6#31826634C214yesmaximal, complemented
7#31826583C211yesmaximal, characteristic, complemented
8#443×C283C211yesmaximal, characteristic, complemented
9#22583656C2214yescomplemented
10#22583638—24no—
11#22583638C2214yescomplemented
12#22583637—24no—
13#22583635C2214yescomplemented
14#22583633—22no—
15#22583632—22no—
16#22583632C2211yescharacteristic
17C2×D2264—22no—
18C2×D2264C2212yescomplemented
19C2×D2264C2212yescomplemented
20D8×D283—22no—
21#22583616C2212yescomplemented
22#22583607C2212yescomplemented
23#22583546C2211yescharacteristic, complemented
24#22583538C2212yescomplemented
25#22583526C2214yescomplemented
26#135×C283C2211yescharacteristic
27#131×C283C2212yescomplemented
28#108×C283C2211yescharacteristic, complemented
29#104×C283C2212yescomplemented
30C4×C8×C283C2211yescharacteristic, complemented
31#17985393—24no—
32#17985392—24no—
33#17985392C2312yes—
34C2×D1132—24no—
35C2×D1132—42no—
36C2×D1132C2314yes—
37D4×D283—24no—
38D4×D283—44no—
39Dic1132D411yescharacteristic, complemented
40#17985381—24no—
41#17985381D412yescomplemented
42#17985379—24no—
43D2264—24no—
44D2264—22no—
45D2264—22no—
46D2264—22no—
47D2264—22no—
48D2264D411yescharacteristic, complemented
49#17985377—24no—
50#17985376C2312yes—
51#17985375—22no—
52C8×D283—22no—
53#41×C283C2312yes—
54Q16×C283D411yescharacteristic, complemented
55SD16×C283D412yescomplemented
56D8×C283D411yescharacteristic, complemented
57M16×C283—22no—
58M16×C283C2312yes—
59C2×C8×C283—22no—
60C2×C8×C283C2311yescharacteristic
61C2×C8×C283C2311yescharacteristic
62C42×C283C2311yescharacteristic
63C22×D283—24no—
64C22×D283—44no—
65Dic566C2×D412yes—
66D1132—24no—
67D1132—24no—
68D1132—44no—
69D1132—84no—
70D1132C2×D412yes—
71C22⋊D283—22no—
72C22⋊D283—44no—
73C4×D283—22no—
74C4×D283—44no—
75C283⋊C8—22no—
76Q8×C283C2×D412yes—
77D4×C283—22no—
78D4×C283C2×D412yes—
79C2×C4×C283—21no—
80C2×C4×C283—22no—
81C2×C4×C283C2411yescharacteristic
82C8×C283—22no—
83C8×C283C2×D411yescharacteristic
84C8×C283C2×D411yescharacteristic
85C8×C283C2×D411yescharacteristic
86C8×C283C2×D411yescharacteristic
87C22×C283—22no—
88C22×C283H(5,𝔽2)11yescharacteristic
89C2×D283—22no—
90C2×D283—44no—
91C2×D283—84no—
92C2×D283—164no—
93C4×C283—22no—
94C4×C283—22no—
95C4×C283C22×D411yescharacteristic
96C4×C283C22×D411yescharacteristic
97Dic283—22no—
98C2×C283—21no—
99C2×C283—42no—
100C2×C283D4211yescharacteristic
101D283—42no—
102D283—84no—
103C283#329211yescharacteristic, complemented, Sylow
104#3292—2831nomaximal, Sylow
105#576—2832no—
106#576—2832no—
107#493—2831no—
108#472—2832no—
109#472—2832no—
110#453—2834no—
111#443—2831no—
112#443D28311yescharacteristic, complemented
113H(5,𝔽2)—2834no—
114AGL(1,ℤ/8)—2834no—
115AGL(1,ℤ/8)—2834no—
116AGL(1,ℤ/8)—5664no—
117AGL(1,ℤ/8)—5664no—
118#135—2831no—
119#135—5662no—
120#135—5662no—
121#135C2×D28311yescharacteristic
122C2×D8—2832no—
123C2×D8—2832no—
124C2×D8—5662no—
125C2×D8—5662no—
126#131—2832no—
127#131C2×D28312yescomplemented
128#127—2832no—
129#108—2831no—
130#108C2×D28311yescharacteristic, complemented
131#104—2832no—
132#104C2×D28312yescomplemented
133#100—2834no—
134C4×C8C2×D28311yescharacteristic, complemented
135#41—2832no—
136#41—5664no—
137#41—5664no—
138#41C22×D28312yes—
139C2×D4—2834no—
140C2×D4—5664no—
141C2×D4—5664no—
142C2×D4—11322no—
143C2×D4—11324no—
144Q16—2831no—
145Q16D113211yescharacteristic, complemented
146SD16—2832no—
147SD16—5664no—
148SD16—5664no—
149SD16D113212yescomplemented
150D8—2831no—
151D8—5664no—
152D8—5664no—
153D8—5662no—
154D8—5662no—
155D8—5662no—
156D8—5662no—
157D8D113211yescharacteristic, complemented
158M16—22no—
159M16—2832no—
160M16—5662no—
161M16C22×D28312yes—
162C2×C8—22no—
163C2×C8—5662no—
164C2×C8C22×D28311yescharacteristic
165C2×C8C22×D28311yescharacteristic
166C42C22×D28311yescharacteristic
167C23—5664no—
168C23—11324no—
169Q8—2832no—
170Q8C2×D113212yes—
171D4—22no—
172D4—2832no—
173D4—5662no—
174D4—5664no—
175D4—5664no—
176D4—11324no—
177D4—11324no—
178D4—22644no—
179D4C2×D113212yes—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—5662no—
183C2×C4—11324no—
184C2×C4C23×D28311yescharacteristic
185C8—22no—
186C8—5662no—
187C8D4×D28311yescharacteristic
188C8D4×D28311yescharacteristic
189C8C2×D113211yescharacteristic
190C8C2×D113211yescharacteristic
191C22—22no—
192C22—5662no—
193C22—11324no—
194C22—45284no—
195C22—22644no—
196C22#2258365611yescharacteristic
197C4—22no—
198C4—22no—
199C4—5662no—
200C4C2×D4×D28311yescharacteristic
201C4C22×D113211yescharacteristic
202C2—21no—
203C2—42no—
204C2—11322no—
205C2—22644no—
206C2D4×D113211yescharacteristic
subgroupG/H
centerC2D4×D1132
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#443×C283C2
socleC2×C283D42
hypercenter#443D283
hypocenterC283#3292
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #31827270C2semidirect
2. #31827267C2semidirect
3. #31826842C2semidirect
4. #31826736C2semidirect
5. #31826735C2semidirect
6. #31826634C2semidirect
7. #31826583C2semidirect
8. #443×C283C2semidirect
9. #22583656C22semidirect
11. #22583638C22semidirect
13. #22583635C22semidirect
18. C2×D2264C22semidirect
19. C2×D2264C22semidirect
21. #22583616C22semidirect
22. #22583607C22semidirect
23. #22583546C22semidirect
24. #22583538C22semidirect
25. #22583526C22semidirect
27. #131×C283C22semidirect
28. #108×C283C22semidirect
29. #104×C283C22semidirect
30. C4×C8×C283C22semidirect
39. Dic1132D4semidirect
41. #17985381D4semidirect
48. D2264D4semidirect
54. Q16×C283D4semidirect
55. SD16×C283D4semidirect
56. D8×C283D4semidirect
103. C283#3292semidirect
112. #443D283semidirect
127. #131C2×D283semidirect
130. #108C2×D283semidirect
132. #104C2×D283semidirect
134. C4×C8C2×D283semidirect
145. Q16D1132semidirect
149. SD16D1132semidirect
157. D8D1132semidirect

Maximal subgroups

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

subgroupindex
#318272702
#318272672
#318268422
#318267362
#318267352
#318266342
#318265832
#443×C2832
#3292283