Finite Group Databasebeta

← #49133054

Subgroups — #49133054

Order 36224.

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

normal subgroups only

This group has 460 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#31827271C214yesmaximal, complemented
2#31826725C214yesmaximal, complemented
3#31826647C212yesmaximal, complemented
4#31826632C212yesmaximal, complemented
5#31826583C212yesmaximal, complemented
6#492×C283C211yesmaximal, characteristic, complemented
7#22583658C2212yes—
8#22583655C2212yescomplemented
9#22583641C2214yescomplemented
10#22583639—24no—
11#22583637—24no—
12#22583636C2214yes—
13#22583632C2212yes—
14#22583631—24no—
15SD16×D283—24no—
16#22583616C2214yescomplemented
17#22583546C2212yescomplemented
18#22583538C2214yescomplemented
19#22583529C2212yescomplemented
20#22583527C2212yescomplemented
21C2×SD16×C283C2214yescomplemented
22C4⋊Q8×C283C2211yescharacteristic, complemented
23#127×C283C2211yescharacteristic
24C4×C8×C283C2211yescharacteristic, complemented
25#17985393—24no—
26#17985392—42no—
27#17985392C2314yes—
28#17985391—24no—
29Q8×D283—24no—
30C2×C22⋊D283—42no—
31D4×D283—24no—
32Dic1132D412yescomplemented
33#17985382—24no—
34#17985381D414yescomplemented
35#17985380—24no—
36#17985379—24no—
37D2264D412yescomplemented
38#17985377—24no—
39#17985376C2314yes—
40#17985375—24no—
41C8×D283—24no—
42C2×Q8×C283C2312yes—
43C2×D4×C283—41no—
44C2×D4×C283C2312yes—
45SD16×C283—24no—
46SD16×C283—24no—
47C2×C8×C283C2312yes—
48C4⋊C4×C283—41no—
49C42×C283C2311yescharacteristic
50C22×D283—42no—
51Dic566—42no—
52Dic566C2×D414yes—
53D1132—42no—
54D1132C2×D414yes—
55C22⋊D283—24no—
56C22⋊D283—44no—
57C2×Dic283—42no—
58C4×D283—24no—
59C4×D283—44no—
60C283⋊C8—24no—
61C23×C283—22no—
62Q8×C283—24no—
63D4×C283—24no—
64D4×C283—82no—
65C2×C4×C283—21no—
66C2×C4×C283—22no—
67C2×C4×C283C2411yescharacteristic
68C8×C283C2×D412yes—
69C8×C283C2×D412yes—
70C22×C283—42no—
71C22×C283—82no—
72C22×C283H(5,𝔽2)11yescharacteristic
73C2×D283—24no—
74C2×D283—162no—
75C4×C283—22no—
76C4×C283—42no—
77C4×C283C22×D412yes—
78Dic283—24no—
79C2×C283—21no—
80C2×C283—82no—
81C2×C283D4211yescharacteristic
82D283—44no—
83C283#329111yescharacteristic, complemented, Sylow
84#3291—2831nomaximal, Sylow
85#577—2834no—
86#492D28311yescharacteristic, complemented
87#471—2834no—
88#456—2832no—
89#453—2832no—
90#443—2832no—
912−1+4—2832no—
92H(5,𝔽2)—2832no—
93#137—2834no—
94#137—5664no—
95AGL(1,ℤ/8)—2834no—
96AGL(1,ℤ/8)—5664no—
97#135—2832no—
98#135—5664no—
99C2×SD16—5664no—
100C2×SD16C2×D28314yescomplemented
101#131—2834no—
102C4⋊Q8C2×D28311yescharacteristic, complemented
103#127C2×D28311yescharacteristic
104#108—2832no—
105#104—2834no—
106#101—2832no—
107#100—2832no—
108C4×C8C2×D28311yescharacteristic, complemented
109#41—2834no—
110#41—5664no—
111#41—5664no—
112#41—11322no—
113C2×Q8—5664no—
114C2×Q8C22×D28312yes—
115C2×D4—41no—
116C2×D4—5664no—
117C2×D4—11322no—
118C2×D4C22×D28312yes—
119Q16—2832no—
120Q16—5664no—
121SD16—24no—
122SD16—24no—
123SD16—2834no—
124SD16—5664no—
125SD16—5664no—
126D8—2832no—
127D8—5664no—
128M16—2834no—
129M16—5664no—
130C2×C8—5664no—
131C2×C8C22×D28312yes—
132C4⋊C4—41no—
133C42C22×D28311yescharacteristic
134C23—22no—
135C23—11322no—
136Q8—24no—
137Q8—2834no—
138Q8—11322no—
139D4—24no—
140D4—82no—
141D4—2834no—
142D4—5664no—
143D4—11322no—
144D4—11324no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—5664no—
148C2×C4—11324no—
149C2×C4—11322no—
150C2×C4C23×D28311yescharacteristic
151C8—5664no—
152C8D4×D28312yes—
153C8D4×D28312yes—
154C22—42no—
155C22—82no—
156C22—5664no—
157C22—45282no—
158C22#2258365511yescharacteristic
159C4—22no—
160C4—42no—
161C4—5664no—
162C4C2×D4×D28312yes—
163C2—21no—
164C2—82no—
165C2—11324no—
166C2#3182710011yescharacteristic
subgroupG/H
centerC2#31827100
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#492×C283C2
socleC2×C283D42
hypercenter#492D283
hypocenterC283#3291
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #31827271C2semidirect
2. #31826725C2semidirect
3. #31826647C2semidirect
4. #31826632C2semidirect
5. #31826583C2semidirect
6. #492×C283C2semidirect
8. #22583655C22semidirect
9. #22583641C22semidirect
16. #22583616C22semidirect
17. #22583546C22semidirect
18. #22583538C22semidirect
19. #22583529C22semidirect
20. #22583527C22semidirect
21. C2×SD16×C283C22semidirect
22. C4⋊Q8×C283C22semidirect
24. C4×C8×C283C22semidirect
32. Dic1132D4semidirect
34. #17985381D4semidirect
37. D2264D4semidirect
83. C283#3291semidirect
86. #492D283semidirect
100. C2×SD16C2×D283semidirect
102. C4⋊Q8C2×D283semidirect
108. C4×C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272712
#318267252
#318266472
#318266322
#318265832
#492×C2832
#3291283