Finite Group Databasebeta

← #46822678

Subgroups — #46822678

Order 34432.

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#31771411C214yesmaximal, complemented
2#31770865C214yesmaximal, complemented
3#31770787C212yesmaximal, complemented
4#31770772C212yesmaximal, complemented
5#31770723C212yesmaximal, complemented
6#492×C269C211yesmaximal, characteristic, complemented
7#22575501C2212yes—
8#22575498C2212yescomplemented
9#22575484C2214yescomplemented
10#22575482—24no—
11#22575480—24no—
12#22575479C2214yes—
13#22575475C2212yes—
14#22575474—24no—
15SD16×D269—24no—
16#22575459C2214yescomplemented
17#22575389C2212yescomplemented
18#22575381C2214yescomplemented
19#22575372C2212yescomplemented
20#22575370C2212yescomplemented
21C2×SD16×C269C2214yescomplemented
22C4⋊Q8×C269C2211yescharacteristic, complemented
23#127×C269C2211yescharacteristic
24C4×C8×C269C2211yescharacteristic, complemented
25#16864403—24no—
26#16864402—42no—
27#16864402C2314yes—
28#16864401—24no—
29Q8×D269—24no—
30C2×C22⋊D269—42no—
31D4×D269—24no—
32Dic1076D412yescomplemented
33#16864392—24no—
34#16864391D414yescomplemented
35#16864390—24no—
36#16864389—24no—
37D2152D412yescomplemented
38#16864387—24no—
39#16864386C2314yes—
40#16864385—24no—
41C8×D269—24no—
42C2×Q8×C269C2312yes—
43C2×D4×C269—41no—
44C2×D4×C269C2312yes—
45SD16×C269—24no—
46SD16×C269—24no—
47C2×C8×C269C2312yes—
48C4⋊C4×C269—41no—
49C42×C269C2311yescharacteristic
50C22×D269—42no—
51Dic538—42no—
52Dic538C2×D414yes—
53D1076—42no—
54D1076C2×D414yes—
55C22⋊D269—24no—
56C22⋊D269—44no—
57C2×Dic269—42no—
58C4×D269—24no—
59C4×D269—44no—
60#14524130—24no—
61C23×C269—22no—
62Q8×C269—24no—
63D4×C269—24no—
64D4×C269—82no—
65C2×C4×C269—21no—
66C2×C4×C269—22no—
67C2×C4×C269C2411yescharacteristic
68C8×C269C2×D412yes—
69C8×C269C2×D412yes—
70C22×C269—42no—
71C22×C269—82no—
72C22×C269H(5,𝔽2)11yescharacteristic
73C2×D269—24no—
74C2×D269—162no—
75C4×C269—22no—
76C4×C269—42no—
77C4×C269C22×D412yes—
78Dic269—24no—
79C2×C269—21no—
80C2×C269—82no—
81C2×C269D4211yescharacteristic
82D269—44no—
83C269#329111yescharacteristic, complemented, Sylow
84#3291—2691nomaximal, Sylow
85#577—2694no—
86#492D26911yescharacteristic, complemented
87#471—2694no—
88#456—2692no—
89#453—2692no—
90#443—2692no—
912−1+4—2692no—
92H(5,𝔽2)—2692no—
93#137—2694no—
94#137—5384no—
95AGL(1,ℤ/8)—2694no—
96AGL(1,ℤ/8)—5384no—
97#135—2692no—
98#135—5384no—
99C2×SD16—5384no—
100C2×SD16C2×D26914yescomplemented
101#131—2694no—
102C4⋊Q8C2×D26911yescharacteristic, complemented
103#127C2×D26911yescharacteristic
104#108—2692no—
105#104—2694no—
106#101—2692no—
107#100—2692no—
108C4×C8C2×D26911yescharacteristic, complemented
109#41—2694no—
110#41—5384no—
111#41—5384no—
112#41—10762no—
113C2×Q8—5384no—
114C2×Q8C22×D26912yes—
115C2×D4—41no—
116C2×D4—5384no—
117C2×D4—10762no—
118C2×D4C22×D26912yes—
119Q16—2692no—
120Q16—5384no—
121SD16—24no—
122SD16—24no—
123SD16—2694no—
124SD16—5384no—
125SD16—5384no—
126D8—2692no—
127D8—5384no—
128M16—2694no—
129M16—5384no—
130C2×C8—5384no—
131C2×C8C22×D26912yes—
132C4⋊C4—41no—
133C42C22×D26911yescharacteristic
134C23—22no—
135C23—10762no—
136Q8—24no—
137Q8—2694no—
138Q8—10762no—
139D4—24no—
140D4—82no—
141D4—2694no—
142D4—5384no—
143D4—10762no—
144D4—10764no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—5384no—
148C2×C4—10764no—
149C2×C4—10762no—
150C2×C4C23×D26911yescharacteristic
151C8—5384no—
152C8D4×D26912yes—
153C8D4×D26912yes—
154C22—42no—
155C22—82no—
156C22—5384no—
157C22—43042no—
158C22#2257549811yescharacteristic
159C4—22no—
160C4—42no—
161C4—5384no—
162C4C2×D4×D26912yes—
163C2—21no—
164C2—82no—
165C2—10764no—
166C2#3177124011yescharacteristic
subgroupG/H
centerC2#31771240
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#492×C269C2
socleC2×C269D42
hypercenter#492D269
hypocenterC269#3291
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #31771411C2semidirect
2. #31770865C2semidirect
3. #31770787C2semidirect
4. #31770772C2semidirect
5. #31770723C2semidirect
6. #492×C269C2semidirect
8. #22575498C22semidirect
9. #22575484C22semidirect
16. #22575459C22semidirect
17. #22575389C22semidirect
18. #22575381C22semidirect
19. #22575372C22semidirect
20. #22575370C22semidirect
21. C2×SD16×C269C22semidirect
22. C4⋊Q8×C269C22semidirect
24. C4×C8×C269C22semidirect
32. Dic1076D4semidirect
34. #16864391D4semidirect
37. D2152D4semidirect
83. C269#3291semidirect
86. #492D269semidirect
100. C2×SD16C2×D269semidirect
102. C4⋊Q8C2×D269semidirect
108. C4×C8C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714112
#317708652
#317707872
#317707722
#317707232
#492×C2692
#3291269