Finite Group Databasebeta

← #46824023

Subgroups — #46824023

Order 34432.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#31771407C212yesmaximal, complemented
2#31771042C214yesmaximal, complemented
3#31771040C212yesmaximal, complemented
4#31771038C212yesmaximal, complemented
5C2×D4304C211yesmaximal, characteristic, complemented
6D16×D269C212yesmaximal, complemented
7#31771021C211yesmaximal, characteristic, complemented
8#508×C269C211yesmaximal, characteristic, complemented
9#22575499—22no—
10#22575480—24no—
11#22575478—22no—
12#22575476C2214yescomplemented
13C2×D2152C2212yescomplemented
14D8×D269C2214yescomplemented
15#22575459C2211yescharacteristic
16#22575396C2214yescomplemented
17D4304C2212yescomplemented
18D4304C2212yescomplemented
19#22575394C2214yescomplemented
20#22575393C2211yescharacteristic, complemented
21#22575392C2212yescomplemented
22C16×D269C2212yescomplemented
23#135×C269C2212yescomplemented
24Q32×C269C2211yescharacteristic, complemented
25SD32×C269C2212yescomplemented
26D16×C269C2211yescharacteristic, complemented
27C2×C16×C269C2211yescharacteristic, complemented
28#16864403—24no—
29#16864402—42no—
30#16864402D411yescharacteristic
31C2×D1076—22no—
32C2×D1076—42no—
33D4×D269—24no—
34D4×D269—44no—
35#16864389—24no—
36D2152—22no—
37D2152C2314yes—
38#16864387—24no—
39#16864386D411yescharacteristic
40#16864385D412yes—
41C8×D269C2312yes—
42#16864377C2312yes—
43#41×C269—22no—
44Q16×C269C2312yes—
45SD16×C269—22no—
46D8×C269C2312yes—
47C2×C8×C269C2311yescharacteristic
48C16×C269C2311yescharacteristic
49C16×C269C2311yescharacteristic
50C22×D269—42no—
51C22×D269—44no—
52Dic538D811yescharacteristic
53D1076—24no—
54D1076—44no—
55D1076—42no—
56D1076D811yescharacteristic
57C22⋊D269—44no—
58C22⋊D269D812yes—
59C4×D269—44no—
60C4×D269C2×D412yes—
61#14524130C2×D412yes—
62Q8×C269—22no—
63D4×C269—22no—
64D4×C269—42no—
65C2×C4×C269—42no—
66C2×C4×C269C2×D411yescharacteristic
67C8×C269C2×D411yescharacteristic
68C8×C269C2411yescharacteristic
69C22×C269—42no—
70C22×C269C2×D811yescharacteristic
71C2×D269—44no—
72C2×D269—164no—
73C2×D269—162no—
74C2×D269C2×D812yes—
75C4×C269—42no—
76C4×C269C2×D811yescharacteristic
77C4×C269C22×D411yescharacteristic
78Dic269C2×D812yes—
79C2×C269—21no—
80C2×C269—82no—
81C2×C269C22×D811yescharacteristic
82D269—22no—
83D269—84no—
84C269#341511yescharacteristic, complemented, Sylow
85#3415—2691nomaximal, Sylow
86#576—2692no—
87#509—2692no—
88#509—2694no—
89#508—2692no—
90#508D26911yescharacteristic, complemented
91C2×D16—2692no—
92C2×D16—2691no—
93#504—2691no—
94H(5,𝔽2)—5382no—
95AGL(1,ℤ/8)—5384no—
96AGL(1,ℤ/8)—5382no—
97#135—2694no—
98#135C2×D26912yescomplemented
99C2×D8—2694no—
100C2×D8—2692no—
101#131—2691no—
102Q32C2×D26911yescharacteristic, complemented
103SD32—2694no—
104SD32C2×D26912yescomplemented
105D16—2694no—
106D16—2692no—
107D16—2692no—
108D16C2×D26911yescharacteristic, complemented
109M32—2691no—
110M32—2692no—
111C2×C16—2692no—
112C2×C16C2×D26911yescharacteristic, complemented
113#41—22no—
114#41—2691no—
115#41—5384no—
116#41—10762no—
117C2×D4—5384no—
118C2×D4—5382no—
119C2×D4—10762no—
120C2×D4—10764no—
121Q16C22×D26912yes—
122SD16—22no—
123SD16—5384no—
124D8—2694no—
125D8—5384no—
126D8—5382no—
127D8C22×D26912yes—
128M16—2691no—
129M16—2692no—
130C2×C8—2692no—
131C2×C8C22×D26911yescharacteristic
132C16—2692no—
133C16C22×D26911yescharacteristic
134C16C22×D26911yescharacteristic
135C23—10764no—
136C23—10762no—
137Q8—22no—
138Q8—2691no—
139D4—22no—
140D4—42no—
141D4—2692no—
142D4—2691no—
143D4—5384no—
144D4—10764no—
145D4—10764no—
146D4—10762no—
147C2×C4—42no—
148C2×C4—2692no—
149C2×C4—10764no—
150C2×C4D4×D26911yescharacteristic
151C8—2692no—
152C8D4×D26911yescharacteristic
153C8C23×D26911yescharacteristic
154C22—42no—
155C22—2692no—
156C22—10764no—
157C22—43042no—
158C22—43044no—
159C22D8×D26911yescharacteristic
160C4—42no—
161C4—2692no—
162C4D8×D26911yescharacteristic
163C4C2×D4×D26911yescharacteristic
164C2—21no—
165C2—82no—
166C2—5382no—
167C2—21524no—
168C2C2×D8×D26911yescharacteristic
subgroupG/H
centerC2C2×D8×D269
commutator, cosocleC8×C269C24
FrattiniC8C23×D269
Fitting#508×C269C2
socleC2×C269C22×D8
hypercenter#508D269
hypocenterC269#3415
LCSG ▷ C8×C269 ▷ C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #31771407C2semidirect
2. #31771042C2semidirect
3. #31771040C2semidirect
4. #31771038C2semidirect
5. C2×D4304C2semidirect
6. D16×D269C2semidirect
7. #31771021C2semidirect
8. #508×C269C2semidirect
12. #22575476C22semidirect
13. C2×D2152C22semidirect
14. D8×D269C22semidirect
16. #22575396C22semidirect
17. D4304C22semidirect
18. D4304C22semidirect
19. #22575394C22semidirect
20. #22575393C22semidirect
21. #22575392C22semidirect
22. C16×D269C22semidirect
23. #135×C269C22semidirect
24. Q32×C269C22semidirect
25. SD32×C269C22semidirect
26. D16×C269C22semidirect
27. C2×C16×C269C22semidirect
84. C269#3415semidirect
90. #508D269semidirect
98. #135C2×D269semidirect
102. Q32C2×D269semidirect
104. SD32C2×D269semidirect
108. D16C2×D269semidirect
112. C2×C16C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714072
#317710422
#317710402
#317710382
C2×D43042
D16×D2692
#317710212
#508×C2692
#3415269