Finite Group Databasebeta

← #46809999

Subgroups — #46809999

Order 34432.

314 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 57 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#31771240C211yesmaximal, characteristic
2#31770892C212yesmaximal, complemented
3#31770349C211yesmaximal, characteristic, complemented
4#31770334C212yesmaximal, complemented
5#492×C269C211yesmaximal, characteristic, complemented
6C2×D4×D269—22no—
7#22575424—22no—
8#22575421—22no—
9C4×D1076C2212yes—
10#22575384—22no—
11C4⋊#14524130C2212yescomplemented
12#22575373—22no—
13C2×SD16×C269—22no—
14C4⋊Q8×C269C2211yescharacteristic, complemented
15#127×C269C2211yescharacteristic
16C4×C8×C269C2211yescharacteristic, complemented
17C23×D269—22no—
18C2×D1076D412yes—
19C2×C22⋊D269—24no—
20D4×D269—44no—
21D4×D269—44no—
22C2×C4×D269—22no—
23C2×#14524130—22no—
24C4⋊Dic269D412yes—
25C22⋊Dic269—22no—
26#16864379—22no—
27C2×Q8×C269—21no—
28C2×D4×C269—22no—
29C2×D4×C269D412yes—
30SD16×C269—44no—
31C2×C8×C269—21no—
32C4⋊C4×C269—22no—
33C42×C269C2311yescharacteristic
34C22×D269—22no—
35C22×D269—48no—
36C22×D269—82no—
37D1076—22no—
38D1076SD1614yescomplemented
39C22⋊D269—48no—
40C2×Dic269—22no—
41C4×D269—42no—
42C4×D269—42no—
43#14524130—42no—
44C23×C269—22no—
45Q8×C269—42no—
46D4×C269—42no—
47D4×C269—42no—
48D4×C269—42no—
49C2×C4×C269—41no—
50C2×C4×C269C2×D412yes—
51C2×C4×C269C2×D411yescharacteristic
52C8×C269—22no—
53C22×C269—44no—
54C22×C269—42no—
55C22×C269#12011yescharacteristic
56C2×D269—24no—
57C2×D269—42no—
58C2×D269—42no—
59C2×D269—88no—
60C4×C269—21no—
61C4×C269—81no—
62C4×C269C2×SD1612yes—
63C4×C269AGL(1,ℤ/8)12yes—
64Dic269—42no—
65C2×C269—82no—
66C2×C269#45012yes—
67C2×C269#45311yescharacteristic
68D269—44no—
69C269#162911yescharacteristic, complemented, Sylow
70#1629—2691nomaximal, Sylow
71D42—2691no—
72#492D26911yescharacteristic, complemented
73#476—2692no—
74#331—2691no—
75D4⋊C8—2692no—
76C22×D4—5382no—
77C2×SD16—22no—
78C4⋊Q8C2×D26911yescharacteristic, complemented
79#127C2×D26911yescharacteristic
80#121—5382no—
81#120—5382no—
82C4×D4—2692no—
83#106—5382no—
84C4⋊C8—2692no—
85#102—5382no—
86C4×C8C2×D26911yescharacteristic, complemented
87C24—5382no—
88C2×Q8—21no—
89C2×D4—22no—
90C2×D4—2692no—
91C2×D4—5384no—
92C2×D4—10764no—
93C2×D4—10764no—
94C2×D4C22⋊D26912yes—
95C22×C4—5382no—
96SD16—44no—
97C2×C8—21no—
98C2×C8—5382no—
99C4⋊C4—22no—
100C4⋊C4—2692no—
101C22⋊C4—5382no—
102C22⋊C4—5382no—
103C42C22×D26911yescharacteristic
104C23—22no—
105C23—5382no—
106C23—10768no—
107C23—21522no—
108Q8—42no—
109D4—42no—
110D4—42no—
111D4—42no—
112D4—2694no—
113D4—5382no—
114D4—10768no—
115C2×C4—41no—
116C2×C4—5382no—
117C2×C4—10762no—
118C2×C4—10762no—
119C2×C4D4×D26912yes—
120C2×C4C2×C22⋊D26911yescharacteristic
121C8—22no—
122C8—10762no—
123C22—44no—
124C22—42no—
125C22—5384no—
126C22—10762no—
127C22—10762no—
128C22—21528no—
129C22#2257542111yescharacteristic
130C4—21no—
131C4—81no—
132C4—10762no—
133C4SD16×D26912yes—
134C4#2257548012yes—
135C2—82no—
136C2—10764no—
137C2#3177075412yes—
138C2#3177077211yescharacteristic
subgroupG/H
centerC22#22575421
commutator, cosocleC42×C269C23
FrattiniC42C22×D269
Fitting#492×C269C2
socleC22×C269#120
prime radical#31771240C2
hypercenter#492D269
hypocenterC269#1629
LCSG ▷ C42×C269 ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
2. #31770892C2semidirect
3. #31770349C2semidirect
4. #31770334C2semidirect
5. #492×C269C2semidirect
11. C4⋊#14524130C22semidirect
14. C4⋊Q8×C269C22semidirect
16. C4×C8×C269C22semidirect
38. D1076SD16semidirect
69. C269#1629semidirect
72. #492D269semidirect
78. C4⋊Q8C2×D269semidirect
86. C4×C8C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317712402
#317708922
#317703492
#317703342
#492×C2692
#1629269