Finite Group Databasebeta

← #46811415

Subgroups — #46811415

Order 34432.

362 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 85 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#31771172C211yesmaximal, characteristic
2#31771080C211yesmaximal, characteristic, complemented
3#31770621C212yesmaximal, complemented
4#31770362C212yesmaximal, complemented
5C2×#100×C269C211yesmaximal, characteristic, complemented
6C2×#16864402C2211yescharacteristic
7#22575449C412yescomplemented
8#22575440C412yescomplemented
9#22575424—22no—
10#22575422—22no—
11D4×Dic269—24no—
12#22575412C2211yescharacteristic
13#22575412C2212yes—
14C2×C22⋊Dic269—22no—
15#22575379—24no—
16C22×D4×C269C2211yescharacteristic
17C2×M16×C269C2212yes—
18#100×C269—22no—
19#16864402—22no—
20#16864402—22no—
21C2×Dic538C2×C411yescharacteristic, complemented
22C2×D1076C2×C411yescharacteristic, complemented
23C2×C22⋊D269—24no—
24C2×C22⋊D269—21no—
25C22×Dic269—24no—
26C2×C4×D269—21no—
27C4⋊Dic269—22no—
28C4⋊Dic269—21no—
29C22⋊Dic269—24no—
30C22⋊Dic269—24no—
31C22⋊Dic269—22no—
32C22⋊Dic269—21no—
33C4×Dic269C2×C412yescomplemented
34C4×Dic269D414yescomplemented
35C24×C269D412yes—
36C2×D4×C269—22no—
37C2×D4×C269—22no—
38C2×D4×C269C2×C412yes—
39C2×D4×C269D412yes—
40C22×C4×C269C2311yescharacteristic
41M16×C269—22no—
42M16×C269—22no—
43C2×C8×C269—22no—
44C22×D269—21no—
45Dic538—22no—
46D1076—22no—
47C22⋊D269—41no—
48C22⋊D269—41no—
49C22⋊D269—84no—
50C2×Dic269—22no—
51C2×Dic269—21no—
52C2×Dic269—24no—
53C2×Dic269—84no—
54C4×D269—42no—
55C23×C269—22no—
56C23×C269—22no—
57C23×C269—82no—
58C23×C269C22⋊C414yes—
59C23×C269C2×D411yescharacteristic
60D4×C269—24no—
61D4×C269—24no—
62C2×C4×C269C22×C411yescharacteristic
63C2×C4×C269C2×D412yes—
64C2×C4×C269C2×D411yescharacteristic
65C2×C4×C269#4112yes—
66C8×C269—42no—
67C22×C269—24no—
68C22×C269—41no—
69C22×C269—44no—
70C22×C269—44no—
71C22×C269—82no—
72C22×C269C2×C22⋊C411yescharacteristic
73C22×C269#12011yescharacteristic
74C22×C269#12311yescharacteristic
75C2×D269—42no—
76C2×D269—41no—
77C4×C269C4×D412yes—
78C4×C269#12112yes—
79Dic269—44no—
80Dic269—42no—
81Dic269—41no—
82C2×C269—21no—
83C2×C269—21no—
84C2×C269—21no—
85C2×C269—44no—
86C2×C269#38611yescharacteristic
87D269—81no—
88C269#191111yescharacteristic, complemented, Sylow
89#1911—2691nomaximal, Sylow
90#535—2691no—
91#518—2691no—
92#421—2692no—
93C2×#100D26911yescharacteristic, complemented
94H(3,ℤ/4)—2692no—
95C2×#41—2691no—
96C22×D4C2×D26911yescharacteristic
97C2×M16C2×D26912yes—
98#127—2692no—
99#124—2692no—
100#121—5382no—
101#120—5382no—
102C4×D4—5384no—
103#117—2692no—
104#117—2691no—
105C2×C22⋊C4—5382no—
106#104—5384no—
107#100—22no—
108C24D107612yes—
109#41—5382no—
110#41—5382no—
111C2×Q8—2691no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—2691no—
115C2×D4—5381no—
116C2×D4—5384no—
117C2×D4C4×D26912yes—
118C2×D4C22⋊D26912yes—
119C22×C4—5384no—
120C22×C4—5381no—
121C22×C4C22×D26911yescharacteristic
122M16—22no—
123M16—22no—
124C2×C8—22no—
125C4⋊C4—5382no—
126C4⋊C4—5381no—
127C22⋊C4—5384no—
128C22⋊C4—5384no—
129C22⋊C4—5382no—
130C22⋊C4—5381no—
131C42—2692no—
132C42—2694no—
133C23—22no—
134C23—22no—
135C23—82no—
136C23—5381no—
137C23#1686437914yes—
138C23C2×D107611yescharacteristic
139Q8—5382no—
140D4—24no—
141D4—24no—
142D4—5382no—
143D4—10761no—
144D4—10761no—
145D4—21524no—
146C2×C4—5382no—
147C2×C4—5381no—
148C2×C4—5384no—
149C2×C4—10762no—
150C2×C4—21524no—
151C2×C4C2×C4×D26911yescharacteristic
152C2×C4D4×D26912yes—
153C2×C4C2×C22⋊D26911yescharacteristic
154C2×C4#1686440112yes—
155C8—42no—
156C22—24no—
157C22—41no—
158C22—44no—
159C22—44no—
160C22—82no—
161C22—10762no—
162C22—10761no—
163C22C2×#1686437911yescharacteristic
164C22#2257542311yescharacteristic
165C22#2257543711yescharacteristic
166C4—10764no—
167C4—10762no—
168C4—10761no—
169C4#2257541312yes—
170C4#2257542512yes—
171C2—21no—
172C2—21no—
173C2—21no—
174C2—44no—
175C2—21521no—
176C2#3177048211yescharacteristic
subgroupG/H
centerC2#31770482
commutatorC2×C4×C269C22×C4
FrattiniC22×C4C22×D269
FittingC2×#100×C269C2
socleC2×C269#386
prime radical#31771172C2
cosocleC22×C4×C269C23
hypercenterC2×#100D269
hypocenterC269#1911
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. #31771080C2semidirect
3. #31770621C2semidirect
4. #31770362C2semidirect
5. C2×#100×C269C2semidirect
7. #22575449C4semidirect
8. #22575440C4semidirect
21. C2×Dic538C2×C4semidirect
22. C2×D1076C2×C4semidirect
33. C4×Dic269C2×C4semidirect
34. C4×Dic269D4semidirect
88. C269#1911semidirect
93. C2×#100D269semidirect

Maximal subgroups

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

subgroupindex
#317711722
#317710802
#317706212
#317703622
C2×#100×C2692
#1911269