Finite Group Databasebeta

← #149883

Subgroups — #149883

Order 512.

204 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 54 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#22258C211yesmaximal, characteristic
2#11540C212yesmaximal
3#7368C212yesmaximal
4#7360C212yesmaximal
5#3160C411yescharacteristic
6#2961C411yescharacteristic
7#2961C2211yescharacteristic
8#2166—22no—
9#1995C411yescharacteristic
10#1995C2211yescharacteristic
11#1922—22no—
12#1922—22no—
13#1837C2211yescharacteristic
14#1746C411yescharacteristic
15#1328—22no—
16#1327C2214yes—
17#533C2×C411yescharacteristic
18#533C2×C411yescharacteristic
19#500—21no—
20#499—22no—
21#498—21no—
22#446—42no—
23#446—42no—
24#427—22no—
25#424—21no—
26#424D412yes—
27#423C2×C411yescharacteristic
28#423C2311yescharacteristic
29#405—21no—
30#405D412yes—
31C2×#97C2×C411yescharacteristic
32C2×C4×C8C2×C411yescharacteristic
33#376C2×C411yescharacteristic
34#364—42no—
35#349—24no—
36#347—42no—
37#341—22no—
38#339—22no—
39#339—22no—
40#336—21no—
41#336—21no—
42C2×M16—21no—
43C2×M16C2×D412yes—
44C22×C8—21no—
45C22×C8C22⋊C411yescharacteristic
46C22×C8C22⋊C411yescharacteristic
47C4⋊Q8—21no—
48C4⋊Q8—21no—
49#125—21no—
50#125M1612yes—
51C22⋊Q8—41no—
52C22⋊Q8—41no—
53C4×Q8—41no—
54C4×Q8—41no—
55#117—21no—
56#117C22⋊C412yes—
57C2×C42—21no—
58C2×C42C22×C411yescharacteristic
59M32—44no—
60#107—24no—
61#106—24no—
62C4⋊C8—41no—
63C4⋊C8—42no—
64C4⋊C8—41no—
65C4⋊C8—42no—
66#97—22no—
67#97—22no—
68#97—41no—
69C4×C8—22no—
70C4×C8—21no—
71C4×C8—41no—
72C4×C8M1611yescharacteristic
73C4×C8M1611yescharacteristic
74#95—41no—
75C2×Q8—41no—
76C2×Q8—41no—
77C22×C4—21no—
78C22×C4—41no—
79C22×C4C2×C22⋊C411yescharacteristic
80M16—42no—
81M16—42no—
82C2×C8—21no—
83C2×C8—21no—
84C2×C8—22no—
85C2×C8—21no—
86C2×C8—41no—
87C2×C8—41no—
88C2×C8—41no—
89C2×C8—41no—
90C2×C8—41no—
91C2×C8#9912yes—
92C4⋊C4—22no—
93C4⋊C4—22no—
94C4⋊C4—42no—
95C4⋊C4—42no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C22⋊C4—22no—
99C22⋊C4—41no—
100C42—21no—
101C42—22no—
102C42—41no—
103C42—41no—
104C42—41no—
105C42C2×M1611yescharacteristic
106C42C2×M1611yescharacteristic
107C16—84no—
108C23#41311yescharacteristic
109Q8—161no—
110Q8—161no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—81no—
118C2×C4—81no—
119C2×C4—82no—
120C2×C4#40711yescharacteristic
121C2×C4C2×#9911yescharacteristic
122C8—22no—
123C8—41no—
124C8—41no—
125C8—42no—
126C8—41no—
127C8—81no—
128C22—21no—
129C22—41no—
130C22#146511yescharacteristic
131C4—21no—
132C4—41no—
133C4—41no—
134C4—82no—
135C4—82no—
136C4—161no—
137C4#212911yescharacteristic
138C4#213311yescharacteristic
139C2—21no—
140C2—41no—
141C2#1153611yescharacteristic
subgroupG/H
centerC4#2129
commutatorC2×C42C22×C4
Frattini, cosocle#423C23
socleC2#11536
prime radicalC23#413
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C8 ◁ #2961 ◁ G
LECSG ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#222582
#115402
#73682
#73602