Finite Group Databasebeta

← #149884

Subgroups — #149884

Order 512.

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

normal subgroups only

This group has 208 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#22259C211yesmaximal, characteristic
2#11540C212yesmaximal
3#7369C212yesmaximal
4#7361C212yesmaximal
5#3162C411yescharacteristic
6#2961C411yescharacteristic
7#2961C2211yescharacteristic
8#2166—22no—
9#1994C411yescharacteristic
10#1994C2211yescharacteristic
11#1922—22no—
12#1922—22no—
13#1838C2211yescharacteristic
14#1745C411yescharacteristic
15#1328—22no—
16#1327C2214yes—
17#533C2×C411yescharacteristic
18#533C2×C411yescharacteristic
19#501—22no—
20#501—22no—
21#446—42no—
22#446—42no—
23#427—22no—
24#424—21no—
25#424D412yes—
26#423C2×C411yescharacteristic
27#423C2311yescharacteristic
28#405—21no—
29#405D412yes—
30C2×#97C2×C411yescharacteristic
31C2×#97C2×C411yescharacteristic
32C43C2×C411yescharacteristic
33#364—42no—
34#349—24no—
35#347—42no—
36#341—22no—
37#339—22no—
38#339—22no—
39#336—21no—
40#336—21no—
41C2×M16—21no—
42C2×M16C2×D412yes—
43C22×C8—21no—
44C22×C8C22⋊C411yescharacteristic
45C22×C8C22⋊C411yescharacteristic
46C4⋊Q8—21no—
47C4⋊Q8—21no—
48#125—21no—
49#125M1612yes—
50C22⋊Q8—41no—
51C22⋊Q8—41no—
52C4×Q8—41no—
53C4×Q8—41no—
54#117—21no—
55#117C22⋊C412yes—
56C2×C42—21no—
57C2×C42—41no—
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—21no—
69#97—41no—
70#97M1612yes—
71C4×C8—22no—
72C4×C8—41no—
73C2×Q8—41no—
74C2×Q8—41no—
75C22×C4—21no—
76C22×C4—41no—
77C22×C4C2×C22⋊C411yescharacteristic
78M16—42no—
79M16—42no—
80C2×C8—22no—
81C2×C8—22no—
82C2×C8—21no—
83C2×C8—41no—
84C2×C8—41no—
85C2×C8—41no—
86C2×C8—41no—
87C2×C8—41no—
88C2×C8#9912yes—
89C4⋊C4—22no—
90C4⋊C4—22no—
91C4⋊C4—42no—
92C4⋊C4—42no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C22⋊C4—22no—
96C22⋊C4—41no—
97C42—21no—
98C42—22no—
99C42—41no—
100C42—41no—
101C42—41no—
102C42—82no—
103C42C2×M1611yescharacteristic
104C42C2×M1611yescharacteristic
105C16—84no—
106C23#41311yescharacteristic
107Q8—161no—
108Q8—161no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—81no—
117C2×C4—81no—
118C2×C4—81no—
119C2×C4#40711yescharacteristic
120C2×C4C2×#9911yescharacteristic
121C8—22no—
122C8—42no—
123C8—42no—
124C8—41no—
125C8—81no—
126C22—21no—
127C22—41no—
128C22#146511yescharacteristic
129C4—21no—
130C4—41no—
131C4—41no—
132C4—82no—
133C4—82no—
134C4—82no—
135C4#212911yescharacteristic
136C4#213311yescharacteristic
137C2—21no—
138C2—41no—
139C2#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
#222592
#115402
#73692
#73612