Finite Group Databasebeta

← #125438

Subgroups — #125438

Order 512.

246 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 66 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#33295C211yesmaximal, characteristic
2#12896C211yesmaximal, characteristic
3#12595C211yesmaximal, characteristic
4#7135C214yesmaximal
5#2859C411yescharacteristic
6#2859C411yescharacteristic
7C2×#347—21no—
8#1992—21no—
9#1992—21no—
10#1991C412yes—
11#1991C2212yes—
12#1836—21no—
13#1836—21no—
14C4×#97C2211yescharacteristic
15#1377—21no—
16#1377—21no—
17#1376C2212yes—
18#1376C2212yes—
19#1375—21no—
20C2×#125—21no—
21C2×#125C2×C412yes—
22C2×M32—22no—
23C2×M32—22no—
24C2×#97D411yescharacteristic
25C2×#97D411yescharacteristic
26C2×#97D411yescharacteristic
27C2×#97D411yescharacteristic
28C2×C4×C8C2×C411yescharacteristic
29C2×C4×C8C2311yescharacteristic
30#389—42no—
31#389—42no—
32#384—21no—
33#384—21no—
34#382—21no—
35#382C2×C412yes—
36C43C2×C411yescharacteristic
37#347—24no—
38#340—22no—
39#340—22no—
40#340—22no—
41#340—22no—
42C22×C8C22⋊C411yescharacteristic
43C22×C8C22⋊C411yescharacteristic
44C22×C8C2×D411yescharacteristic
45C22×C8C2×D411yescharacteristic
46#125—24no—
47#125—42no—
48#125—42no—
49#125—42no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—42no—
53C2×C4⋊C4—42no—
54C2×C4⋊C4—42no—
55C2×C4⋊C4—42no—
56C2×C42—22no—
57C2×C42C22⋊C411yescharacteristic
58C2×C42C22⋊C411yescharacteristic
59C2×C42C22×C411yescharacteristic
60M32—24no—
61M32—42no—
62M32—42no—
63M32—42no—
64C2×C16—42no—
65C2×C16—42no—
66#97—22no—
67#97—22no—
68#97—22no—
69#97—41no—
70C4×C8—22no—
71C4×C8—41no—
72#95—42no—
73#95—42no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—42no—
77C22×C4—42no—
78C22×C4#10012yes—
79C22×C4C2×C22⋊C411yescharacteristic
80C2×C8—21no—
81C2×C8—21no—
82C2×C8—21no—
83C2×C8—21no—
84C2×C8—21no—
85C2×C8—21no—
86C2×C8—21no—
87C2×C8—21no—
88C2×C8—21no—
89C2×C8—21no—
90C2×C8#13712yes—
91C2×C8#13712yes—
92C4⋊C4—42no—
93C4⋊C4—42no—
94C4⋊C4—42no—
95C4⋊C4—42no—
96C4⋊C4—84no—
97C4⋊C4—84no—
98C4⋊C4—84no—
99C4⋊C4—84no—
100C42—21no—
101C42—21no—
102C42—21no—
103C42—21no—
104C42—82no—
105C42#10412yes—
106C42#13112yes—
107C16—44no—
108C16—82no—
109C23C2×#10011yescharacteristic
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—22no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—84no—
117C2×C4—84no—
118C2×C4—82no—
119C2×C4—82no—
120C2×C4#40811yescharacteristic
121C2×C4#41011yescharacteristic
122C2×C4#41911yescharacteristic
123C2×C4#41911yescharacteristic
124C2×C4C2×#10411yescharacteristic
125C2×C4#42111yescharacteristic
126C8—22no—
127C8—22no—
128C8—41no—
129C8—41no—
130C22—22no—
131C22#146911yescharacteristic
132C22#152711yescharacteristic
133C22#155811yescharacteristic
134C4—42no—
135C4—42no—
136C4—42no—
137C4—162no—
138C4—162no—
139C4#148511yescharacteristic
140C4#149711yescharacteristic
141C4#150611yescharacteristic
142C4#151511yescharacteristic
143C2—21no—
144C2—21no—
145C2#770211yescharacteristic
146C2#1199412yes—
subgroupG/H
center, socleC22#1558
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23C2×#100
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332952
#128962
#125952
#71352