Finite Group Databasebeta

← #124647

Subgroups — #124647

Order 512.

262 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 80 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#21728C211yesmaximal, characteristic
2#12896C211yesmaximal, characteristic
3#12896C211yesmaximal, characteristic
4#7135C214yesmaximal
5C2×#499C411yescharacteristic
6C4×#125C412yes—
7#1991C2212yes—
8#1944—21no—
9#1943—21no—
10#1846C411yescharacteristic
11#1776—21no—
12#1776—21no—
13C4×#97C2211yescharacteristic
14#1377—21no—
15#1377—21no—
16#1376C2212yes—
17#1376C2212yes—
18#1375—21no—
19#1375—21no—
20C2×#125C2×C412yes—
21C2×M32—22no—
22C2×M32—22no—
23#499—22no—
24#499C814yes—
25C2×#107—21no—
26C2×#107—21no—
27C2×#106—21no—
28C2×#106—21no—
29C2×#97D411yescharacteristic
30C2×#97D411yescharacteristic
31C2×#97D411yescharacteristic
32C2×#97D411yescharacteristic
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8C2311yescharacteristic
35#389—22no—
36#387—42no—
37#382C2×C412yes—
38C4×C4⋊C4—22no—
39C4×C4⋊C4—42no—
40C43C2×C411yescharacteristic
41#340—22no—
42#340—22no—
43C22×C8C22⋊C411yescharacteristic
44C22×C8C22⋊C411yescharacteristic
45C22×C8C2×D411yescharacteristic
46C22×C8C2×D411yescharacteristic
47#125—22no—
48#125—42no—
49#125C2×C814yes—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—42no—
53C2×C4⋊C4—42no—
54C2×C42—22no—
55C2×C42—42no—
56C2×C42C22⋊C411yescharacteristic
57C2×C42C22⋊C411yescharacteristic
58C2×C42C22×C411yescharacteristic
59M32—42no—
60M32—42no—
61M32—42no—
62M32—42no—
63C2×C16—42no—
64C2×C16—42no—
65#107—24no—
66#107—42no—
67#106—24no—
68#106—42no—
69#97—22no—
70#97—22no—
71#97—41no—
72#97—41no—
73C4×C8—41no—
74C4×C8C2×C812yes—
75C4×C8M1612yes—
76#95—42no—
77#95—42no—
78#95—42no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4#10012yes—
84C22×C4C2×C22⋊C411yescharacteristic
85C2×C8—21no—
86C2×C8—21no—
87C2×C8—21no—
88C2×C8—21no—
89C2×C8—21no—
90C2×C8—21no—
91C2×C8—21no—
92C2×C8—21no—
93C2×C8C22⋊C812yes—
94C2×C8C22⋊C812yes—
95C2×C8#13712yes—
96C2×C8#13712yes—
97C4⋊C4—24no—
98C4⋊C4—44no—
99C4⋊C4—44no—
100C4⋊C4—42no—
101C4⋊C4—42no—
102C4⋊C4—42no—
103C4⋊C4—84no—
104C42—21no—
105C42—21no—
106C42—21no—
107C42—21no—
108C42—21no—
109C42—82no—
110C42—84no—
111C42C22×C811yescharacteristic
112C42C2×M1611yescharacteristic
113C16—82no—
114C16—82no—
115C23C2×#10011yescharacteristic
116C2×C4—22no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—41no—
123C2×C4—44no—
124C2×C4—82no—
125C2×C4—82no—
126C2×C4—82no—
127C2×C4—82no—
128C2×C4—82no—
129C2×C4C2×C22⋊C811yescharacteristic
130C2×C4#41011yescharacteristic
131C2×C4#41911yescharacteristic
132C2×C4#41911yescharacteristic
133C2×C4#42111yescharacteristic
134C2×C4#42111yescharacteristic
135C8—22no—
136C8—22no—
137C8—22no—
138C8—22no—
139C22—22no—
140C22#146011yescharacteristic
141C22#153611yescharacteristic
142C22#155811yescharacteristic
143C4—41no—
144C4—41no—
145C4—42no—
146C4—81no—
147C4—84no—
148C4—162no—
149C4#147911yescharacteristic
150C4#147911yescharacteristic
151C4#151511yescharacteristic
152C4#151511yescharacteristic
153C2—21no—
154C2—21no—
155C2#760511yescharacteristic
156C2#1199412yes—
subgroupG/H
center, socleC22#1558
commutatorC42C22×C8
Frattini, cosocleC2×C4×C8C23
prime radicalC23C2×#100
LCSG ▷ 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
#217282
#128962
#71352