Finite Group Databasebeta

← #125463

Subgroups — #125463

Order 512.

290 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 68 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#33214C211yesmaximal, characteristic
2#12850C211yesmaximal, characteristic
3#12606C211yesmaximal, characteristic
4#7130C214yesmaximal, complemented
5C2×#499C411yescharacteristic
6C2×#405C2211yescharacteristic
7#2629C412yes—
8C2×#349—21no—
9#2112—21no—
10C2×#347—21no—
11#1992C2212yes—
12#1944—21no—
13#1941—21no—
14#1844C411yescharacteristic
15#1809—21no—
16#1807—21no—
17#1376—21no—
18#1375C2212yes—
19#1375C2212yes—
20C22×M16D411yescharacteristic
21C23×C8D411yescharacteristic
22C2×#125C2×C412yes—
23C2×#117C2×C411yescharacteristic
24C2×M32—22no—
25C2×M32—22no—
26#499—42no—
27C2×#107—21no—
28C2×#106—21no—
29C2×#106—21no—
30C2×#106—21no—
31#405—42no—
32C2×#97D411yescharacteristic
33C2×#97C2311yescharacteristic
34C2×C4×C8C2×C411yescharacteristic
35C2×C4×C8D411yescharacteristic
36#398—42no—
37#396—42no—
38#389—22no—
39#384C2×C412yes—
40#380—22no—
41#349—42no—
42C22⋊C16—24no—
43#347—42no—
44#340—22no—
45#340—22no—
46C23×C4C22⋊C411yescharacteristic
47C2×M16—22no—
48C2×M16—22no—
49C2×M16—41no—
50C22×C8—22no—
51C22×C8—22no—
52C22×C8—41no—
53C22×C8C22⋊C411yescharacteristic
54C22×C8C22⋊C411yescharacteristic
55C22×C8C2×D411yescharacteristic
56C22×C8C2×D411yescharacteristic
57#125—24no—
58#125—42no—
59#117—22no—
60#117—41no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4—42no—
66C2×C4⋊C4—42no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—42no—
69C2×C42C22⋊C411yescharacteristic
70C2×C42C22×C411yescharacteristic
71M32—42no—
72M32—42no—
73M32—42no—
74M32—42no—
75C2×C16—24no—
76C2×C16—42no—
77#107—42no—
78#106—42no—
79#106—42no—
80#106—42no—
81#97—22no—
82#97—41no—
83C4×C8—22no—
84C4×C8—41no—
85#95—42no—
86#95—42no—
87C24#10011yescharacteristic
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4#10011yescharacteristic
99C22×C4C2×C22⋊C411yescharacteristic
100M16—44no—
101C2×C8—21no—
102C2×C8—21no—
103C2×C8—21no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—44no—
110C2×C8#9912yes—
111C2×C8#10412yes—
112C2×C8#13712yes—
113C2×C8#13712yes—
114C4⋊C4—42no—
115C4⋊C4—42no—
116C4⋊C4—42no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—84no—
120C4⋊C4—162no—
121C4⋊C4—162no—
122C22⋊C4—42no—
123C22⋊C4—162no—
124C42—21no—
125C42—21no—
126C42—21no—
127C42#13112yes—
128C16—44no—
129C16—82no—
130C23—21no—
131C23—21no—
132C23—21no—
133C23—42no—
134C23C2×#10011yescharacteristic
135C2×C4—21no—
136C2×C4—22no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—42no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4—41no—
146C2×C4—82no—
147C2×C4—82no—
148C2×C4—82no—
149C2×C4—82no—
150C2×C4—82no—
151C2×C4—82no—
152C2×C4#40811yescharacteristic
153C2×C4C2×#9911yescharacteristic
154C2×C4#41911yescharacteristic
155C2×C4#41911yescharacteristic
156C2×C4C2×#10411yescharacteristic
157C2×C4#42111yescharacteristic
158C8—22no—
159C8—22no—
160C8—22no—
161C8—22no—
162C22—22no—
163C22—42no—
164C22—41no—
165C22—42no—
166C22—81no—
167C22#147111yescharacteristic
168C22#153011yescharacteristic
169C22#155811yescharacteristic
170C4—42no—
171C4—41no—
172C4—81no—
173C4—162no—
174C4—162no—
175C4#148511yescharacteristic
176C4#149711yescharacteristic
177C4#150811yescharacteristic
178C4#151111yescharacteristic
179C2—21no—
180C2—21no—
181C2—81no—
182C2#770711yescharacteristic
183C2#1191212yes—
subgroupG/H
center, socleC22#1530
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
prime radicalC24#100
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7130C2semidirect

Maximal subgroups

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

subgroupindex
#332142
#128502
#126062
#71302