Finite Group Databasebeta

← #149586

Subgroups — #149586

Order 512.

386 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 82 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#21318C211yesmaximal, characteristic
2#11447C212yesmaximal
3#7359C214yesmaximal, complemented
4C2×#533C411yescharacteristic
5C22×C4×C8C411yescharacteristic
6#2285C2211yescharacteristic
7#1991C412yes—
8#1991C2212yes—
9#1990—22no—
10#1735—22no—
11#1458—22no—
12#1325—22no—
13#1324C2214yes—
14C23×C8—21no—
15#533—22no—
16#533—42no—
17#533C814yescomplemented
18C2×C4⋊Q8—21no—
19C2×#125C2×C412yes—
20C2×C22⋊Q8—41no—
21C2×C4×Q8—41no—
22C2×#117—21no—
23C2×#117D412yes—
24C22×C42C2×C411yescharacteristic
25#431—42no—
26C2×C4×C8—21no—
27C2×C4×C8—42no—
28C2×C4×C8C2×C412yes—
29#384D412yes—
30#382C2×C411yescharacteristic
31#382C2311yescharacteristic
32#380—41no—
33C4×C4⋊C4—41no—
34#340—24no—
35#340—24no—
36#324—24no—
37#323—42no—
38C22×Q8—41no—
39C23×C4—21no—
40C23×C4C22⋊C411yescharacteristic
41C22×C8—24no—
42C22×C8—22no—
43C22×C8—41no—
44C22×C8—41no—
45C22×C8—42no—
46C4⋊Q8—24no—
47C4⋊Q8—42no—
48#125—22no—
49#125—42no—
50#125C2×C814yes—
51C22⋊Q8—44no—
52C22⋊Q8—44no—
53C4×Q8—44no—
54C4×Q8—44no—
55#117—24no—
56#117—42no—
57#117—42no—
58#117—42no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—42no—
61C2×C4⋊C4—41no—
62C2×C4⋊C4—42no—
63C2×C4⋊C4C22⋊C412yes—
64C2×C4⋊C4C2×D412yes—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—41no—
67C2×C42—21no—
68C2×C42—22no—
69C2×C42—22no—
70C2×C42—41no—
71C2×C42C22⋊C411yescharacteristic
72C2×C42C2×C812yes—
73C2×C42M1612yes—
74C2×C42C22×C411yescharacteristic
75C4⋊C8—82no—
76C22⋊C8—44no—
77C4×C8—42no—
78C4×C8—42no—
79C4×C8—82no—
80#95—41no—
81#95—41no—
82C24#9911yescharacteristic
83C2×Q8—44no—
84C2×Q8—84no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—44no—
95C22×C4—42no—
96C22×C4C22⋊C812yes—
97C22×C4#9911yescharacteristic
98C22×C4#10012yes—
99C22×C4C2×C22⋊C411yescharacteristic
100C2×C8—24no—
101C2×C8—42no—
102C2×C8—44no—
103C2×C8—44no—
104C2×C8—42no—
105C2×C8—84no—
106C4⋊C4—24no—
107C4⋊C4—44no—
108C4⋊C4—42no—
109C4⋊C4—42no—
110C4⋊C4—44no—
111C4⋊C4—44no—
112C4⋊C4—42no—
113C4⋊C4—84no—
114C22⋊C4—44no—
115C22⋊C4—82no—
116C42—22no—
117C42—22no—
118C42—21no—
119C42—21no—
120C42—44no—
121C42—42no—
122C42—42no—
123C42C22⋊C812yes—
124C42C22×C811yescharacteristic
125C42C2×M1611yescharacteristic
126C23—21no—
127C23—21no—
128C23—42no—
129C23#32312yes—
130C23C2×#9911yescharacteristic
131Q8—88no—
132C2×C4—24no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—44no—
143C2×C4—44no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—44no—
148C2×C4—84no—
149C2×C4—82no—
150C2×C4—82no—
151C2×C4#32312yes—
152C2×C4#35411yescharacteristic
153C2×C4#35411yescharacteristic
154C2×C4#35612yes—
155C2×C4C2×C22⋊C811yescharacteristic
156C2×C4C2×#10011yescharacteristic
157C8—44no—
158C8—164no—
159C22—22no—
160C22—22no—
161C22—42no—
162C22—42no—
163C22—42no—
164C22C2×#32311yescharacteristic
165C22C2×#35411yescharacteristic
166C22C2×#35611yescharacteristic
167C4—24no—
168C4—21no—
169C4—21no—
170C4—41no—
171C4—41no—
172C4—84no—
173C4—82no—
174C4—82no—
175C4#131912yes—
176C4#131912yes—
177C2—21no—
178C2—21no—
179C2—42no—
180C2C2×#131911yescharacteristic
181C2#1360212yes—
subgroupG/H
centerC2×C4#354
commutatorC42C22×C8
Frattini, cosocle#382C23
socleC22C2×#354
prime radicalC24#99
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G
LECSG ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7359C2semidirect
17. #533C8semidirect

Maximal subgroups

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

subgroupindex
#213182
#114472
#73592