Finite Group Databasebeta

← #149587

Subgroups — #149587

Order 512.

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

normal subgroups only

This group has 390 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#21323C211yesmaximal, characteristic
2C2×#1324C212yesmaximal
3#7359C214yesmaximal, complemented
4C2×#533C411yescharacteristic
5C2×C4×M16C411yescharacteristic
6C2×#382C2211yescharacteristic
7#1993—22no—
8#1991C412yes—
9#1991C2212yes—
10#1736—22no—
11C2×#324—22no—
12#1324—22no—
13#1324C2214yes—
14C22×M16—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—
23C22×C4⋊C4D412yes—
24C22×C42C2×C411yescharacteristic
25C2×C22⋊C8—42no—
26C4×M16—42no—
27C2×#97—21no—
28C2×C4×C8C2×C412yes—
29#382—41no—
30#382C2×C411yescharacteristic
31#382D412yes—
32#382C2311yescharacteristic
33C2×#95—41no—
34#340—24no—
35#340—24no—
36#324—24no—
37#324—42no—
38C22×Q8—41no—
39C23×C4—21no—
40C23×C4—22no—
41C23×C4C22⋊C411yescharacteristic
42C2×M16—24no—
43C2×M16—42no—
44C2×M16—42no—
45C22×C8—22no—
46C22×C8—82no—
47C4⋊Q8—24no—
48C4⋊Q8—42no—
49#125—22no—
50#125—42no—
51#125C2×C814yescomplemented
52C22⋊Q8—44no—
53C22⋊Q8—44no—
54C4×Q8—44no—
55C4×Q8—44no—
56#117—24no—
57#117—42no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—42no—
61C2×C4⋊C4—41no—
62C2×C4⋊C4—42no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4C22⋊C412yes—
65C2×C4⋊C4C2×D412yes—
66C2×C22⋊C4—41no—
67C2×C42—21no—
68C2×C42—22no—
69C2×C42—41no—
70C2×C42C22⋊C411yescharacteristic
71C2×C42C2×C812yes—
72C2×C42M1612yes—
73C2×C42C22×C411yescharacteristic
74C22⋊C8—44no—
75C22⋊C8—82no—
76#97—42no—
77C4×C8—42no—
78#95—41no—
79#95—42no—
80#95—41no—
81C24#9911yescharacteristic
82C2×Q8—44no—
83C2×Q8—84no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—42no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4—82no—
96C22×C4—84no—
97C22×C4C22⋊C812yes—
98C22×C4#9911yescharacteristic
99C22×C4#10012yes—
100C22×C4C2×C22⋊C411yescharacteristic
101M16—48no—
102C2×C8—24no—
103C2×C8—42no—
104C2×C8—42no—
105C2×C8—84no—
106C2×C8—164no—
107C4⋊C4—24no—
108C4⋊C4—42no—
109C4⋊C4—42no—
110C4⋊C4—44no—
111C4⋊C4—44no—
112C4⋊C4—44no—
113C4⋊C4—42no—
114C4⋊C4—82no—
115C22⋊C4—44no—
116C42—22no—
117C42—22no—
118C42—21no—
119C42—21no—
120C42—44no—
121C42C22⋊C812yes—
122C42C22×C811yescharacteristic
123C42C2×M1611yescharacteristic
124C23—42no—
125C23#32312yes—
126C23#35412yes—
127C23#35612yes—
128C23C2×#9911yescharacteristic
129Q8—88no—
130C2×C4—24no—
131C2×C4—24no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—22no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—44no—
144C2×C4—82no—
145C2×C4—82no—
146C2×C4—84no—
147C2×C4—84no—
148C2×C4—82no—
149C2×C4—82no—
150C2×C4—162no—
151C2×C4#32312yes—
152C2×C4C2×C22⋊C811yescharacteristic
153C2×C4C2×#10011yescharacteristic
154C8—44no—
155C8—164no—
156C22—22no—
157C22—22no—
158C22—22no—
159C22—42no—
160C22—42no—
161C22#131914yes—
162C22C2×#32311yescharacteristic
163C22C2×#35411yescharacteristic
164C22C2×#35611yescharacteristic
165C4—21no—
166C4—22no—
167C4—21no—
168C4—41no—
169C4—41no—
170C4—42no—
171C4—84no—
172C4—82no—
173C4—82no—
174C2—21no—
175C2—24no—
176C2—21no—
177C2C2×#131911yescharacteristic
178C2#1360312yes—
subgroupG/H
center, socleC22C2×#354
commutatorC42C22×C8
Frattini, cosocle#382C23
prime radicalC24#99
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#382 ◁ 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
51. #125C2×C8semidirect

Maximal subgroups

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

subgroupindex
#213232
C2×#13242
#73592