Finite Group Databasebeta

← #149526

Subgroups — #149526

Order 512.

365 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 96 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#33333C211yesmaximal, characteristic
2C2×#1400C212yesmaximal, complemented
3#7337C214yesmaximal, complemented
4C22×#131C411yescharacteristic
5C22×M32C411yescharacteristic
6C22×M32C2211yescharacteristic
7C2×M64—22no—
8#2113—22no—
9#2113C412yes—
10#2113C2212yes—
11#2112—22no—
12#1400—22no—
13#1400C2214yescomplemented
14C22×#41C811yescharacteristic
15C2×#131—21no—
16C2×#131—21no—
17C2×#131—42no—
18C2×#131—42no—
19C2×#131C812yes—
20C2×#131C2×C412yes—
21C22×M16—21no—
22C22×M16C811yescharacteristic
23C23×C8—21no—
24C23×C8C2×C411yescharacteristic
25C2×M32—21no—
26C2×M32—21no—
27C2×M32—42no—
28C2×M32—42no—
29C2×M32C2×C411yescharacteristic
30C2×M32D412yes—
31C2×M32C2311yescharacteristic
32C22×C16C2×C412yes—
33C22×C16D412yes—
34M64—24no—
35M64—42no—
36C2×C32—42no—
37C22⋊C16—24no—
38C22⋊C16—24no—
39C2×#41—21no—
40C2×#41—21no—
41C2×#41—42no—
42C2×#41—42no—
43C2×#41C1612yes—
44C2×#41C2×C812yes—
45C22×Q8C1611yescharacteristic
46C22×D4—21no—
47C22×D4C1611yescharacteristic
48C23×C4—21no—
49C23×C4C2×C811yescharacteristic
50#131—44no—
51#131—44no—
52#131—42no—
53#131—42no—
54#131—42no—
55#131—42no—
56C2×M16—21no—
57C2×M16—22no—
58C2×M16—22no—
59C2×M16—44no—
60C2×M16—42no—
61C2×M16C2×C812yes—
62C22×C8—22no—
63C22×C8—22no—
64C22×C8—21no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—42no—
68C22×C8—42no—
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22⋊C411yescharacteristic
71C22×C8C2×C811yescharacteristic
72C22×C8M1611yescharacteristic
73C22×C8M1611yescharacteristic
74C22×C8C22×C411yescharacteristic
75M32—41no—
76M32—42no—
77M32—41no—
78M32—42no—
79M32—42no—
80C2×C16—22no—
81C2×C16—22no—
82C2×C16—22no—
83C2×C16—42no—
84C2×C16C22⋊C412yes—
85C2×C16C2×D412yes—
86C32—44no—
87C24—21no—
88C24M3211yescharacteristic
89#41—44no—
90#41—44no—
91#41—42no—
92#41—42no—
93#41—42no—
94#41—42no—
95C2×Q8—21no—
96C2×Q8—42no—
97C2×Q8C2×C1612yes—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—21no—
101C2×D4—42no—
102C2×D4—44no—
103C2×D4C2×C1612yes—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4C22⋊C811yescharacteristic
112C22×C4C22⋊C811yescharacteristic
113C22×C4C2×C1611yescharacteristic
114C22×C4C2×C1611yescharacteristic
115C22×C4M3211yescharacteristic
116C22×C4C22×C811yescharacteristic
117M16—44no—
118M16—42no—
119M16—44no—
120M16—42no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—22no—
129C2×C8—44no—
130C2×C8—42no—
131C2×C8—42no—
132C2×C8—42no—
133C2×C8C22⋊C811yescharacteristic
134C2×C8C22⋊C811yescharacteristic
135C2×C8#9912yes—
136C2×C8#10012yes—
137C2×C8C2×C22⋊C411yescharacteristic
138C2×C8C2×M1611yescharacteristic
139C16—22no—
140C16—22no—
141C16—42no—
142C23—22no—
143C23—22no—
144C23—42no—
145C23—81no—
146C23—82no—
147C23C22⋊C1611yescharacteristic
148C23C22⋊C1611yescharacteristic
149C23C2×M3211yescharacteristic
150Q8—42no—
151Q8—42no—
152D4—42no—
153D4—44no—
154D4—44no—
155D4—42no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—44no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4#32312yes—
169C2×C4C22⋊C1611yescharacteristic
170C2×C4C22⋊C1611yescharacteristic
171C2×C4#34912yes—
172C2×C4C2×C22⋊C811yescharacteristic
173C2×C4C22×C1611yescharacteristic
174C8—21no—
175C8—21no—
176C8—42no—
177C8#32312yes—
178C8C2×#9911yescharacteristic
179C8C2×#10011yescharacteristic
180C22—21no—
181C22—21no—
182C22—41no—
183C22—41no—
184C22—41no—
185C22—44no—
186C22—42no—
187C22—82no—
188C22—84no—
189C22#131412yes—
190C22C2×C22⋊C1611yescharacteristic
191C4—21no—
192C4—21no—
193C4—42no—
194C4#131412yes—
195C4C2×#32311yescharacteristic
196C4C2×#34911yescharacteristic
197C2—21no—
198C2—22no—
199C2—41no—
200C2—41no—
201C2—82no—
202C2C2×#131411yescharacteristic
subgroupG/H
centerC8C2×#99
commutatorC2×C4C22×C16
Frattini, cosocleC2×M32C23
socleC2C2×#1314
prime radicalC22×#41C8
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C8 ◁ C22×C8 ◁ C22×M32 ◁ G
LECSG ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1400C2semidirect
3. #7337C2semidirect
13. #1400C22semidirect

Maximal subgroups

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

subgroupindex
#333332
C2×#14002
#73372