Finite Group Databasebeta

← #122245

Subgroups — #122245

Order 512.

409 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 100 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#21818C211yesmaximal, characteristic
2#11459C212yesmaximal
3#7057C214yesmaximal, complemented
4C22×C4×Q8C411yescharacteristic
5C2×#431C411yescharacteristic
6C2×#431C2211yescharacteristic
7#2162—22no—
8#1984—22no—
9#1984C412yes—
10#1984C2212yes—
11#1817—22no—
12#1322C2214yes—
13#1314—22no—
14C23×Q8C811yescharacteristic
15C2×C4×Q8—21no—
16C2×C4×Q8—21no—
17C2×C4×Q8—42no—
18C2×C4×Q8—42no—
19C2×C4×Q8C812yes—
20C2×C4×Q8C2×C412yes—
21C22×C4⋊C4—21no—
22C22×C4⋊C4C811yescharacteristic
23C22×C42—21no—
24C22×C42C2×C411yescharacteristic
25#431—42no—
26#431—42no—
27C2×C4⋊C8—21no—
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C4⋊C8C2311yescharacteristic
30C2×C22⋊C8—21no—
31C2×C22⋊C8D412yes—
32C2×C4×C8C2×C412yes—
33C2×C4×C8D412yes—
34C4⋊C16—42no—
35C22⋊C16—24no—
36C4×C16—42no—
37#336—24no—
38#336—24no—
39C22×Q8—21no—
40C22×Q8—22no—
41C22×Q8—42no—
42C22×Q8—42no—
43C22×Q8C1614yescomplemented
44C22×Q8C2×C812yes—
45C23×C4—21no—
46C23×C4—21no—
47C23×C4C22⋊C411yescharacteristic
48C23×C4C2×C811yescharacteristic
49C23×C4M1611yescharacteristic
50C22×C8—22no—
51C22×C8C22⋊C412yes—
52C22×C8C2×D412yes—
53C4×Q8—44no—
54C4×Q8—44no—
55C4×Q8—42no—
56C4×Q8—42no—
57C4×Q8—42no—
58C4×Q8—42no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—44no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4C2×C812yes—
65C2×C42—22no—
66C2×C42—22no—
67C2×C42—22no—
68C2×C42—22no—
69C2×C42—42no—
70C2×C42—42no—
71C2×C42C22⋊C411yescharacteristic
72C2×C42C2×C811yescharacteristic
73C2×C42M1611yescharacteristic
74C2×C42C22×C411yescharacteristic
75C2×C16—44no—
76C4⋊C8—41no—
77C4⋊C8—41no—
78C4⋊C8—42no—
79C22⋊C8—42no—
80C22⋊C8—42no—
81C4×C8—22no—
82C4×C8—22no—
83C4×C8—42no—
84C24C22⋊C811yescharacteristic
85C2×Q8—24no—
86C2×Q8—22no—
87C2×Q8—48no—
88C2×Q8—48no—
89C2×Q8—42no—
90C2×Q8—42no—
91C2×Q8—42no—
92C2×Q8—42no—
93C2×Q8C2×C1614yes—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—24no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—82no—
108C22×C4—81no—
109C22×C4C22⋊C811yescharacteristic
110C22×C4C22⋊C811yescharacteristic
111C22×C4C22⋊C811yescharacteristic
112C22×C4C2×C1612yes—
113C22×C4M3212yes—
114C22×C4C2×C22⋊C411yescharacteristic
115C22×C4C22×C811yescharacteristic
116C22×C4C2×M1611yescharacteristic
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—42no—
120C2×C8—42no—
121C2×C8—42no—
122C2×C8—42no—
123C2×C8—42no—
124C4⋊C4—42no—
125C4⋊C4—44no—
126C4⋊C4—44no—
127C4⋊C4—42no—
128C42—22no—
129C42—22no—
130C42—21no—
131C42—21no—
132C42—44no—
133C42—42no—
134C42—42no—
135C42#9912yes—
136C42#10012yes—
137C16—84no—
138C23—21no—
139C23—21no—
140C23—41no—
141C23—41no—
142C23C22⋊C1612yes—
143C23C2×C22⋊C811yescharacteristic
144Q8—44no—
145Q8—44no—
146Q8—48no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—24no—
150C2×C4—24no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—24no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—48no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—44no—
166C2×C4—42no—
167C2×C4—82no—
168C2×C4—84no—
169C2×C4#32312yes—
170C2×C4#32312yes—
171C2×C4C22⋊C1612yes—
172C2×C4#34912yes—
173C2×C4C2×#9911yescharacteristic
174C2×C4C2×#10011yescharacteristic
175C2×C4C22×C1611yescharacteristic
176C2×C4C2×M3211yescharacteristic
177C8—42no—
178C8—42no—
179C8—82no—
180C22—21no—
181C22—21no—
182C22—22no—
183C22—42no—
184C22—42no—
185C22—41no—
186C22—41no—
187C22C2×#32311yescharacteristic
188C22C2×C22⋊C1611yescharacteristic
189C22C2×#34911yescharacteristic
190C4—21no—
191C4—21no—
192C4—21no—
193C4—22no—
194C4—44no—
195C4—41no—
196C4—41no—
197C4—82no—
198C4#131414yes—
199C4#212911yescharacteristic
200C4#213511yescharacteristic
201C2—21no—
202C2—21no—
203C2—42no—
204C2C2×#131411yescharacteristic
205C2#1143411yescharacteristic
206C2#1143811yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC2×C4C22×C16
Frattini, cosocleC2×C4⋊C8C23
socleC22C2×#323
prime radicalC24C22⋊C8
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7057C2semidirect
43. C22×Q8C16semidirect

Maximal subgroups

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

subgroupindex
#218182
#114592
#70572