Finite Group Databasebeta

← #122313

Subgroups — #122313

Order 512.

340 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 102 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#21349C211yesmaximal, characteristic
2#11459C212yesmaximal
3#7064C214yesmaximal
4C2×#431C2211yescharacteristic
5C22×C4×C8C411yescharacteristic
6C4×#117C411yescharacteristic
7#2162—22no—
8#1939—22no—
9#1850C412yes—
10#1850C2212yes—
11C4×C4⋊C8—22no—
12#1322—22no—
13#1314C2214yes—
14C23×C8C2×C412yes—
15C2×#117C811yescharacteristic
16C2×#117C811yescharacteristic
17C22×C42C2×C411yescharacteristic
18#431—41no—
19#431—41no—
20C2×C4⋊C8—22no—
21C2×C4⋊C8—22no—
22C2×C4⋊C8D412yes—
23C2×C22⋊C8C2×C411yescharacteristic
24C2×C22⋊C8C2311yescharacteristic
25C2×C4×C8—21no—
26C2×C4×C8—21no—
27C2×C4×C8—41no—
28C2×C4×C8—41no—
29C2×C4×C8D412yes—
30C4×C4⋊C4—21no—
31C4×C4⋊C4—21no—
32C4×C22⋊C4C812yes—
33C4×C22⋊C4C2×C412yes—
34#376—21no—
35C43—21no—
36C4⋊C16—42no—
37C22⋊C16—24no—
38C4×C16—42no—
39#336—22no—
40#336—22no—
41C23×C4C2×C811yescharacteristic
42C23×C4C2×C811yescharacteristic
43C23×C4C22×C411yescharacteristic
44C22×C8—22no—
45C22×C8—44no—
46C22×C8—42no—
47C22×C8C22⋊C412yes—
48C22×C8C2×D412yes—
49#117—22no—
50#117—42no—
51#117C1614yescomplemented
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C22⋊C4C2×C812yes—
55C2×C22⋊C4C2×C812yes—
56C2×C42—21no—
57C2×C42—21no—
58C2×C42—21no—
59C2×C42—21no—
60C2×C42—21no—
61C2×C42—21no—
62C2×C42—22no—
63C2×C42—22no—
64C2×C42C22⋊C411yescharacteristic
65C2×C42C22⋊C411yescharacteristic
66C2×C42M1611yescharacteristic
67C2×C42M1611yescharacteristic
68C2×C16—44no—
69C4⋊C8—42no—
70C4⋊C8—44no—
71C4⋊C8—44no—
72C22⋊C8—41no—
73C22⋊C8—41no—
74C4×C8—22no—
75C4×C8—22no—
76C4×C8—42no—
77C4×C8—41no—
78C4×C8—41no—
79#95—22no—
80#95—22no—
81C24C22×C811yescharacteristic
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—22no—
91C22×C4—42no—
92C22×C4—42no—
93C22×C4C22⋊C811yescharacteristic
94C22×C4C22⋊C811yescharacteristic
95C22×C4C22⋊C811yescharacteristic
96C22×C4C22⋊C811yescharacteristic
97C22×C4C2×C1612yes—
98C22×C4M3212yes—
99C22×C4C2×C22⋊C411yescharacteristic
100C22×C4C2×M1611yescharacteristic
101C2×C8—24no—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—42no—
105C2×C8—44no—
106C2×C8—42no—
107C2×C8—42no—
108C2×C8—42no—
109C2×C8—44no—
110C4⋊C4—24no—
111C4⋊C4—42no—
112C22⋊C4—22no—
113C22⋊C4—42no—
114C22⋊C4C2×C1614yes—
115C42—21no—
116C42—21no—
117C42—21no—
118C42—21no—
119C42—24no—
120C42—21no—
121C42—21no—
122C42—42no—
123C42—42no—
124C42—44no—
125C42—42no—
126C42#9911yescharacteristic
127C42#9911yescharacteristic
128C42#10111yescharacteristic
129C42#10111yescharacteristic
130C16—84no—
131C23—21no—
132C23—21no—
133C23—41no—
134C23—41no—
135C23C2×C22⋊C811yescharacteristic
136C23C22×C1611yescharacteristic
137C23C2×M3211yescharacteristic
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—24no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4#32411yescharacteristic
161C2×C4#32411yescharacteristic
162C2×C4#32411yescharacteristic
163C2×C4#32411yescharacteristic
164C2×C4C22⋊C1612yes—
165C2×C4C22⋊C1612yes—
166C2×C4#34912yes—
167C2×C4C2×#9911yescharacteristic
168C2×C4C2×#10111yescharacteristic
169C8—42no—
170C8—42no—
171C8—44no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—41no—
177C22—41no—
178C22—42no—
179C22—41no—
180C22—41no—
181C22C2×#32411yescharacteristic
182C22C2×C22⋊C1611yescharacteristic
183C22C2×#34911yescharacteristic
184C4—22no—
185C4—21no—
186C4—21no—
187C4—44no—
188C4—41no—
189C4—41no—
190C4—82no—
191C4#132212yes—
192C4#132212yes—
193C4#132812yes—
194C4#212012yes—
195C2—21no—
196C2—21no—
197C2—41no—
198C2—41no—
199C2C2×#132211yescharacteristic
200C2#1145211yescharacteristic
201C2C2×#132811yescharacteristic
subgroupG/H
centerC42#99
commutatorC23C22×C16
Frattini, cosocleC2×C22⋊C8C23
socleC22C2×#324
prime radicalC24C22×C8
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C42 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
51. #117C16semidirect

Maximal subgroups

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

subgroupindex
#213492
#114592
#70642