Finite Group Databasebeta

← #124148

Subgroups — #124148

Order 512.

382 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 106 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#12599C212yesmaximal
3#7118C214yesmaximal, complemented
4C2×#431C411yescharacteristic
5C22×C4×C8C2211yescharacteristic
6C4×#117C411yescharacteristic
7C2×C22⋊C16—22no—
8C2×#346—22no—
9#1939C412yes—
10#1939C2212yes—
11#1850—22no—
12C4×C4⋊C8—22no—
13#1374C2214yes—
14C23×C8D412yes—
15C2×#117C811yescharacteristic
16C2×#117C811yescharacteristic
17C22×C42C2×C411yescharacteristic
18C22×C16—24no—
19#431—42no—
20C2×C4⋊C8—22no—
21C2×C4⋊C8—22no—
22C2×C4⋊C8C2×C412yes—
23C2×C22⋊C8—21no—
24C2×C4×C8—21no—
25C2×C4×C8—42no—
26C2×C4×C8C2×C411yescharacteristic
27C2×C4×C8D412yes—
28C2×C4×C8C2311yescharacteristic
29C4×C4⋊C4C812yes—
30C4×C4⋊C4C2×C412yes—
31C4×C22⋊C4—21no—
32C4×C22⋊C4—21no—
33#376—21no—
34C43—21no—
35C22⋊C16—28no—
36#346—44no—
37#336—22no—
38#336—22no—
39C23×C4C22⋊C411yescharacteristic
40C23×C4C2×C811yescharacteristic
41C23×C4M1611yescharacteristic
42C22×C8—24no—
43C22×C8—24no—
44C22×C8—22no—
45C22×C8—42no—
46C22×C8C22⋊C412yes—
47C22×C8C2×D412yes—
48#117—24no—
49#117—42no—
50C2×C4⋊C4C2×C812yes—
51C2×C4⋊C4C2×C812yes—
52C2×C22⋊C4—21no—
53C2×C22⋊C4—21no—
54C2×C42—21no—
55C2×C42—22no—
56C2×C42—22no—
57C2×C42—21no—
58C2×C42—22no—
59C2×C42—22no—
60C2×C42C22⋊C411yescharacteristic
61C2×C42C2×C811yescharacteristic
62C2×C42M1611yescharacteristic
63C2×C42C22×C411yescharacteristic
64C2×C16—28no—
65C2×C16—44no—
66C2×C16—44no—
67C4⋊C8—42no—
68C4⋊C8—44no—
69C4⋊C8—44no—
70C22⋊C8—42no—
71C4×C8—24no—
72C4×C8—41no—
73C4×C8—41no—
74C4×C8—42no—
75#95—22no—
76#95—22no—
77C24C22⋊C811yescharacteristic
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—42no—
89C22×C4C22⋊C811yescharacteristic
90C22×C4C22⋊C811yescharacteristic
91C22×C4C22⋊C811yescharacteristic
92C22×C4C2×C22⋊C411yescharacteristic
93C22×C4C22×C811yescharacteristic
94C22×C4C2×M1611yescharacteristic
95C2×C8—22no—
96C2×C8—22no—
97C2×C8—22no—
98C2×C8—22no—
99C2×C8—48no—
100C2×C8—42no—
101C2×C8—42no—
102C2×C8—42no—
103C2×C8#10414yes—
104C2×C8#13514yes—
105C4⋊C4—22no—
106C4⋊C4—42no—
107C4⋊C4M3214yes—
108C22⋊C4—24no—
109C22⋊C4—42no—
110C42—24no—
111C42—22no—
112C42—22no—
113C42—21no—
114C42—21no—
115C42—44no—
116C42—42no—
117C42—42no—
118C42—42no—
119C42#9912yes—
120C42#10112yes—
121C16—48no—
122C23—21no—
123C23—22no—
124C23—21no—
125C23—22no—
126C23#34912yes—
127C23C2×C22⋊C811yescharacteristic
128C2×C4—28no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—24no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—44no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—42no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4#32412yes—
149C2×C4#32412yes—
150C2×C4C2×#9911yescharacteristic
151C2×C4C2×#10111yescharacteristic
152C2×C4#41612yes—
153C2×C4C2×#10412yes—
154C2×C4C2×M3212yes—
155C8—24no—
156C8—24no—
157C8—82no—
158C22—24no—
159C22—22no—
160C22—42no—
161C22—41no—
162C22—41no—
163C22C2×#32411yescharacteristic
164C22#147612yes—
165C22#211211yescharacteristic
166C22C2×#34911yescharacteristic
167C22#211712yes—
168C4—22no—
169C4—22no—
170C4—22no—
171C4—44no—
172C4—41no—
173C4—41no—
174C4—82no—
175C4#150112yes—
176C4#152611yescharacteristic
177C4#154411yescharacteristic
178C2—42no—
179C2#756411yescharacteristic
180C2#1144011yescharacteristic
181C2#1144311yescharacteristic
182C2#1161512yes—
183C2#1161912yes—
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7118C2semidirect

Maximal subgroups

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

subgroupindex
#213492
#125992
#71182