Finite Group Databasebeta

← #135582

Subgroups — #135582

Order 512.

462 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 66 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#14289C211yesmaximal, characteristic
2#8554C212yesmaximal
3#7326C214yesmaximal
4C22×#125C2211yescharacteristic
5C2×#383C411yescharacteristic
6C2×C4×C4⋊C4C411yescharacteristic
7#1821—22no—
8#1463—22no—
9#1463C2214yes—
10#1445C412yes—
11#1445C2212yes—
12#1441—22no—
13#1436—22no—
14C2×#125—24no—
15C2×#125—42no—
16C2×#125—42no—
17C22×C4⋊C4—21no—
18C22×C4⋊C4C2×C411yescharacteristic
19C22×C4⋊C4D412yes—
20C22×C4⋊C4D412yes—
21C22×C4⋊C4C2311yescharacteristic
22C22×C42—21no—
23C22×C42C2×C411yescharacteristic
24#407—24no—
25#383—42no—
26C4×C4⋊C4—42no—
27C2×#95—21no—
28C2×#95—21no—
29C2×#95—24no—
30C2×#95—24no—
31C2×#95C2×C412yes—
32C2×#95C2×C412yes—
33#343—44no—
34#324—28no—
35#324—44no—
36C23×C4—22no—
37C23×C4—22no—
38C23×C4C22⋊C412yes—
39C23×C4C22⋊C411yescharacteristic
40C23×C4C22⋊C411yescharacteristic
41C23×C4C22×C411yescharacteristic
42C23×C4C2×D412yes—
43C2×M16—44no—
44#125—416no—
45C2×C4⋊C4—24no—
46C2×C4⋊C4—24no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—24no—
49C2×C4⋊C4—24no—
50C2×C4⋊C4—41no—
51C2×C4⋊C4—41no—
52C2×C4⋊C4—42no—
53C2×C4⋊C4—42no—
54C2×C4⋊C4—41no—
55C2×C4⋊C4—41no—
56C2×C4⋊C4—44no—
57C2×C4⋊C4—42no—
58C2×C4⋊C4#4114yes—
59C2×C42—22no—
60C2×C42—22no—
61C2×C42—22no—
62C2×C42—42no—
63C2×C42—82no—
64C22⋊C8—44no—
65C22⋊C8—44no—
66#95—42no—
67#95—42no—
68#95—48no—
69#95—42no—
70#95—42no—
71#95—84no—
72C24C2×C22⋊C411yescharacteristic
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—24no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—42no—
86C22×C4—41no—
87C22×C4—42no—
88C22×C4—41no—
89C22×C4—44no—
90C22×C4—42no—
91C22×C4—42no—
92C22×C4—44no—
93C22×C4—42no—
94C22×C4—44no—
95C22×C4—42no—
96C22×C4—44no—
97C22×C4—42no—
98C22×C4—44no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4#9912yes—
102C22×C4#9912yes—
103C22×C4#11712yes—
104C22×C4#12314yes—
105M16—88no—
106C2×C8—84no—
107C4⋊C4—44no—
108C4⋊C4—48no—
109C4⋊C4—48no—
110C4⋊C4—44no—
111C4⋊C4—48no—
112C42—28no—
113C42—84no—
114C23—22no—
115C23—22no—
116C23—21no—
117C23—21no—
118C23#38111yescharacteristic
119C23C2×#9911yescharacteristic
120C23C2×#9911yescharacteristic
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—24no—
125C2×C4—24no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—44no—
131C2×C4—42no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—44no—
137C2×C4—44no—
138C2×C4—48no—
139C2×C4—44no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—82no—
144C2×C4—84no—
145C2×C4—82no—
146C2×C4—82no—
147C2×C4—82no—
148C2×C4—84no—
149C2×C4—84no—
150C2×C4—82no—
151C2×C4—82no—
152C2×C4—82no—
153C8—164no—
154C22—24no—
155C22—24no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22#181911yescharacteristic
163C22#183012yes—
164C22#212612yes—
165C22#213411yescharacteristic
166C22#213511yescharacteristic
167C4—42no—
168C4—44no—
169C4—42no—
170C4—42no—
171C4—84no—
172C4—82no—
173C4—162no—
174C2—21no—
175C2—22no—
176C2—22no—
177C2—21no—
178C2#875411yescharacteristic
179C2#1285711yescharacteristic
180C2#1286311yescharacteristic
subgroupG/H
center, socleC22#1819
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#125 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#142892
#85542
#73262