Finite Group Databasebeta

← #135579

Subgroups — #135579

Order 512.

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

normal subgroups only

This group has 490 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#14663C211yesmaximal, characteristic
2#8551C212yesmaximal
3#7326C214yesmaximal
4C22×C4⋊Q8C2211yescharacteristic
5C2×#383C411yescharacteristic
6C2×#383C411yescharacteristic
7#1784—22no—
8#1463—22no—
9#1463C2214yes—
10#1446—22no—
11#1445—22no—
12#1445C412yes—
13#1445C2212yes—
14C23×Q8D412yes—
15C2×C4⋊Q8—24no—
16C2×C4⋊Q8—42no—
17C2×C4⋊Q8—42no—
18C22×C4⋊C4C2×C411yescharacteristic
19C22×C4⋊C4D412yes—
20C22×C4⋊C4C2311yescharacteristic
21C22×C42C2×C411yescharacteristic
22C2×#101—44no—
23#407—24no—
24#383—42no—
25#383—42no—
26C2×#95—21no—
27C2×#95—21no—
28C2×#95—21no—
29C2×#95—21no—
30C2×#95—24no—
31C2×#95—24no—
32C2×#95C2×C412yes—
33C2×#95C2×C412yes—
34#324—28no—
35#324—44no—
36C22×Q8—24no—
37C22×Q8—24no—
38C22×Q8—42no—
39C22×Q8—84no—
40C23×C4—22no—
41C23×C4—22no—
42C23×C4C22⋊C412yes—
43C23×C4C22⋊C411yescharacteristic
44C23×C4C22⋊C411yescharacteristic
45C23×C4C22×C411yescharacteristic
46C23×C4C2×D412yes—
47C2×M16—44no—
48C4⋊Q8—416no—
49C2×C4⋊C4—24no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—24no—
52C2×C4⋊C4—41no—
53C2×C4⋊C4—41no—
54C2×C4⋊C4—42no—
55C2×C4⋊C4—41no—
56C2×C4⋊C4—41no—
57C2×C4⋊C4#4114yes—
58C2×C42—22no—
59C2×C42—22no—
60C2×C42—22no—
61#101—88no—
62C22⋊C8—44no—
63C22⋊C8—44no—
64#95—42no—
65#95—42no—
66#95—42no—
67#95—42no—
68#95—42no—
69#95—42no—
70#95—84no—
71#95—84no—
72C24C2×C22⋊C411yescharacteristic
73C2×Q8—48no—
74C2×Q8—48no—
75C2×Q8—84no—
76C2×Q8—88no—
77C2×Q8—88no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4—42no—
92C22×C4—44no—
93C22×C4—44no—
94C22×C4—42no—
95C22×C4—42no—
96C22×C4—44no—
97C22×C4—44no—
98C22×C4—44no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4—44no—
102C22×C4—42no—
103C22×C4—42no—
104C22×C4#9912yes—
105C22×C4#9912yes—
106C22×C4#11712yes—
107C22×C4#12314yes—
108M16—88no—
109C2×C8—84no—
110C4⋊C4—44no—
111C4⋊C4—48no—
112C4⋊C4—44no—
113C42—28no—
114C23—22no—
115C23—22no—
116C23—22no—
117C23#38111yescharacteristic
118C23C2×#9911yescharacteristic
119C23C2×#9911yescharacteristic
120Q8—816no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—24no—
125C2×C4—24no—
126C2×C4—24no—
127C2×C4—24no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—44no—
135C2×C4—44no—
136C2×C4—44no—
137C2×C4—42no—
138C2×C4—44no—
139C2×C4—82no—
140C2×C4—82no—
141C2×C4—84no—
142C2×C4—84no—
143C2×C4—84no—
144C2×C4—84no—
145C2×C4—82no—
146C2×C4—82no—
147C2×C4—82no—
148C2×C4—82no—
149C2×C4—82no—
150C2×C4—82no—
151C8—164no—
152C22—24no—
153C22—24no—
154C22—21no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—21no—
160C22#181911yescharacteristic
161C22#183012yes—
162C22#212712yes—
163C22#213312yes—
164C4—42no—
165C4—44no—
166C4—42no—
167C4—42no—
168C4—82no—
169C4—162no—
170C4—162no—
171C2—22no—
172C2—22no—
173C2—22no—
174C2#875311yescharacteristic
175C2#1285811yescharacteristic
176C2#1286211yescharacteristic
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×C4⋊Q8 ◁ 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
#146632
#85512
#73262