Finite Group Databasebeta

← #135550

Subgroups — #135550

Order 512.

532 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 70 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#14282C211yesmaximal, characteristic
2#8551C212yesmaximal
3#7308C214yesmaximal
4C22×C4⋊Q8C2211yescharacteristic
5C2×#382C411yescharacteristic
6C2×#376C411yescharacteristic
7#1784C2214yes—
8#1463—22no—
9#1463—22no—
10#1446C412yes—
11#1446C2212yes—
12#1445—22no—
13C4⋊#95—22no—
14C23×Q8C2×C411yescharacteristic
15C23×Q8C2311yescharacteristic
16C2×C4⋊Q8—24no—
17C2×C4⋊Q8—42no—
18C2×C4⋊Q8—41no—
19C2×C4⋊Q8—41no—
20C22×C4⋊C4—21no—
21C22×C4⋊C4D412yes—
22C22×C4⋊C4D412yes—
23C22×C42—21no—
24C22×C42C2×C411yescharacteristic
25C2×#101—28no—
26#407—24no—
27#382—41no—
28#382—41no—
29#376—42no—
30C2×#95—21no—
31C2×#95—21no—
32C2×#95—24no—
33C2×#95—24no—
34C2×#95C2×C412yes—
35C2×#95C2×C412yes—
36#324—44no—
37#324—42no—
38#324—42no—
39C22×Q8—22no—
40C22×Q8—41no—
41C22×Q8—41no—
42C22×Q8—41no—
43C22×Q8—41no—
44C22×Q8—82no—
45C22×Q8—82no—
46C22×Q8#4114yes—
47C23×C4—22no—
48C23×C4—22no—
49C23×C4C22⋊C411yescharacteristic
50C23×C4C22⋊C412yes—
51C23×C4C22⋊C411yescharacteristic
52C23×C4C22×C411yescharacteristic
53C23×C4C2×D412yes—
54C2×M16—44no—
55C4⋊Q8—48no—
56C4⋊Q8—48no—
57C2×C4⋊C4—24no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—42no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—44no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C42—22no—
68C2×C42—21no—
69C2×C42—21no—
70C2×C42—22no—
71C2×C42—24no—
72C2×C42—42no—
73C2×C42—42no—
74#101—88no—
75C22⋊C8—44no—
76C22⋊C8—44no—
77#95—42no—
78#95—42no—
79#95—42no—
80#95—42no—
81#95—84no—
82#95—84no—
83C24C2×C22⋊C411yescharacteristic
84C2×Q8—48no—
85C2×Q8—44no—
86C2×Q8—44no—
87C2×Q8—84no—
88C2×Q8—84no—
89C2×Q8—84no—
90C2×Q8—84no—
91C2×Q8—84no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—24no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—44no—
109C22×C4—44no—
110C22×C4—42no—
111C22×C4—44no—
112C22×C4—42no—
113C22×C4—44no—
114C22×C4—44no—
115C22×C4—44no—
116C22×C4—44no—
117C22×C4—42no—
118C22×C4#9912yes—
119C22×C4#9911yescharacteristic
120C22×C4#9911yescharacteristic
121C22×C4#11712yes—
122C22×C4#12314yes—
123M16—88no—
124C2×C8—84no—
125C4⋊C4—44no—
126C4⋊C4—44no—
127C4⋊C4—48no—
128C4⋊C4—82no—
129C4⋊C4—82no—
130C42—24no—
131C42—24no—
132C42—48no—
133C23—22no—
134C23—21no—
135C23—21no—
136C23—22no—
137C23#38111yescharacteristic
138C23C2×#9911yescharacteristic
139C23C2×#9911yescharacteristic
140Q8—88no—
141Q8—88no—
142C2×C4—24no—
143C2×C4—24no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—24no—
151C2×C4—28no—
152C2×C4—24no—
153C2×C4—24no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—44no—
159C2×C4—48no—
160C2×C4—42no—
161C2×C4—44no—
162C2×C4—82no—
163C2×C4—82no—
164C2×C4—84no—
165C2×C4—82no—
166C2×C4—82no—
167C2×C4—82no—
168C2×C4—82no—
169C2×C4—88no—
170C2×C4—82no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4#35512yes—
174C2×C4#35612yes—
175C8—164no—
176C22—24no—
177C22—24no—
178C22—21no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22#181911yescharacteristic
185C22#182812yes—
186C22C2×#35511yescharacteristic
187C22C2×#35611yescharacteristic
188C22#213312yes—
189C4—22no—
190C4—22no—
191C4—44no—
192C4—42no—
193C4—42no—
194C4—84no—
195C4—82no—
196C4—162no—
197C2—22no—
198C2—21no—
199C2—22no—
200C2—21no—
201C2#874911yescharacteristic
202C2#1286011yescharacteristic
203C2#1286211yescharacteristic
subgroupG/H
center, socleC22#1819
commutatorC23×C4C22×C4
Frattini, cosocleC23×Q8C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#142822
#85512
#73082