Finite Group Databasebeta

← #135583

Subgroups — #135583

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#7326C214yesmaximal
4C22×C4⋊Q8C2211yescharacteristic
5C2×#382C411yescharacteristic
6C2×#376C411yescharacteristic
7#1784—22no—
8#1463—22no—
9#1463C2214yes—
10#1446—22no—
11#1445C412yes—
12#1445C2212yes—
13C4⋊#95—22no—
14C23×Q8D412yes—
15C2×C4⋊Q8—24no—
16C2×C4⋊Q8—41no—
17C2×C4⋊Q8—41no—
18C2×C4⋊Q8—42no—
19C22×C4⋊C4—21no—
20C22×C4⋊C4C2×C411yescharacteristic
21C22×C4⋊C4D412yes—
22C22×C4⋊C4C2311yescharacteristic
23C22×C42—21no—
24C22×C42C2×C411yescharacteristic
25C2×#101—44no—
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—28no—
37#324—42no—
38#324—42no—
39C22×Q8—24no—
40C22×Q8—24no—
41C22×Q8—42no—
42C22×Q8—84no—
43C23×C4—22no—
44C23×C4—22no—
45C23×C4C22⋊C411yescharacteristic
46C23×C4C22⋊C412yes—
47C23×C4C22⋊C411yescharacteristic
48C23×C4C22×C411yescharacteristic
49C23×C4C2×D412yes—
50C2×M16—44no—
51C4⋊Q8—48no—
52C4⋊Q8—48no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—41no—
58C2×C4⋊C4—41no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4—41no—
61C2×C4⋊C4—41no—
62C2×C4⋊C4—44no—
63C2×C4⋊C4—41no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4#4114yes—
66C2×C42—22no—
67C2×C42—21no—
68C2×C42—21no—
69C2×C42—22no—
70C2×C42—24no—
71C2×C42—42no—
72C2×C42—42no—
73#101—88no—
74C22⋊C8—44no—
75C22⋊C8—44no—
76#95—42no—
77#95—42no—
78#95—42no—
79#95—42no—
80#95—84no—
81#95—84no—
82C24C2×C22⋊C411yescharacteristic
83C2×Q8—48no—
84C2×Q8—48no—
85C2×Q8—84no—
86C2×Q8—88no—
87C2×Q8—88no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—24no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—42no—
103C22×C4—42no—
104C22×C4—44no—
105C22×C4—44no—
106C22×C4—44no—
107C22×C4—42no—
108C22×C4—44no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—44no—
113C22×C4—44no—
114C22×C4—42no—
115C22×C4#9912yes—
116C22×C4#9911yescharacteristic
117C22×C4#9911yescharacteristic
118C22×C4#11712yes—
119C22×C4#12314yes—
120M16—88no—
121C2×C8—84no—
122C4⋊C4—44no—
123C4⋊C4—44no—
124C4⋊C4—44no—
125C4⋊C4—44no—
126C4⋊C4—82no—
127C4⋊C4—82no—
128C42—24no—
129C42—24no—
130C42—48no—
131C23—22no—
132C23—22no—
133C23—21no—
134C23—21no—
135C23#38111yescharacteristic
136C23C2×#9911yescharacteristic
137C23C2×#9911yescharacteristic
138Q8—816no—
139C2×C4—24no—
140C2×C4—24no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—24no—
148C2×C4—24no—
149C2×C4—24no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—44no—
155C2×C4—44no—
156C2×C4—44no—
157C2×C4—48no—
158C2×C4—44no—
159C2×C4—42no—
160C2×C4—44no—
161C2×C4—82no—
162C2×C4—84no—
163C2×C4—82no—
164C2×C4—82no—
165C2×C4—82no—
166C2×C4—84no—
167C2×C4—84no—
168C2×C4—82no—
169C2×C4—82no—
170C2×C4—82no—
171C2×C4#35512yes—
172C2×C4#35612yes—
173C8—164no—
174C22—24no—
175C22—24no—
176C22—21no—
177C22—21no—
178C22—21no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22#181911yescharacteristic
183C22#183012yes—
184C22#212712yes—
185C22C2×#35511yescharacteristic
186C22C2×#35611yescharacteristic
187C4—22no—
188C4—22no—
189C4—42no—
190C4—44no—
191C4—42no—
192C4—84no—
193C4—82no—
194C4—162no—
195C2—22no—
196C2—21no—
197C2—22no—
198C2—21no—
199C2#875411yescharacteristic
200C2#1285811yescharacteristic
201C2#1286011yescharacteristic
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
#142822
#85512
#73262