Finite Group Databasebeta

← #135568

Subgroups — #135568

Order 512.

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

normal subgroups only

This group has 554 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#14292C211yesmaximal, characteristic
2#8555C212yesmaximal
3#7326C214yesmaximal, complemented
4C22×C22⋊Q8C2211yescharacteristic
5C2×#381C411yescharacteristic
6C2×C4×C22⋊C4C411yescharacteristic
7#1821—22no—
8#1463C2214yes—
9#1462—22no—
10#1446C412yes—
11#1445C2212yes—
12#1440—22no—
13#1436—22no—
14C23×Q8C2×C411yescharacteristic
15C24×C4C2×C411yescharacteristic
16C2×C22⋊Q8—24no—
17C2×C22⋊Q8—42no—
18C2×C22⋊Q8—42no—
19C22×C4⋊C4D412yes—
20C22×C4⋊C4C2311yescharacteristic
21C22×C22⋊C4—21no—
22C22×C22⋊C4—21no—
23C22×C22⋊C4D412yes—
24C22×C42—21no—
25#407—24no—
26#381—42no—
27C4×C22⋊C4—42no—
28C2×#95—21no—
29C2×#95—24no—
30C2×#95—24no—
31C2×#95C2×C412yes—
32C2×#95C2×C412yes—
33#343—44no—
34#324—28no—
35#323—44no—
36C25C22⋊C411yescharacteristic
37C22×Q8—22no—
38C22×Q8—41no—
39C22×Q8—41no—
40C22×Q8—82no—
41C23×C4—22no—
42C23×C4—22no—
43C23×C4—22no—
44C23×C4—22no—
45C23×C4—22no—
46C23×C4—82no—
47C23×C4C22⋊C412yes—
48C23×C4C22⋊C411yescharacteristic
49C23×C4C22×C411yescharacteristic
50C23×C4C2×D412yes—
51C2×M16—44no—
52C22⋊Q8—416no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—24no—
55C2×C4⋊C4—42no—
56C2×C4⋊C4—41no—
57C2×C4⋊C4—41no—
58C2×C4⋊C4#4114yes—
59C2×C22⋊C4—24no—
60C2×C22⋊C4—24no—
61C2×C22⋊C4—42no—
62C2×C22⋊C4—44no—
63C2×C22⋊C4—42no—
64C2×C22⋊C4—42no—
65C2×C22⋊C4—82no—
66C2×C42—42no—
67C2×C42—82no—
68C22⋊C8—44no—
69C22⋊C8—44no—
70#95—42no—
71#95—42no—
72#95—48no—
73#95—42no—
74#95—84no—
75C24—21no—
76C24—21no—
77C24—42no—
78C24—42no—
79C24—42no—
80C24#9912yes—
81C24C2×C22⋊C411yescharacteristic
82C2×Q8—44no—
83C2×Q8—84no—
84C2×Q8—84no—
85C2×Q8—84no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—22no—
91C22×C4—28no—
92C22×C4—22no—
93C22×C4—24no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4—44no—
100C22×C4—44no—
101C22×C4—42no—
102C22×C4—44no—
103C22×C4—42no—
104C22×C4—44no—
105C22×C4—44no—
106C22×C4—44no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—84no—
111C22×C4—84no—
112C22×C4—84no—
113C22×C4#9912yes—
114C22×C4#11712yes—
115C22×C4#12314yes—
116M16—88no—
117C2×C8—84no—
118C4⋊C4—44no—
119C4⋊C4—48no—
120C22⋊C4—48no—
121C22⋊C4—48no—
122C22⋊C4—84no—
123C42—84no—
124C23—22no—
125C23—22no—
126C23—22no—
127C23—22no—
128C23—24no—
129C23—42no—
130C23—42no—
131C23—42no—
132C23—42no—
133C23—42no—
134C23—42no—
135C23—42no—
136C23—42no—
137C23—84no—
138C23—84no—
139C23#38111yescharacteristic
140C23C2×#9911yescharacteristic
141C23C2×#9911yescharacteristic
142Q8—88no—
143C2×C4—24no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—24no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—44no—
152C2×C4—42no—
153C2×C4—44no—
154C2×C4—44no—
155C2×C4—48no—
156C2×C4—42no—
157C2×C4—44no—
158C2×C4—82no—
159C2×C4—88no—
160C2×C4—82no—
161C2×C4—84no—
162C2×C4—82no—
163C2×C4—82no—
164C2×C4—82no—
165C2×C4—82no—
166C2×C4—84no—
167C2×C4—84no—
168C2×C4—82no—
169C2×C4—82no—
170C8—164no—
171C22—24no—
172C22—24no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—21no—
179C22—42no—
180C22—42no—
181C22—42no—
182C22—84no—
183C22—84no—
184C22—84no—
185C22#181911yescharacteristic
186C22#182912yes—
187C22#212612yes—
188C22#213312yes—
189C4—42no—
190C4—42no—
191C4—44no—
192C4—84no—
193C4—82no—
194C4—162no—
195C2—22no—
196C2—22no—
197C2—22no—
198C2—82no—
199C2#875211yescharacteristic
200C2#1285711yescharacteristic
201C2#1286211yescharacteristic
subgroupG/H
center, socleC22#1819
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7326C2semidirect

Maximal subgroups

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

subgroupindex
#142922
#85552
#73262