Finite Group Databasebeta

← #135232

Subgroups — #135232

Order 512.

596 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 86 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#13939C211yesmaximal, characteristic
2#8300C212yesmaximal
3#7189C214yesmaximal, complemented
4C22×#117C2211yescharacteristic
5C2×#380C411yescharacteristic
6C2×C4×C22⋊C4C411yescharacteristic
7#1817—22no—
8C2×#324C2214yes—
9C2×#323—22no—
10#1440—22no—
11C4⋊#95C412yes—
12C4⋊#95C2212yes—
13#1434—22no—
14C24×C4C2×C411yescharacteristic
15C2×#117—24no—
16C2×#117—42no—
17C2×#117—42no—
18C22×C4⋊C4C2×C412yes—
19C22×C4⋊C4C2×C411yescharacteristic
20C22×C4⋊C4C2311yescharacteristic
21C22×C22⋊C4—21no—
22C22×C22⋊C4—21no—
23C22×C22⋊C4D412yes—
24C22×C42C2×C412yes—
25C22×C42D412yes—
26C2×C22⋊C8—24no—
27#380—42no—
28C4×C22⋊C4—42no—
29C2×#95—21no—
30C2×#95—21no—
31C2×#95—24no—
32C2×#95—24no—
33#336—44no—
34#324—28no—
35#323—44no—
36C25C22⋊C411yescharacteristic
37C23×C4—22no—
38C23×C4—22no—
39C23×C4—22no—
40C23×C4—22no—
41C23×C4—24no—
42C23×C4—22no—
43C23×C4—42no—
44C23×C4C22⋊C412yes—
45C23×C4C22⋊C411yescharacteristic
46C23×C4C22×C411yescharacteristic
47C23×C4C2×D412yes—
48C22×C8—44no—
49#117—416no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—24no—
52C2×C4⋊C4—41no—
53C2×C4⋊C4—41no—
54C2×C4⋊C4—41no—
55C2×C4⋊C4—41no—
56C2×C4⋊C4—42no—
57C2×C4⋊C4—42no—
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—24no—
67C2×C42—24no—
68C2×C42—24no—
69C2×C42—42no—
70C2×C42—42no—
71C2×C42—42no—
72C22⋊C8—44no—
73C22⋊C8—44no—
74#95—48no—
75#95—42no—
76#95—42no—
77#95—84no—
78C24—21no—
79C24—21no—
80C24—22no—
81C24—22no—
82C24—42no—
83C24—41no—
84C24—41no—
85C24#9912yes—
86C24C2×C22⋊C411yescharacteristic
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—22no—
90C22×C4—28no—
91C22×C4—21no—
92C22×C4—22no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—24no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—48no—
102C22×C4—48no—
103C22×C4—44no—
104C22×C4—44no—
105C22×C4—44no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—44no—
115C22×C4—44no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4#9912yes—
120C22×C4#11712yes—
121C22×C4#12314yes—
122C22×C4#13114yes—
123C2×C8—84no—
124C2×C8—88no—
125C4⋊C4—44no—
126C4⋊C4—44no—
127C4⋊C4—84no—
128C22⋊C4—48no—
129C22⋊C4—48no—
130C22⋊C4—84no—
131C42—48no—
132C42—84no—
133C23—22no—
134C23—21no—
135C23—21no—
136C23—22no—
137C23—24no—
138C23—42no—
139C23—42no—
140C23—42no—
141C23—42no—
142C23—41no—
143C23—41no—
144C23—42no—
145C23—41no—
146C23—41no—
147C23—42no—
148C23—82no—
149C23—82no—
150C23—82no—
151C23—82no—
152C23#38111yescharacteristic
153C23#40812yes—
154C23C2×#9911yescharacteristic
155C23C2×#9911yescharacteristic
156C23#41312yes—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—24no—
160C2×C4—24no—
161C2×C4—22no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—416no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—44no—
168C2×C4—42no—
169C2×C4—44no—
170C2×C4—44no—
171C2×C4—44no—
172C2×C4—44no—
173C2×C4—44no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—84no—
184C2×C4—84no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4#43314yes—
189C8—164no—
190C22—22no—
191C22—22no—
192C22—22no—
193C22—22no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—84no—
202C22—82no—
203C22—82no—
204C22—82no—
205C22—82no—
206C22#146812yes—
207C22#146812yes—
208C22#181811yescharacteristic
209C22#181911yescharacteristic
210C22#182211yescharacteristic
211C22#212612yes—
212C22#213411yescharacteristic
213C22#213511yescharacteristic
214C4—24no—
215C4—44no—
216C4—42no—
217C4—84no—
218C4—82no—
219C4—162no—
220C2—22no—
221C2—21no—
222C2—21no—
223C2—82no—
224C2#870211yescharacteristic
225C2#1155012yes—
226C2#1155212yes—
227C2#1285711yescharacteristic
228C2#1286311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7189C2semidirect

Maximal subgroups

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

subgroupindex
#139392
#83002
#71892