Finite Group Databasebeta

← #135233

Subgroups — #135233

Order 512.

692 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 90 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#13924C211yesmaximal, characteristic
2#8299C212yesmaximal
3#7189C214yesmaximal, complemented
4C23×C4⋊C4C2211yescharacteristic
5C2×#380C411yescharacteristic
6C2×C4×C22⋊C4C411yescharacteristic
7#1779—22no—
8C2×#324—22no—
9C2×#324C2214yes—
10#1445—22no—
11C4⋊#95C412yes—
12C4⋊#95C2212yes—
13C22⋊#95—22no—
14C24×C4C2×C411yescharacteristic
15C24×C4D412yes—
16C22×C4⋊C4—24no—
17C22×C4⋊C4—42no—
18C22×C4⋊C4—41no—
19C22×C4⋊C4—41no—
20C22×C4⋊C4C2×C412yes—
21C22×C4⋊C4C2×C411yescharacteristic
22C22×C4⋊C4D412yes—
23C22×C4⋊C4C2311yescharacteristic
24C22×C22⋊C4—21no—
25C22×C22⋊C4—21no—
26C22×C42C2×C412yes—
27C2×C22⋊C8—24no—
28C2×C22⋊C8—42no—
29C2×C22⋊C8—42no—
30#380—41no—
31#380—41no—
32C4×C22⋊C4—42no—
33C2×#95—21no—
34C2×#95—21no—
35C2×#95—22no—
36C2×#95—22no—
37C2×#95—22no—
38C2×#95—22no—
39#324—28no—
40#324—44no—
41C25C22⋊C411yescharacteristic
42C23×C4—21no—
43C23×C4—21no—
44C23×C4—22no—
45C23×C4—22no—
46C23×C4—22no—
47C23×C4—22no—
48C23×C4—22no—
49C23×C4—24no—
50C23×C4—24no—
51C23×C4—22no—
52C23×C4—42no—
53C23×C4—42no—
54C23×C4—82no—
55C23×C4—82no—
56C23×C4C22⋊C411yescharacteristic
57C23×C4C22⋊C412yes—
58C23×C4C22×C411yescharacteristic
59C23×C4C2×D412yes—
60C22×C8—44no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—48no—
71C2×C4⋊C4—48no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—42no—
74C2×C4⋊C4#4114yes—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—42no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4—44no—
79C2×C22⋊C4—41no—
80C2×C22⋊C4—41no—
81C2×C42—24no—
82C2×C42—42no—
83C2×C42—42no—
84C22⋊C8—44no—
85C22⋊C8—44no—
86C22⋊C8—84no—
87C22⋊C8—84no—
88#95—42no—
89#95—42no—
90#95—82no—
91#95—82no—
92#95—82no—
93#95—82no—
94C24—22no—
95C24—22no—
96C24—21no—
97C24—21no—
98C24—22no—
99C24—22no—
100C24—42no—
101C24#9911yescharacteristic
102C24#9911yescharacteristic
103C24C2×C22⋊C411yescharacteristic
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—24no—
108C22×C4—24no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—24no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—48no—
120C22×C4—44no—
121C22×C4—44no—
122C22×C4—44no—
123C22×C4—44no—
124C22×C4—44no—
125C22×C4—44no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—44no—
133C22×C4—44no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—82no—
146C22×C4—82no—
147C22×C4—88no—
148C22×C4—88no—
149C22×C4#9912yes—
150C22×C4#11712yes—
151C22×C4#12314yes—
152C22×C4#13114yes—
153C2×C8—84no—
154C2×C8—84no—
155C2×C8—84no—
156C4⋊C4—44no—
157C4⋊C4—48no—
158C4⋊C4—44no—
159C4⋊C4—84no—
160C22⋊C4—48no—
161C22⋊C4—82no—
162C22⋊C4—82no—
163C42—84no—
164C23—22no—
165C23—21no—
166C23—21no—
167C23—24no—
168C23—24no—
169C23—21no—
170C23—21no—
171C23—22no—
172C23—22no—
173C23—21no—
174C23—21no—
175C23—22no—
176C23—22no—
177C23—44no—
178C23—44no—
179C23—44no—
180C23—44no—
181C23—42no—
182C23—42no—
183C23—41no—
184C23—41no—
185C23—42no—
186C23#35512yes—
187C23#35612yes—
188C23#38111yescharacteristic
189C23#40812yes—
190C23C2×#9911yescharacteristic
191C23C2×#9911yescharacteristic
192C23#41312yes—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—48no—
201C2×C4—48no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—82no—
216C2×C4—88no—
217C2×C4—88no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4#43314yes—
231C8—164no—
232C22—22no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—24no—
241C22—24no—
242C22—22no—
243C22—22no—
244C22—48no—
245C22—44no—
246C22—44no—
247C22—42no—
248C22—42no—
249C22—41no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22#146812yes—
254C22#146812yes—
255C22#181811yescharacteristic
256C22#181911yescharacteristic
257C22#182211yescharacteristic
258C22#212712yes—
259C22C2×#35511yescharacteristic
260C22C2×#35611yescharacteristic
261C4—24no—
262C4—44no—
263C4—42no—
264C4—84no—
265C4—82no—
266C4—162no—
267C2—22no—
268C2—21no—
269C2—21no—
270C2—42no—
271C2—42no—
272C2#870211yescharacteristic
273C2#1155312yes—
274C2#1155512yes—
275C2#1285811yescharacteristic
276C2#1286011yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ 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
#139242
#82992
#71892