Finite Group Databasebeta

← #135231

Subgroups — #135231

Order 512.

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

normal subgroups only

This group has 720 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#13904C211yesmaximal, characteristic
2#8299C212yesmaximal
3#7189C214yesmaximal, complemented
4C23×C4⋊C4C2211yescharacteristic
5C2×#381—22no—
6C2×#380C411yescharacteristic
7C2×C4×C22⋊C4C411yescharacteristic
8#1779—22no—
9C2×#324—22no—
10C2×#324C2214yes—
11#1445—22no—
12C4⋊#95C412yes—
13C4⋊#95C2212yes—
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#381—48no—
31#380—41no—
32#380—41no—
33C4×C22⋊C4—42no—
34C2×#95—24no—
35C2×#95—21no—
36C2×#95—21no—
37C2×#95—24no—
38#324—28no—
39#324—44no—
40C25C22⋊C411yescharacteristic
41C23×C4—21no—
42C23×C4—21no—
43C23×C4—22no—
44C23×C4—22no—
45C23×C4—22no—
46C23×C4—22no—
47C23×C4—22no—
48C23×C4—24no—
49C23×C4—24no—
50C23×C4—22no—
51C23×C4—42no—
52C23×C4—42no—
53C23×C4—44no—
54C23×C4—44no—
55C23×C4C22⋊C411yescharacteristic
56C23×C4C22⋊C412yes—
57C23×C4C22×C411yescharacteristic
58C23×C4C2×D412yes—
59C22×C8—44no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—24no—
63C2×C4⋊C4—24no—
64C2×C4⋊C4—41no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—48no—
70C2×C4⋊C4—48no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4#4114yes—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—42no—
76C2×C22⋊C4—42no—
77C2×C22⋊C4—44no—
78C2×C22⋊C4—41no—
79C2×C22⋊C4—41no—
80C2×C42—24no—
81C2×C42—42no—
82C2×C42—42no—
83C22⋊C8—44no—
84C22⋊C8—44no—
85C22⋊C8—84no—
86C22⋊C8—84no—
87#95—42no—
88#95—48no—
89#95—48no—
90#95—42no—
91C24—22no—
92C24—22no—
93C24—21no—
94C24—21no—
95C24—22no—
96C24—22no—
97C24—42no—
98C24#9911yescharacteristic
99C24#9911yescharacteristic
100C24C2×C22⋊C411yescharacteristic
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—24no—
105C22×C4—24no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—24no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—48no—
117C22×C4—44no—
118C22×C4—44no—
119C22×C4—44no—
120C22×C4—44no—
121C22×C4—44no—
122C22×C4—44no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—416no—
131C22×C4—44no—
132C22×C4—44no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—44no—
137C22×C4—44no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—82no—
142C22×C4—82no—
143C22×C4—88no—
144C22×C4#9912yes—
145C22×C4#11712yes—
146C22×C4#12314yes—
147C22×C4#13114yes—
148C2×C8—84no—
149C2×C8—84no—
150C2×C8—84no—
151C4⋊C4—44no—
152C4⋊C4—48no—
153C4⋊C4—44no—
154C4⋊C4—84no—
155C22⋊C4—48no—
156C22⋊C4—82no—
157C22⋊C4—82no—
158C42—84no—
159C23—21no—
160C23—21no—
161C23—22no—
162C23—24no—
163C23—24no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—22no—
169C23—22no—
170C23—22no—
171C23—22no—
172C23—44no—
173C23—44no—
174C23—44no—
175C23—44no—
176C23—42no—
177C23—42no—
178C23—41no—
179C23—41no—
180C23—42no—
181C23#35312yes—
182C23#35412yes—
183C23#38111yescharacteristic
184C23#40812yes—
185C23C2×#9911yescharacteristic
186C23C2×#9911yescharacteristic
187C23#41312yes—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—48no—
196C2×C4—48no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—82no—
211C2×C4—88no—
212C2×C4—88no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—84no—
219C2×C4—84no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4#43314yes—
224C8—164no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—22no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—24no—
234C22—24no—
235C22—22no—
236C22—22no—
237C22—48no—
238C22—44no—
239C22—44no—
240C22—42no—
241C22—42no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22#146812yes—
247C22#146812yes—
248C22#181811yescharacteristic
249C22#181911yescharacteristic
250C22#182211yescharacteristic
251C22C2×#35311yescharacteristic
252C22C2×#35411yescharacteristic
253C22#213312yes—
254C4—24no—
255C4—44no—
256C4—42no—
257C4—84no—
258C4—82no—
259C4—162no—
260C2—22no—
261C2—21no—
262C2—21no—
263C2—42no—
264C2—42no—
265C2#870211yescharacteristic
266C2#1154912yes—
267C2#1155112yes—
268C2#1285611yescharacteristic
269C2#1286211yescharacteristic
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
#139042
#82992
#71892