Finite Group Databasebeta

← #116354

Subgroups — #116354

Order 512.

748 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 748 conjugacy classes of subgroups — too many to draw. Only its 134 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#21931C211yesmaximal, characteristic
2#20688C211yesmaximal, characteristic
3#20685C212yesmaximal
4#8112C211yesmaximal, characteristic
5#7927C212yesmaximal
6C22×#117C2211yescharacteristic
7C2×#407C2212yes—
8C22×C22⋊C8C2212yes—
9#2280C2211yescharacteristic
10#2280C2211yescharacteristic
11#1850—24no—
12#1820—22no—
13#1819C414yes—
14#1818—24no—
15#1817—24no—
16#1798—22no—
17#1740—22no—
18#1729—24no—
19#1306—22no—
20#1282—24no—
21C24×C4C2311yescharacteristic
22C22×M16—22no—
23C23×C8D414yes—
24C2×#117—22no—
25C2×#117—21no—
26C2×#117—22no—
27C2×#117—21no—
28C2×#117—21no—
29C2×#117—21no—
30C2×#117—22no—
31C2×#117—22no—
32C2×#117—21no—
33C2×#117D412yes—
34C22×C4⋊C4Q812yes—
35C22×C22⋊C4C2×C412yes—
36C22×C42D412yes—
37C2×C4⋊C8—24no—
38#407—22no—
39#407—22no—
40#407—22no—
41#407—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8D414yes—
50#381—21no—
51#381D412yes—
52#380—22no—
53#380—22no—
54#379C2×C412yes—
55#379C2×C412yes—
56C4×C22⋊C4—22no—
57C4×C22⋊C4—22no—
58#342—28no—
59#336—24no—
60#336—216no—
61C25C22×C411yescharacteristic
62C23×C4—44no—
63C23×C4C4⋊C414yes—
64C23×C4C2×D412yes—
65C23×C4C2×D411yescharacteristic
66C23×C4C2×D411yescharacteristic
67C23×C4C2×D412yes—
68C23×C4C2×D411yescharacteristic
69C23×C4C2×Q811yescharacteristic
70C23×C4#4111yescharacteristic
71C23×C4#4112yes—
72C23×C4#4111yescharacteristic
73C23×C4#4112yes—
74C2×M16—48no—
75C2×M16—42no—
76C2×M16—42no—
77C22×C8—24no—
78C22×C8—24no—
79C22×C8—24no—
80C22×C8—24no—
81C22×C8—48no—
82#117—41no—
83#117—42no—
84#117—42no—
85#117—42no—
86#117—42no—
87#117—41no—
88#117—42no—
89#117—41no—
90#117—42no—
91#117—41no—
92#117—44no—
93#117—44no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—41no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4C22⋊C414yes—
110C2×C42—24no—
111C2×C42—22no—
112C2×C42—22no—
113C2×C42—22no—
114C2×C42—24no—
115C2×C42—41no—
116C4⋊C8—84no—
117C22⋊C8—24no—
118C22⋊C8—24no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125#95—24no—
126#95—42no—
127C24—42no—
128C24—44no—
129C24C2×C22⋊C411yescharacteristic
130C24C2×C4⋊C411yescharacteristic
131C24#11711yescharacteristic
132C24C4×D412yes—
133C24C4×D412yes—
134C22×C4—24no—
135C22×C4—28no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—44no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—44no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—44no—
148C22×C4—48no—
149C22×C4—42no—
150C22×C4#12012yes—
151C22×C4#12011yescharacteristic
152C22×C4#12112yes—
153C22×C4#12111yescharacteristic
154C22×C4#12112yes—
155C22×C4#12111yescharacteristic
156C22×C4#12112yes—
157C22×C4C22⋊Q812yes—
158C22×C4C22⋊Q811yescharacteristic
159C22×C4C22⋊Q811yescharacteristic
160C22×C4C22⋊Q812yes—
161C22×C4#12311yescharacteristic
162C22×C4#12312yes—
163C22×C4#12312yes—
164C22×C4#12412yes—
165C22×C4#12411yescharacteristic
166C22×C4#12612yes—
167C22×C4#12711yescharacteristic
168M16—84no—
169M16—84no—
170C2×C8—48no—
171C2×C8—48no—
172C2×C8—44no—
173C2×C8—44no—
174C2×C8—48no—
175C2×C8—84no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—82no—
181C4⋊C4—82no—
182C22⋊C4—48no—
183C22⋊C4—42no—
184C22⋊C4—44no—
185C22⋊C4—42no—
186C22⋊C4—44no—
187C22⋊C4—48no—
188C22⋊C4—44no—
189C42—42no—
190C42—44no—
191C42—42no—
192C42—44no—
193C42—82no—
194C23—24no—
195C23—24no—
196C23—24no—
197C23—24no—
198C23—22no—
199C23—41no—
200C23—42no—
201C23—41no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—88no—
207C23#38011yescharacteristic
208C23#38512yes—
209C23#38611yescharacteristic
210C23#38812yes—
211C23#39011yescharacteristic
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—48no—
230C2×C4—48no—
231C2×C4—48no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—84no—
235C2×C4—82no—
236C2×C4—84no—
237C2×C4#39211yescharacteristic
238C2×C4#39311yescharacteristic
239C2×C4#39412yes—
240C2×C4#39612yes—
241C2×C4#39711yescharacteristic
242C2×C4#39911yescharacteristic
243C2×C4#45814yes—
244C8—84no—
245C8—84no—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—21no—
250C22—21no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—48no—
257C22—44no—
258C22—82no—
259C22—84no—
260C22—82no—
261C22#143911yescharacteristic
262C22#186814yes—
263C22#189012yes—
264C4—22no—
265C4—22no—
266C4—21no—
267C4—21no—
268C4—84no—
269C4—84no—
270C4—162no—
271C4#202212yes—
272C4#203012yes—
273C2—22no—
274C2—24no—
275C2—41no—
276C2—41no—
277C2—42no—
278C2#1107011yescharacteristic
279C2#1108812yes—
subgroupG/H
centerC2×C4#392
commutator, prime radicalC25C22×C4
Frattini, cosocleC24×C4C23
socleC22#1439
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#219312
#206882
#206852
#81122
#79272