Finite Group Databasebeta

← #213935

Subgroups — #213935

Order 512.

777 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 777 conjugacy classes of subgroups — too many to draw. Only its 186 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#36962C211yesmaximal, characteristic
2#21832C212yesmaximal, complemented
3#9514C214yesmaximal, complemented
4#9512C214yesmaximal
5#8755C212yesmaximal, complemented
6#8556C212yesmaximal
7C2×#544C2211yescharacteristic
8C22×#126C2211yescharacteristic
9#2921C2214yes—
10#2855C412yes—
11#2721—24no—
12#2716—24no—
13#2626C412yes—
14#2561C412yes—
15#2502C412yes—
16#2484C2212yes—
17#2482C2212yes—
18#2313C2211yescharacteristic
19#1998—24no—
20#1997—24no—
21#1822C2214yes—
22#1821C2214yes—
23#1463C2214yes—
24#1462C2214yes—
25#1306C2218yes—
26#544—44no—
27C2×C4⋊Q8D414yes—
28C2×#126—24no—
29C2×#126—44no—
30C2×#125D414yes—
31C2×C22⋊Q8C2×C412yes—
32C2×C22⋊Q8C2×C412yes—
33C2×#117C2312yes—
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2311yescharacteristic
36C22×C4⋊C4C2311yescharacteristic
37C22×C22⋊C4C2×C411yescharacteristic
38C22×C22⋊C4C2×C411yescharacteristic
39C22×C22⋊C4C2311yescharacteristic
40C22×C42C2×C411yescharacteristic
41#433—48no—
42#423—24no—
43#407C2318yes—
44C4×M16—24no—
45#400—22no—
46#399—22no—
47#399—22no—
48#398—24no—
49#398—22no—
50#397—22no—
51#397—22no—
52#396—24no—
53#393—22no—
54#393—22no—
55#391—22no—
56#391—22no—
57#389—22no—
58#388—24no—
59#388—24no—
60#387—24no—
61#387—24no—
62#387—24no—
63#385C2×C414yes—
64#385C2×C414yes—
65#384—22no—
66#381C2×C414yes—
67#381C2312yes—
68#380C2×C412yes—
69#379C2×C414yes—
70C4×C22⋊C4C2×C412yes—
71#343—28no—
72#343—24no—
73#343—24no—
74#324—44no—
75#323—44no—
76C25C22×C411yescharacteristic
77C22×Q8C22⋊C412yes—
78C22×Q8C22⋊C412yes—
79C23×C4C22×C411yescharacteristic
80C23×C4C22×C411yescharacteristic
81C23×C4C22×C411yescharacteristic
82C23×C4C22×C411yescharacteristic
83C23×C4C22×C411yescharacteristic
84C23×C4C2411yescharacteristic
85C2×M16—28no—
86C4⋊Q8—44no—
87C4⋊Q8—44no—
88#126—48no—
89#126—48no—
90#125—44no—
91#125—44no—
92C22⋊Q8—44no—
93C22⋊Q8—44no—
94#117—28no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4C22⋊C412yes—
104C2×C4⋊C4C22⋊C412yes—
105C2×C4⋊C4C22⋊C414yes—
106C2×C4⋊C4C22⋊C414yes—
107C2×C4⋊C4C2×D414yes—
108C2×C4⋊C4C2×D412yes—
109C2×C4⋊C4C2×D412yes—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4C22×C414yes—
119C2×C22⋊C4C22×C414yes—
120C2×C42—22no—
121C2×C42—22no—
122C2×C42—42no—
123C2×C42C2×D414yes—
124C4⋊C8—48no—
125C22⋊C8—28no—
126C22⋊C8—28no—
127#97—44no—
128C4×C8—44no—
129#95—22no—
130#95—22no—
131#95—24no—
132#95—24no—
133#95—24no—
134#95—24no—
135#95—22no—
136#95—22no—
137C24—21no—
138C24—22no—
139C24—42no—
140C24—81no—
141C24—81no—
142C24C23×C411yescharacteristic
143C2×Q8—42no—
144C2×Q8—42no—
145C2×Q8—82no—
146C2×Q8—82no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—24no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—81no—
159C22×C4—81no—
160C22×C4—81no—
161C22×C4—81no—
162C22×C4C2×C22⋊C414yes—
163C22×C4C2×C22⋊C412yes—
164C22×C4C2×C22⋊C412yes—
165C22×C4C2×C22⋊C414yes—
166C22×C4C22×D412yes—
167C22×C4H(5,𝔽2)14yes—
168M16—416no—
169C2×C8—48no—
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—44no—
180C4⋊C4—84no—
181C4⋊C4—84no—
182C4⋊C4—82no—
183C22⋊C4—28no—
184C22⋊C4—44no—
185C22⋊C4—44no—
186C22⋊C4—44no—
187C22⋊C4—44no—
188C22⋊C4—84no—
189C22⋊C4—84no—
190C42—24no—
191C42—24no—
192C42—44no—
193C42—82no—
194C23—22no—
195C23—41no—
196C23—41no—
197C23—42no—
198C23—42no—
199C23—44no—
200C23—41no—
201C23—42no—
202C23—82no—
203C23—82no—
204C23—164no—
205C23C22×C22⋊C411yescharacteristic
206C23#51812yes—
207Q8—162no—
208Q8—162no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4#53414yes—
234C8—88no—
235C22—21no—
236C22—21no—
237C22—22no—
238C22—22no—
239C22—22no—
240C22—44no—
241C22—84no—
242C22—81no—
243C22—81no—
244C22—81no—
245C22—164no—
246C22#230811yescharacteristic
247C22#288412yes—
248C4—24no—
249C4—42no—
250C4—42no—
251C4—44no—
252C4—44no—
253C4—84no—
254C4—81no—
255C4—81no—
256C4—81no—
257C2—22no—
258C2—42no—
259C2—161no—
260C2#2175111yescharacteristic
261C2#2177911yescharacteristic
262C2#2178111yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21832C2semidirect
3. #9514C2semidirect
5. #8755C2semidirect

Maximal subgroups

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

subgroupindex
#369622
#218322
#95142
#95122
#87552
#85562