Finite Group Databasebeta

← #96966

Subgroups — #96966

Order 512.

916 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 916 conjugacy classes of subgroups — too many to draw. Only its 146 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#34249C211yesmaximal, characteristic
2#11380C211yesmaximal, characteristic
3#8303C211yesmaximal, characteristic
4#7579C214yesmaximal
5C2×#519C411yescharacteristic
6C2×#519C411yescharacteristic
7C22×C4×Q8C2211yescharacteristic
8#2351—24no—
9Q8×C4⋊C4—24no—
10#2317—22no—
11#2317C412yes—
12#2296—22no—
13#2296C2212yes—
14#1817C2212yes—
15C2×#336—22no—
16C2×#336C2212yes—
17C2×#324—22no—
18#1271—24no—
19#1271—24no—
20C23×Q8C2×C411yescharacteristic
21C23×C8—22no—
22#519—22no—
23#519—24no—
24#519—44no—
25#519C814yescomplemented
26C2×C4×Q8—24no—
27C2×C4×Q8—22no—
28C2×C4×Q8—22no—
29C2×C4×Q8—24no—
30C2×C4×Q8—44no—
31C2×#117—22no—
32C2×#117—22no—
33C2×#117C2×C412yes—
34C2×#117C2×C412yes—
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4D412yes—
37C22×C42D412yes—
38C22×C42C2311yescharacteristic
39C2×C22⋊C8—22no—
40#389—28no—
41#389—24no—
42#389—24no—
43#389—24no—
44#389D414yes—
45#389D414yes—
46#385—24no—
47#385—22no—
48#385—22no—
49#384—24no—
50#384—22no—
51#382—24no—
52#382—22no—
53C4×C4⋊C4—28no—
54C4×C4⋊C4—24no—
55#336—24no—
56#336—24no—
57#336—44no—
58#324—44no—
59C22×Q8—24no—
60C22×Q8—24no—
61C22×Q8—22no—
62C22×Q8—22no—
63C22×Q8C2×C812yes—
64C22×Q8M1612yes—
65C23×C4C22⋊C412yes—
66C23×C4C22⋊C412yes—
67C23×C4C22×C411yescharacteristic
68C23×C4C2×D411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C22×C8—24no—
71C22×C8—42no—
72C22×C8—44no—
73C22×C8—44no—
74C22×C8—42no—
75C22×C8—42no—
76C4×Q8—216no—
77C4×Q8—216no—
78C4×Q8—416no—
79C4×Q8—48no—
80C4×Q8—48no—
81#117—28no—
82#117—28no—
83#117—22no—
84#117—22no—
85#117—44no—
86#117—44no—
87#117—44no—
88#117C2×C814yes—
89#117M1614yes—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—28no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4C22⋊C414yes—
102C2×C4⋊C4C22⋊C412yes—
103C2×C4⋊C4C22⋊C412yes—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—22no—
106C2×C42—24no—
107C2×C42—22no—
108C2×C42—24no—
109C2×C42—24no—
110C2×C42—22no—
111C2×C42—24no—
112C2×C42—42no—
113C2×C42—42no—
114C2×C42C2×D414yes—
115C22⋊C8—44no—
116#95—28no—
117#95—24no—
118#95—24no—
119C24C2×C22⋊C411yescharacteristic
120C2×Q8—28no—
121C2×Q8—216no—
122C2×Q8—28no—
123C2×Q8—28no—
124C2×Q8—28no—
125C2×Q8—24no—
126C2×Q8—24no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—42no—
144C22×C4—44no—
145C22×C4C22⋊C814yes—
146C22×C4#9914yes—
147C22×C4#10114yes—
148C22×C4C2×C22⋊C412yes—
149C22×C4#12012yes—
150C22×C4#12012yes—
151C22×C4C22×C811yescharacteristic
152C22×C4C2×M1612yes—
153C22×C4C2×M1611yescharacteristic
154C2×C8—48no—
155C2×C8—44no—
156C2×C8—84no—
157C2×C8—82no—
158C2×C8—82no—
159C2×C8—82no—
160C2×C8—84no—
161C4⋊C4—216no—
162C4⋊C4—24no—
163C4⋊C4—48no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—44no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—44no—
171C4⋊C4—84no—
172C4⋊C4—88no—
173C4⋊C4C22⋊C818yes—
174C22⋊C4—24no—
175C22⋊C4—24no—
176C22⋊C4—44no—
177C42—216no—
178C42—24no—
179C42—24no—
180C42—44no—
181C42—48no—
182C42—48no—
183C42—44no—
184C23—22no—
185C23—22no—
186C23#37911yescharacteristic
187C23C2×C22⋊C811yescharacteristic
188C23#40711yescharacteristic
189C23C2×#9912yes—
190C23C2×#10112yes—
191Q8—216no—
192Q8—216no—
193C2×C4—24no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—22no—
207C2×C4—24no—
208C2×C4—22no—
209C2×C4—24no—
210C2×C4—28no—
211C2×C4—22no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4#32418yes—
224C2×C4C2×C22⋊C812yes—
225C2×C4#40712yes—
226C2×C4#45412yes—
227C2×C4#45612yes—
228C8—84no—
229C8—162no—
230C22—22no—
231C22—24no—
232C22—22no—
233C22—22no—
234C22—22no—
235C22C2×#32412yes—
236C22#146312yes—
237C22#177911yescharacteristic
238C22#178111yescharacteristic
239C22#178411yescharacteristic
240C22#179312yes—
241C22#179412yes—
242C4—22no—
243C4—22no—
244C4—28no—
245C4—22no—
246C4—22no—
247C4—42no—
248C4—48no—
249C4—82no—
250C4—82no—
251C4—82no—
252C2—22no—
253C2—22no—
254C2#829911yescharacteristic
255C2#831912yes—
256C2#845511yescharacteristic
257C2#846012yes—
258C2#846411yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #519C8semidirect

Maximal subgroups

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

subgroupindex
#342492
#113802
#83032
#75792