Finite Group Databasebeta

← #391919

Subgroups — #391919

Order 512.

854 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 854 conjugacy classes of subgroups — too many to draw. Only its 242 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
1C2×#2647C211yesmaximal, characteristic
2#20753C211yesmaximal, characteristic
3#20753C211yesmaximal, characteristic
4#11355C212yesmaximal
5#11354C212yesmaximal
6#11204C214yesmaximal
7#11204C214yesmaximal
8C2×#423C2212yes—
9C2×#423C2212yes—
10#2647—24no—
11#2647—24no—
12C2×#400C414yes—
13C2×#398C414yes—
14C2×#389C2212yes—
15C2×#389C2212yes—
16C2×#384C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×#376C2211yescharacteristic
19#1832—24no—
20#1832—24no—
21C2×#343C2218yes—
22C2×#343C2218yes—
23#1728C2212yes—
24#1728C2212yes—
25#1728C2212yes—
26#1728C2212yes—
27C22×M16C2314yes—
28C22×M16C2314yes—
29C22×C4⋊C4C2×C414yes—
30C22×C4⋊C4C2×C414yes—
31C22×C4⋊C4C2×C414yes—
32C22×C4⋊C4C2312yes—
33C22×C4⋊C4C2312yes—
34C22×C42C2311yescharacteristic
35C22×C42C2311yescharacteristic
36C22×C42C2311yescharacteristic
37#423—44no—
38#423—44no—
39C2×C4⋊C8—24no—
40C2×C4⋊C8—24no—
41#400—216no—
42#398—216no—
43#389—24no—
44#389—24no—
45#389—24no—
46#389—24no—
47#384—22no—
48#384—22no—
49#382—22no—
50#382—22no—
51#376D414yes—
52#376D414yes—
53C2×#95C2×C414yes—
54C2×#95C2×C414yes—
55C2×#95C2×C414yes—
56C2×#95C2×C412yes—
57C2×#95C2×C412yes—
58#343—48no—
59#343—48no—
60#336—24no—
61#336—24no—
62#336—24no—
63#336—24no—
64C23×C4C22×C414yes—
65C23×C4C22×C414yes—
66C23×C4C22×C411yescharacteristic
67C23×C4C22×C412yes—
68C23×C4C22×C412yes—
69C23×C4C22×C411yescharacteristic
70C23×C4C2411yescharacteristic
71C2×M16—24no—
72C2×M16—24no—
73C2×M16—28no—
74C2×M16—28no—
75C2×M16—28no—
76C2×M16—28no—
77C2×M16—24no—
78C2×M16—24no—
79C22×C8—24no—
80C22×C8—24no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—28no—
90C2×C4⋊C4—28no—
91C2×C4⋊C4—28no—
92C2×C4⋊C4—84no—
93C2×C4⋊C4—84no—
94C2×C4⋊C4—84no—
95C2×C42—22no—
96C2×C42—22no—
97C2×C42—24no—
98C2×C42—22no—
99C2×C42—22no—
100C2×C42C2×D412yes—
101C2×C42C2×D412yes—
102C2×C42C2×D412yes—
103C2×C42C2×D412yes—
104C2×C42C2×D412yes—
105C2×C42C2×D412yes—
106C4⋊C8—48no—
107C4⋊C8—48no—
108#95—28no—
109#95—28no—
110#95—28no—
111#95C22⋊C418yes—
112#95C22⋊C418yes—
113C24C23×C411yescharacteristic
114C22×C4—24no—
115C22×C4—22no—
116C22×C4—24no—
117C22×C4—22no—
118C22×C4—24no—
119C22×C4—24no—
120C22×C4—28no—
121C22×C4—28no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—44no—
129C22×C4—44no—
130C22×C4—44no—
131C22×C4—44no—
132C22×C4C2×C22⋊C412yes—
133C22×C4C2×C22⋊C414yes—
134C22×C4C2×C22⋊C414yes—
135C22×C4C2×C22⋊C412yes—
136C22×C4C22×D411yescharacteristic
137C22×C4C22×D411yescharacteristic
138C22×C4H(5,𝔽2)12yes—
139C22×C4H(5,𝔽2)12yes—
140C22×C42−1+414yes—
141C22×C42−1+412yes—
142C22×C42−1+412yes—
143M16—48no—
144M16—48no—
145M16—48no—
146M16—48no—
147C2×C8—44no—
148C2×C8—44no—
149C2×C8—44no—
150C2×C8—44no—
151C2×C8—44no—
152C2×C8—44no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—44no—
158C4⋊C4—44no—
159C4⋊C4—88no—
160C4⋊C4—88no—
161C4⋊C4—88no—
162C42—22no—
163C42—22no—
164C42—24no—
165C42—24no—
166C42—22no—
167C42—22no—
168C42—24no—
169C42—24no—
170C23—22no—
171C23—22no—
172C23C22×C22⋊C411yescharacteristic
173C23#51912yes—
174C23#52012yes—
175C23#52012yes—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—48no—
187C2×C4—44no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—44no—
192C2×C4—44no—
193C2×C4—84no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—88no—
197C2×C4—88no—
198C2×C4#53511yescharacteristic
199C2×C4#53511yescharacteristic
200C2×C4#53612yes—
201C2×C4#53612yes—
202C2×C4#53611yescharacteristic
203C2×C4#53611yescharacteristic
204C2×C4#53712yes—
205C2×C4#53712yes—
206C2×C4#56214yes—
207C2×C4#56314yes—
208C2×C4#56314yes—
209C2×C4#56414yes—
210C8—84no—
211C8—84no—
212C22—24no—
213C22—24no—
214C22—22no—
215C22—22no—
216C22#231011yescharacteristic
217C22#231111yescharacteristic
218C22#231111yescharacteristic
219C22#238214yes—
220C22#288812yes—
221C22#288912yes—
222C4—44no—
223C4—44no—
224C4—42no—
225C4—42no—
226C4—84no—
227C4—84no—
228C4#282012yes—
229C4#282012yes—
230C4#282212yes—
231C4#282212yes—
232C2—24no—
233C2#1466411yescharacteristic
234C2#2183212yes—
235C2#2188112yes—
236C2#2188211yescharacteristic
237C2#2188311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#26472
#207532
#113552
#113542
#112042