Finite Group Databasebeta

← #93646

Subgroups — #93646

Order 512.

404 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 56 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#8300C211yesmaximal, characteristic
2#8300C211yesmaximal, characteristic
3#8300C211yesmaximal, characteristic
4C22×#117C2211yescharacteristic
5#1817—21no—
6#1817—21no—
7#1817C412yes—
8C2×#324—21no—
9C2×#324—21no—
10C2×#324C412yes—
11C2×#323—21no—
12C2×#323—21no—
13C2×#323C412yes—
14C24×C4Q811yescharacteristic
15C2×#117—42no—
16C2×#117—42no—
17C2×#117—42no—
18C22×C4⋊C4C2×C411yescharacteristic
19C22×C4⋊C4D411yescharacteristic
20C22×C22⋊C4C2×C411yescharacteristic
21C22×C22⋊C4D411yescharacteristic
22C22×C42C2×C411yescharacteristic
23C22×C42D411yescharacteristic
24C2×C22⋊C8—22no—
25C2×C22⋊C8—22no—
26C2×C22⋊C8—22no—
27#336—42no—
28#336—42no—
29#336—42no—
30#324—42no—
31#324—42no—
32#324—42no—
33#323—42no—
34#323—42no—
35#323—42no—
36C25C4⋊C411yescharacteristic
37C23×C4—22no—
38C23×C4—22no—
39C23×C4—22no—
40C23×C4—41no—
41C23×C4—41no—
42C23×C4—41no—
43C23×C4—41no—
44C23×C4C4211yescharacteristic
45C23×C4C22⋊C411yescharacteristic
46C23×C4C22⋊C411yescharacteristic
47C23×C4C22⋊C411yescharacteristic
48C23×C4C4⋊C411yescharacteristic
49C23×C4C4⋊C411yescharacteristic
50C22×C8—42no—
51C22×C8—42no—
52C22×C8—42no—
53#117—416no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—41no—
58C2×C4⋊C4—41no—
59C2×C4⋊C4—41no—
60C2×C22⋊C4—22no—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—41no—
64C2×C22⋊C4—41no—
65C2×C22⋊C4—41no—
66C2×C42—22no—
67C2×C42—22no—
68C2×C42—22no—
69C2×C42—41no—
70C2×C42—41no—
71C2×C42—41no—
72C22⋊C8—42no—
73C22⋊C8—42no—
74C22⋊C8—42no—
75C22⋊C8—42no—
76C22⋊C8—42no—
77C22⋊C8—42no—
78C24—21no—
79C24—21no—
80C24—21no—
81C24—21no—
82C24—21no—
83C24—21no—
84C24—81no—
85C24—81no—
86C24#9511yescharacteristic
87C24#10812yes—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—44no—
110C22×C4—44no—
111C22×C4—44no—
112C22×C4—44no—
113C22×C4—44no—
114C22×C4—44no—
115C22×C4—44no—
116C22×C4—81no—
117C22×C4—81no—
118C22×C4—81no—
119C22×C4—81no—
120C22×C4#9912yes—
121C22×C4#9912yes—
122C22×C4#9912yes—
123C22×C4#10812yes—
124C22×C4#10812yes—
125C2×C8—82no—
126C2×C8—82no—
127C2×C8—82no—
128C2×C8—84no—
129C2×C8—84no—
130C2×C8—84no—
131C4⋊C4—44no—
132C4⋊C4—44no—
133C22⋊C4—44no—
134C22⋊C4—44no—
135C42—44no—
136C42—44no—
137C23—22no—
138C23—22no—
139C23—22no—
140C23—41no—
141C23—41no—
142C23—41no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23—81no—
147C23—81no—
148C23—81no—
149C23—81no—
150C23—81no—
151C23—81no—
152C23—82no—
153C23—82no—
154C23—82no—
155C23—82no—
156C23#33811yescharacteristic
157C23#34111yescharacteristic
158C23#34111yescharacteristic
159C23#34111yescharacteristic
160C23#34211yescharacteristic
161C23#34211yescharacteristic
162C23#34211yescharacteristic
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—416no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C8—162no—
195C8—162no—
196C8—162no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—42no—
208C22—82no—
209C22—82no—
210C22—82no—
211C22—82no—
212C22—82no—
213C22—82no—
214C22—81no—
215C22—81no—
216C22#129311yescharacteristic
217C22#129311yescharacteristic
218C22#129311yescharacteristic
219C22#129811yescharacteristic
220C22#129811yescharacteristic
221C22#129811yescharacteristic
222C22#130411yescharacteristic
223C4—42no—
224C4—42no—
225C4—42no—
226C4—42no—
227C4—42no—
228C4—81no—
229C2—21no—
230C2—21no—
231C2—21no—
232C2—21no—
233C2—21no—
234C2—21no—
235C2—82no—
236C2#704711yescharacteristic
237C2#731211yescharacteristic
238C2#731311yescharacteristic
subgroupG/H
center, socleC22#1304
commutatorC23×C4C42
Frattini, cosocleC22×#117C22
prime radicalC25C4⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C22×#117 ▷ C24 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#83002