Finite Group Databasebeta

← #93175

Subgroups — #93175

Order 512.

471 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 471 conjugacy classes of subgroups — too many to draw. Only its 52 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#11382C211yesmaximal, characteristic
2#10983C211yesmaximal, characteristic
3#8551C211yesmaximal, characteristic
4C22×C4⋊Q8C2211yescharacteristic
5C22×#101—21no—
6C22×#101C412yes—
7#1784—21no—
8C2×#343—21no—
9C2×#343C412yes—
10#1728—21no—
11#1728—21no—
12#1463—21no—
13#1463C412yes—
14C23×Q8C2×C411yescharacteristic
15C23×Q8D411yescharacteristic
16C22×M16—22no—
17C22×M16—22no—
18C2×C4⋊Q8—42no—
19C2×C4⋊Q8—42no—
20C2×C4⋊Q8—42no—
21C22×C4⋊C4C2×C411yescharacteristic
22C22×C4⋊C4C2×C411yescharacteristic
23C22×C4⋊C4D411yescharacteristic
24C22×C4⋊C4Q811yescharacteristic
25C22×C42D411yescharacteristic
26C2×#101—44no—
27C2×#101—42no—
28C2×#101—42no—
29C2×#101—44no—
30C2×#101—42no—
31#407—22no—
32#343—42no—
33#343—42no—
34#336—42no—
35#336—42no—
36#324—42no—
37#324—42no—
38C22×Q8—22no—
39C22×Q8—22no—
40C22×Q8—22no—
41C22×Q8—41no—
42C22×Q8—41no—
43C22×Q8—41no—
44C22×Q8—42no—
45C22×Q8—42no—
46C22×Q8—42no—
47C22×Q8—42no—
48C23×C4C4211yescharacteristic
49C23×C4C22⋊C411yescharacteristic
50C23×C4C22⋊C411yescharacteristic
51C23×C4C22⋊C411yescharacteristic
52C23×C4C4⋊C411yescharacteristic
53C23×C4C4⋊C411yescharacteristic
54C23×C4C4⋊C411yescharacteristic
55C2×M16—24no—
56C2×M16—24no—
57C2×M16—42no—
58C2×M16—42no—
59C2×M16—42no—
60C2×M16—42no—
61C2×M16—42no—
62C2×M16—42no—
63C2×M16—42no—
64C2×M16—42no—
65C2×M16—42no—
66C2×M16—42no—
67C2×M16—42no—
68C22×C8—42no—
69C22×C8—42no—
70C4⋊Q8—416no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C42—22no—
84C2×C42—22no—
85C2×C42—41no—
86#101—48no—
87#101—44no—
88#101—44no—
89#101—82no—
90#101—82no—
91C22⋊C8—42no—
92C22⋊C8—42no—
93C24#9511yescharacteristic
94C2×Q8—44no—
95C2×Q8—42no—
96C2×Q8—44no—
97C2×Q8—44no—
98C2×Q8—42no—
99C2×Q8—44no—
100C2×Q8—42no—
101C2×Q8—42no—
102C2×Q8—42no—
103C2×Q8—42no—
104C2×Q8—82no—
105C2×Q8—82no—
106C2×Q8—82no—
107C2×Q8—81no—
108C2×Q8—81no—
109C2×Q8—81no—
110C2×Q8—82no—
111C2×Q8—81no—
112C2×Q8—81no—
113C2×Q8—81no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4#9912yes—
142C22×C4#9912yes—
143C22×C4#9912yes—
144M16—44no—
145M16—44no—
146M16—44no—
147M16—44no—
148M16—82no—
149M16—82no—
150M16—82no—
151M16—82no—
152M16—84no—
153C2×C8—44no—
154C2×C8—44no—
155C2×C8—82no—
156C2×C8—82no—
157C2×C8—82no—
158C2×C8—82no—
159C2×C8—82no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C42—44no—
165C23—22no—
166C23—22no—
167C23—41no—
168C23#34211yescharacteristic
169C23#34211yescharacteristic
170C23#34211yescharacteristic
171Q8—84no—
172Q8—84no—
173Q8—84no—
174Q8—84no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4AΣL(1,𝔽16)12yes—
215C2×C4#35212yes—
216C2×C4#35512yes—
217C2×C4#35612yes—
218C8—84no—
219C8—84no—
220C8—162no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—44no—
229C22#130411yescharacteristic
230C22#139012yes—
231C22#139412yes—
232C4—22no—
233C4—22no—
234C4—22no—
235C4—22no—
236C4—42no—
237C4—42no—
238C4—81no—
239C4—81no—
240C4—81no—
241C2—22no—
242C2—22no—
243C2—41no—
244C2#730411yescharacteristic
245C2#730811yescharacteristic
246C2#731511yescharacteristic
subgroupG/H
center, socleC22#1304
commutatorC23×C4C42
Frattini, cosocleC22×C4⋊Q8C22
prime radicalC24#95
LCSG ▷ C23×C4 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊Q8 ◁ G
LECSG ▷ C22×C4⋊Q8 ▷ C24 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#113822
#109832
#85512