Finite Group Databasebeta

← #93106

Subgroups — #93106

Order 512.

463 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 46 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#10968C211yesmaximal, characteristic
2#8555C211yesmaximal, characteristic
3#8550C211yesmaximal, characteristic
4C22×C22⋊Q8C2211yescharacteristic
5C22×#101—21no—
6C2×#407C412yes—
7#1821—21no—
8#1784—21no—
9C2×#343—21no—
10#1463—21no—
11#1463—21no—
12#1462C412yes—
13#1462C412yes—
14C23×Q8D411yescharacteristic
15C24×C4C2×C411yescharacteristic
16C22×M16—22no—
17C2×C22⋊Q8—42no—
18C2×C22⋊Q8—42no—
19C2×C22⋊Q8—42no—
20C22×C4⋊C4D411yescharacteristic
21C22×C4⋊C4D411yescharacteristic
22C22×C4⋊C4Q811yescharacteristic
23C22×C22⋊C4C2×C411yescharacteristic
24C22×C22⋊C4C2×C411yescharacteristic
25C2×#101—44no—
26C2×#101—42no—
27C2×#101—42no—
28#407—22no—
29#407—22no—
30#407—44no—
31C2×C22⋊C8—42no—
32#343—42no—
33#343—42no—
34#324—42no—
35#324—42no—
36#323—42no—
37#323—42no—
38C25C4211yescharacteristic
39C22×Q8—22no—
40C22×Q8—22no—
41C22×Q8—41no—
42C22×Q8—42no—
43C22×Q8—42no—
44C23×C4—22no—
45C23×C4—41no—
46C23×C4—41no—
47C23×C4—42no—
48C23×C4—42no—
49C23×C4C22⋊C411yescharacteristic
50C23×C4C22⋊C411yescharacteristic
51C23×C4C22⋊C411yescharacteristic
52C23×C4C4⋊C411yescharacteristic
53C23×C4C4⋊C411yescharacteristic
54C23×C4C4⋊C411yescharacteristic
55C2×M16—24no—
56C2×M16—42no—
57C2×M16—42no—
58C2×M16—42no—
59C2×M16—42no—
60C2×M16—42no—
61C2×M16—42no—
62C2×M16—42no—
63C22×C8—42no—
64C22⋊Q8—416no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—41no—
77C2×C22⋊C4—41no—
78C2×C22⋊C4—41no—
79C2×C22⋊C4—41no—
80#101—48no—
81#101—82no—
82#101—82no—
83C22⋊C8—44no—
84C22⋊C8—44no—
85C22⋊C8—42no—
86C22⋊C8—42no—
87C22⋊C8—42no—
88C22⋊C8—42no—
89C24—21no—
90C24—21no—
91C24—21no—
92C24—42no—
93C24—42no—
94C24—81no—
95C24#9511yescharacteristic
96C2×Q8—44no—
97C2×Q8—42no—
98C2×Q8—44no—
99C2×Q8—42no—
100C2×Q8—42no—
101C2×Q8—42no—
102C2×Q8—82no—
103C2×Q8—82no—
104C2×Q8—81no—
105C2×Q8—81no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—44no—
124C22×C4—44no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—81no—
131C22×C4—82no—
132C22×C4—81no—
133C22×C4—82no—
134C22×C4—81no—
135C22×C4—81no—
136C22×C4—81no—
137C22×C4—81no—
138C22×C4#9912yes—
139C22×C4#9912yes—
140C22×C4#9912yes—
141M16—44no—
142M16—44no—
143M16—82no—
144M16—82no—
145M16—84no—
146M16—84no—
147C2×C8—44no—
148C2×C8—82no—
149C2×C8—82no—
150C2×C8—82no—
151C2×C8—82no—
152C4⋊C4—44no—
153C4⋊C4—44no—
154C4⋊C4—44no—
155C22⋊C4—44no—
156C22⋊C4—44no—
157C23—22no—
158C23—22no—
159C23—22no—
160C23—22no—
161C23—41no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23—42no—
167C23—41no—
168C23—42no—
169C23—81no—
170C23—81no—
171C23—81no—
172C23—81no—
173C23—88no—
174C23#34211yescharacteristic
175C23#34211yescharacteristic
176C23#34211yescharacteristic
177Q8—84no—
178Q8—84no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—84no—
200C2×C4—84no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4AΣL(1,𝔽16)12yes—
210C2×C4#35212yes—
211C8—84no—
212C8—162no—
213C8—162no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—44no—
223C22—44no—
224C22—42no—
225C22—82no—
226C22—82no—
227C22—82no—
228C22—82no—
229C22—82no—
230C22—81no—
231C22—81no—
232C22#130411yescharacteristic
233C22#139012yes—
234C4—22no—
235C4—22no—
236C4—42no—
237C4—42no—
238C4—81no—
239C4—81no—
240C4—81no—
241C2—22no—
242C2—41no—
243C2—41no—
244C2—82no—
245C2#730411yescharacteristic
246C2#731211yescharacteristic
247C2#731711yescharacteristic
subgroupG/H
center, socleC22#1304
commutator, prime radicalC25C42
Frattini, cosocleC22×C22⋊Q8C22
LCSG ▷ C25 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊Q8 ◁ G
LECSG ▷ C22×C22⋊Q8 ▷ C24 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#109682
#85552
#85502