Finite Group Databasebeta

← #93171

Subgroups — #93171

Order 512.

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

normal subgroups only

This group has 536 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#11381C211yesmaximal, characteristic
2#10968C211yesmaximal, characteristic
3#8555C211yesmaximal, characteristic
4C22×C22⋊Q8C2211yescharacteristic
5C22×#101C412yescomplemented
6C2×#407—21no—
7#1821C412yescomplemented
8C2×#343—21no—
9C2×#342—21no—
10C2×#342C412yes—
11#1463—21no—
12#1462—21no—
13C22⋊#95—21no—
14C23×Q8C2×C411yescharacteristic
15C24×C4D411yescharacteristic
16C22×M16—22no—
17C2×C22⋊Q8—42no—
18C2×C22⋊Q8—42no—
19C2×C22⋊Q8—42no—
20C22×C4⋊C4C2×C411yescharacteristic
21C22×C4⋊C4D411yescharacteristic
22C22×C4⋊C4Q811yescharacteristic
23C22×C22⋊C4—22no—
24C22×C22⋊C4C2×C411yescharacteristic
25C22×C22⋊C4D411yescharacteristic
26C2×#101—42no—
27C2×#101—44no—
28#407—22no—
29#407—44no—
30C2×C22⋊C8—42no—
31C2×#95—42no—
32#343—42no—
33#343—42no—
34#342—42no—
35#342—42no—
36#324—42no—
37#323—42no—
38C25C22⋊C411yescharacteristic
39C22×Q8—22no—
40C22×Q8—41no—
41C22×Q8—41no—
42C22×Q8—42no—
43C22×Q8—42no—
44C23×C4—22no—
45C23×C4—22no—
46C23×C4—41no—
47C23×C4—42no—
48C23×C4—42no—
49C23×C4—42no—
50C23×C4C4211yescharacteristic
51C23×C4C22⋊C411yescharacteristic
52C23×C4C22⋊C411yescharacteristic
53C23×C4C4⋊C411yescharacteristic
54C23×C4C4⋊C411yescharacteristic
55C23×C4C4⋊C411yescharacteristic
56C2×M16—24no—
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—41no—
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—22no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—41no—
79C2×C22⋊C4—41no—
80C2×C22⋊C4—41no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4—42no—
83C2×C22⋊C4—42no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—42no—
86#101—44no—
87#101—44no—
88C22⋊C8—48no—
89C22⋊C8—42no—
90C22⋊C8—42no—
91#95—82no—
92#95—82no—
93C24—21no—
94C24—21no—
95C24—22no—
96C24—22no—
97C24—42no—
98C24—81no—
99C24#9511yescharacteristic
100C24#9912yes—
101C2×Q8—44no—
102C2×Q8—44no—
103C2×Q8—42no—
104C2×Q8—42no—
105C2×Q8—81no—
106C2×Q8—82no—
107C2×Q8—82no—
108C2×Q8—81no—
109C2×Q8—81no—
110C2×Q8—81no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—22no—
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—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—44no—
130C22×C4—42no—
131C22×C4—44no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—44no—
139C22×C4—82no—
140C22×C4—82no—
141C22×C4—81no—
142C22×C4—81no—
143C22×C4—81no—
144C22×C4—81no—
145C22×C4—82no—
146C22×C4—82no—
147C22×C4—84no—
148C22×C4—84no—
149C22×C4#9912yes—
150C22×C4#9912yes—
151M16—44no—
152M16—44no—
153M16—82no—
154M16—82no—
155M16—84no—
156C2×C8—44no—
157C2×C8—82no—
158C2×C8—82no—
159C2×C8—82no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—44no—
163C22⋊C4—44no—
164C22⋊C4—44no—
165C22⋊C4—44no—
166C22⋊C4—44no—
167C22⋊C4—82no—
168C22⋊C4—82no—
169C23—22no—
170C23—21no—
171C23—21no—
172C23—22no—
173C23—21no—
174C23—21no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—41no—
180C23—42no—
181C23—42no—
182C23—42no—
183C23—42no—
184C23—81no—
185C23—81no—
186C23—84no—
187C23—84no—
188C23#34211yescharacteristic
189C23#34211yescharacteristic
190C23#34211yescharacteristic
191C23#35512yes—
192C23#35612yes—
193Q8—84no—
194Q8—84no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—84no—
214C2×C4—84no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—84no—
226C2×C4—88no—
227C2×C4—164no—
228C2×C4—162no—
229C2×C4—162no—
230C2×C4AΣL(1,𝔽16)12yes—
231C2×C4#35212yes—
232C8—84no—
233C8—162no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—21no—
241C22—22no—
242C22—21no—
243C22—44no—
244C22—42no—
245C22—42no—
246C22—44no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—82no—
252C22—81no—
253C22—82no—
254C22—81no—
255C22#130411yescharacteristic
256C22#139012yes—
257C22#139412yes—
258C4—22no—
259C4—22no—
260C4—42no—
261C4—81no—
262C4—81no—
263C4—81no—
264C4—81no—
265C4—164no—
266C2—22no—
267C2—22no—
268C2—42no—
269C2—41no—
270C2—42no—
271C2#730411yescharacteristic
272C2#730811yescharacteristic
273C2#731311yescharacteristic
subgroupG/H
center, socleC22#1304
commutatorC23×C4C42
Frattini, cosocleC22×C22⋊Q8C22
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊Q8 ◁ G
LECSG ▷ C22×C22⋊Q8 ▷ C24 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C22×#101C4semidirect
7. #1821C4semidirect

Maximal subgroups

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

subgroupindex
#113812
#109682
#85552