Finite Group Databasebeta

← #94496

Subgroups — #94496

Order 512.

312 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 44 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#11506C211yesmaximal, characteristic
2#11162C211yesmaximal, characteristic
3#7629C211yesmaximal, characteristic
4C22×#101C411yescharacteristic
5#2310C411yescharacteristic
6C2×#391C2211yescharacteristic
7#1749—21no—
8#1736—21no—
9C2×#339—21no—
10#1463—21no—
11#1319—21no—
12#1319C412yescomplemented
13#1272—21no—
14#1272C412yescomplemented
15C23×Q8C2×C411yescharacteristic
16C22×M16—21no—
17C2×#117—21no—
18C22×C4⋊C4D411yescharacteristic
19C22×C42D411yescharacteristic
20C2×#101—41no—
21C2×#101—42no—
22#407—41no—
23C2×C22⋊C8—41no—
24C2×#97—41no—
25#391—41no—
26#391—42no—
27#391C2×C411yescharacteristic
28#391C2×C411yescharacteristic
29#391D411yescharacteristic
30#391Q811yescharacteristic
31#381—41no—
32C2×#95—41no—
33#340—42no—
34#339—42no—
35#336—22no—
36#324—22no—
37#324—41no—
38C22×Q8—21no—
39C22×Q8—41no—
40C22×Q8—41no—
41C22×Q8—41no—
42C22×Q8—41no—
43C22×Q8—41no—
44C22×Q8—41no—
45C22×Q8C4211yescharacteristic
46C22×Q8C4⋊C411yescharacteristic
47C23×C4—21no—
48C23×C4—21no—
49C23×C4—21no—
50C23×C4C22⋊C411yescharacteristic
51C2×M16—22no—
52C2×M16—41no—
53C2×M16—41no—
54C2×M16—41no—
55C2×M16—41no—
56C2×M16—41no—
57C2×M16—81no—
58C22×C8—41no—
59C22×C8—42no—
60#117—41no—
61#117—41no—
62#117—41no—
63#117—41no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4C22⋊C411yescharacteristic
69C2×C4⋊C4C4⋊C411yescharacteristic
70C2×C22⋊C4—41no—
71C2×C42—21no—
72C2×C42—41no—
73C2×C42—41no—
74C2×C42—41no—
75C2×C42C22⋊C411yescharacteristic
76C2×C42C4⋊C411yescharacteristic
77#101—42no—
78#101—42no—
79C22⋊C8—44no—
80C22⋊C8—42no—
81C22⋊C8—81no—
82#97—81no—
83#97—81no—
84#95—41no—
85#95—41no—
86#95—41no—
87#95—41no—
88#95—42no—
89#95—42no—
90C24#9911yescharacteristic
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—42no—
96C2×Q8—42no—
97C2×Q8—41no—
98C2×Q8—41no—
99C2×Q8—41no—
100C2×Q8—41no—
101C2×Q8—41no—
102C2×Q8—41no—
103C2×Q8—81no—
104C2×Q8—81no—
105C2×Q8—81no—
106C2×Q8—81no—
107C2×Q8—81no—
108C2×Q8—81no—
109C2×Q8—81no—
110C2×Q8—81no—
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—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—42no—
132C22×C4—81no—
133C22×C4—82no—
134C22×C4#9511yescharacteristic
135C22×C4#9911yescharacteristic
136M16—42no—
137M16—42no—
138M16—81no—
139M16—81no—
140M16—162no—
141C2×C8—42no—
142C2×C8—44no—
143C2×C8—82no—
144C2×C8—81no—
145C2×C8—81no—
146C2×C8—82no—
147C2×C8—82no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—81no—
153C22⋊C4—81no—
154C22⋊C4—81no—
155C42—41no—
156C42—41no—
157C42—41no—
158C42—41no—
159C42—81no—
160C42—81no—
161C23—21no—
162C23—21no—
163C23—22no—
164C23—41no—
165C23#34211yescharacteristic
166C23AΣL(1,𝔽16)11yescharacteristic
167C23#35211yescharacteristic
168Q8—81no—
169Q8—81no—
170Q8—81no—
171Q8—81no—
172Q8—82no—
173Q8—82no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—84no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—161no—
210C2×C4#34411yescharacteristic
211C2×C4#34411yescharacteristic
212C2×C4AΣL(1,𝔽16)11yescharacteristic
213C2×C4#35211yescharacteristic
214C8—82no—
215C8—84no—
216C8—162no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—42no—
225C22—42no—
226C22#130611yescharacteristic
227C22#139011yescharacteristic
228C22#139011yescharacteristic
229C22#140412yes—
230C4—21no—
231C4—21no—
232C4—21no—
233C4—21no—
234C4—42no—
235C4—81no—
236C4—81no—
237C4—81no—
238C4—81no—
239C4—81no—
240C4—162no—
241C2—21no—
242C2—22no—
243C2—21no—
244C2—41no—
245C2#732411yescharacteristic
246C2#748311yescharacteristic
247C2#748911yescharacteristic
subgroupG/H
center, socleC22#1390
commutatorC22×Q8C42
Frattini, cosocleC2×#391C22
prime radicalC24#99
LCSG ▷ C22×Q8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#391 ◁ G
LECSG ▷ C2×#391 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #1319C4semidirect
14. #1272C4semidirect

Maximal subgroups

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

subgroupindex
#115062
#111622
#76292