Finite Group Databasebeta

← #94076

Subgroups — #94076

Order 512.

488 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C2×#1320C212yesmaximal
2#10727C211yesmaximal, characteristic
3C23×C4⋊C4C411yescharacteristic
4C2×#407C411yescharacteristic
5C2×#407C2211yescharacteristic
6C2×#349—22no—
7C2×#343—22no—
8#1320—22no—
9#1320C414yescomplemented
10C24×C4C812yes—
11C24×C4C2×C411yescharacteristic
12C22×M16—21no—
13C22×M16D412yes—
14C22×C4⋊C4—21no—
15C22×C4⋊C4—21no—
16C22×C4⋊C4—44no—
17C22×C4⋊C4—41no—
18C22×C4⋊C4—41no—
19C2×M32—42no—
20#407—21no—
21#407—41no—
22#407—41no—
23#407C812yes—
24#407C2×C412yes—
25#407D411yescharacteristic
26#407Q811yescharacteristic
27C2×C22⋊C8—41no—
28C2×C22⋊C8—41no—
29#349—24no—
30#349—42no—
31#343—44no—
32C25C2×C811yescharacteristic
33C23×C4—21no—
34C23×C4—21no—
35C23×C4—21no—
36C23×C4—21no—
37C23×C4—22no—
38C23×C4—42no—
39C23×C4—42no—
40C23×C4—44no—
41C23×C4—42no—
42C23×C4—84no—
43C23×C4C4211yescharacteristic
44C23×C4C22⋊C411yescharacteristic
45C23×C4C4⋊C411yescharacteristic
46C23×C4C2×C811yescharacteristic
47C23×C4M1611yescharacteristic
48C23×C4M1611yescharacteristic
49C2×M16—22no—
50C2×M16—22no—
51C2×M16—41no—
52C2×M16—42no—
53C2×M16—42no—
54C2×M16—42no—
55C2×M16—42no—
56C2×M16C22⋊C412yes—
57C2×M16C4⋊C412yes—
58C22×C8—22no—
59C22×C8—41no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4—42no—
62C2×C4⋊C4—42no—
63C2×C4⋊C4—44no—
64C2×C4⋊C4—48no—
65C2×C4⋊C4—44no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4—44no—
69M32—44no—
70M32—82no—
71C2×C16—82no—
72C22⋊C8—42no—
73C22⋊C8—41no—
74C22⋊C8—41no—
75C22⋊C8—42no—
76C22⋊C8—42no—
77C24—21no—
78C24—21no—
79C24—41no—
80C24—41no—
81C24—42no—
82C24—42no—
83C24C4×C811yescharacteristic
84C24C22⋊C811yescharacteristic
85C24C4⋊C811yescharacteristic
86C22×C4—22no—
87C22×C4—22no—
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—24no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4—44no—
102C22×C4—44no—
103C22×C4—42no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—42no—
112C22×C4—44no—
113C22×C4—42no—
114C22×C4—44no—
115C22×C4—84no—
116C22×C4—84no—
117C22×C4—88no—
118C22×C4—84no—
119C22×C4—88no—
120C22×C4#9511yescharacteristic
121C22×C4#9711yescharacteristic
122C22×C4C22⋊C811yescharacteristic
123C22×C4#9911yescharacteristic
124C22×C4#9911yescharacteristic
125C22×C4#10011yescharacteristic
126C22×C4#10111yescharacteristic
127C22×C4C4⋊C811yescharacteristic
128M16—42no—
129M16—42no—
130M16—44no—
131M16—82no—
132M16—82no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—42no—
136C2×C8—42no—
137C2×C8—42no—
138C2×C8—81no—
139C2×C8—81no—
140C4⋊C4—44no—
141C4⋊C4—48no—
142C4⋊C4—44no—
143C16—84no—
144C23—22no—
145C23—22no—
146C23—21no—
147C23—21no—
148C23—21no—
149C23—21no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—42no—
154C23—41no—
155C23—42no—
156C23—42no—
157C23—41no—
158C23—41no—
159C23—42no—
160C23—42no—
161C23—42no—
162C23—42no—
163C23—42no—
164C23—44no—
165C23—44no—
166C23—42no—
167C23—41no—
168C23—41no—
169C23#32311yescharacteristic
170C23#32411yescharacteristic
171C23#33611yescharacteristic
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—44no—
194C2×C4—84no—
195C2×C4—84no—
196C2×C4—88no—
197C2×C4—84no—
198C2×C4—88no—
199C2×C4#32311yescharacteristic
200C2×C4#32411yescharacteristic
201C2×C4#34211yescharacteristic
202C2×C4#34311yescharacteristic
203C2×C4AΣL(1,𝔽16)12yes—
204C2×C4#35212yes—
205C8—42no—
206C8—42no—
207C8—82no—
208C22—21no—
209C22—21no—
210C22—22no—
211C22—22no—
212C22—21no—
213C22—21no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—42no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—44no—
226C22—44no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—42no—
232C22—41no—
233C22#128011yescharacteristic
234C22#131611yescharacteristic
235C22#131711yescharacteristic
236C4—21no—
237C4—21no—
238C4—41no—
239C4—41no—
240C4—84no—
241C4#132612yes—
242C4#139012yes—
243C2—21no—
244C2—21no—
245C2—42no—
246C2—42no—
247C2—41no—
248C2—41no—
249C2#718711yescharacteristic
250C2#734011yescharacteristic
251C2#734111yescharacteristic
subgroupG/H
center, socleC22#1316
commutatorC24C4×C8
Frattini, cosocleC2×#407C22
prime radicalC25C2×C8
LCSG ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#407 ◁ G
LECSG ▷ C2×#407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #1320C4semidirect

Maximal subgroups

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

subgroupindex
C2×#13202
#107272