Finite Group Databasebeta

← #93971

Subgroups — #93971

Order 512.

428 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 48 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#7187C212yesmaximal
3C22×C22⋊Q8C411yescharacteristic
4C2×#342—22no—
5C2×#323—22no—
6C22⋊#95C411yescharacteristic
7C22⋊#95C2211yescharacteristic
8C23×Q8C811yescharacteristic, complemented
9C24×C4C2×C411yescharacteristic
10C2×C22⋊Q8—44no—
11C2×C22⋊Q8—41no—
12C2×C22⋊Q8—41no—
13C22×C4⋊C4—21no—
14C22×C4⋊C4C811yescharacteristic, complemented
15C22×C22⋊C4—21no—
16C22×C22⋊C4—21no—
17C22×C22⋊C4D412yes—
18C2×C22⋊C8—42no—
19C2×#95—41no—
20C2×#95—41no—
21#342—44no—
22#323—24no—
23#323—42no—
24C25C22⋊C411yescharacteristic
25C22×Q8—42no—
26C22×Q8—41no—
27C22×Q8—82no—
28C23×C4—21no—
29C23×C4—21no—
30C23×C4—21no—
31C23×C4—21no—
32C23×C4—22no—
33C23×C4—42no—
34C23×C4—41no—
35C23×C4—82no—
36C23×C4C2×C811yescharacteristic
37C23×C4M1611yescharacteristic
38C22×C8—82no—
39C22⋊Q8—44no—
40C22⋊Q8—44no—
41C22⋊Q8—44no—
42C22⋊Q8—44no—
43C2×C4⋊C4—41no—
44C2×C4⋊C4—42no—
45C2×C4⋊C4—42no—
46C2×C4⋊C4—42no—
47C2×C4⋊C4—42no—
48C2×C22⋊C4—24no—
49C2×C22⋊C4—22no—
50C2×C22⋊C4—22no—
51C2×C22⋊C4—42no—
52C2×C22⋊C4—41no—
53C2×C22⋊C4—41no—
54C2×C22⋊C4—42no—
55C2×C22⋊C4—42no—
56C2×C22⋊C4—42no—
57C2×C22⋊C4—42no—
58C2×C22⋊C4—41no—
59C2×C22⋊C4SD1612yes—
60C2×C22⋊C4Q1612yes—
61C22⋊C8—44no—
62C22⋊C8—82no—
63#95—82no—
64#95—82no—
65C24—22no—
66C24—21no—
67C24—21no—
68C24—81no—
69C24—81no—
70C24C22⋊C811yescharacteristic
71C24#9911yescharacteristic
72C24#10011yescharacteristic
73C24#10312yes—
74C24#10412yes—
75C2×Q8—42no—
76C2×Q8—42no—
77C2×Q8—82no—
78C2×Q8—84no—
79C2×Q8—82no—
80C2×Q8—84no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—42no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4—42no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4—41no—
100C22×C4—42no—
101C22×C4—81no—
102C22×C4—81no—
103C22×C4—81no—
104C22×C4—84no—
105C22×C4—84no—
106C22×C4—82no—
107C22×C4—82no—
108C22×C4—82no—
109C22×C4—82no—
110C22×C4—81no—
111C22×C4—82no—
112C22×C4—82no—
113C22×C4—81no—
114C2×C8—84no—
115C2×C8—162no—
116C2×C8—162no—
117C4⋊C4—42no—
118C4⋊C4—42no—
119C4⋊C4—48no—
120C22⋊C4—44no—
121C22⋊C4—44no—
122C22⋊C4—42no—
123C22⋊C4—42no—
124C22⋊C4—44no—
125C22⋊C4—82no—
126C22⋊C4—82no—
127C23—21no—
128C23—21no—
129C23—21no—
130C23—21no—
131C23—22no—
132C23—22no—
133C23—22no—
134C23—22no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23—82no—
142C23—82no—
143C23—82no—
144C23—82no—
145C23—82no—
146C23—82no—
147C23#32311yescharacteristic
148C23Q8⋊C812yes—
149C23#32812yes—
150C23#32911yescharacteristic
151C23#33311yescharacteristic
152C23AΣL(1,𝔽16)12yes—
153C23#35411yescharacteristic
154C23#35611yescharacteristic
155Q8—84no—
156Q8—84no—
157C2×C4—22no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—84no—
170C2×C4—84no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—82no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—84no—
184C2×C4—162no—
185C2×C4—162no—
186C2×C4—161no—
187C2×C4—161no—
188C8—164no—
189C22—21no—
190C22—22no—
191C22—21no—
192C22—22no—
193C22—21no—
194C22—22no—
195C22—21no—
196C22—42no—
197C22—41no—
198C22—41no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22—42no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—82no—
209C22—82no—
210C22—82no—
211C22—82no—
212C22#127211yescharacteristic
213C22#131611yescharacteristic
214C22#131911yescharacteristic
215C22#135112yes—
216C22#135412yes—
217C4—42no—
218C4—82no—
219C4—82no—
220C4—81no—
221C4—82no—
222C4—82no—
223C4—162no—
224C2—22no—
225C2—21no—
226C2—21no—
227C2—22no—
228C2—41no—
229C2—42no—
230C2—41no—
231C2#713111yescharacteristic
232C2#733811yescharacteristic
233C2#733911yescharacteristic
subgroupG/H
center, socleC22#1316
commutatorC23×C4C2×C8
Frattini, cosocleC22⋊#95C22
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C25 ◁ C22⋊#95 ◁ G
LECSG ▷ C22⋊#95 ▷ C25 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. C23×Q8C8semidirect
14. C22×C4⋊C4C8semidirect

Maximal subgroups

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

subgroupindex
#113812
#71872