Finite Group Databasebeta

← #124505

Subgroups — #124505

Order 512.

642 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 118 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×#1954C211yesmaximal, characteristic
2#12844C212yesmaximal, complemented
3#7102C214yesmaximal, complemented
4C22×C4×D4C411yescharacteristic
5C22×C4⋊C8C411yescharacteristic
6C22×C22⋊C8—22no—
7C22×C4×C8C2211yescharacteristic
8C2×C22⋊C16—22no—
9#1954—22no—
10#1954C412yes—
11#1954C2212yes—
12#1374—22no—
13#1374C2214yescomplemented
14C23×D4C811yescharacteristic
15C24×C4—21no—
16C23×C8—21no—
17C23×C8D412yes—
18C2×C4×D4—21no—
19C2×C4×D4—21no—
20C2×C4×D4—42no—
21C2×C4×D4—42no—
22C2×C4×D4C812yes—
23C2×C4×D4C2×C412yes—
24C22×C4⋊C4C811yescharacteristic
25C22×C22⋊C4—21no—
26C22×C42C2×C411yescharacteristic
27C22×C16—24no—
28C2×C4⋊C8—21no—
29C2×C4⋊C8—42no—
30C2×C4⋊C8C2×C412yes—
31C2×C22⋊C8—24no—
32C2×C22⋊C8—24no—
33C2×C22⋊C8—48no—
34C2×C4×C8—42no—
35C2×C4×C8C2×C411yescharacteristic
36C2×C4×C8D412yes—
37C2×C4×C8C2311yescharacteristic
38C22⋊C16—44no—
39C25—21no—
40C22×D4—21no—
41C22×D4—22no—
42C22×D4—42no—
43C22×D4—42no—
44C22×D4C1614yescomplemented
45C22×D4C2×C812yes—
46C23×C4—21no—
47C23×C4—22no—
48C23×C4—22no—
49C23×C4—44no—
50C23×C4—44no—
51C23×C4—42no—
52C23×C4—42no—
53C23×C4C22⋊C411yescharacteristic
54C23×C4C2×C811yescharacteristic
55C23×C4M1611yescharacteristic
56C22×C8—24no—
57C22×C8—24no—
58C22×C8—22no—
59C22×C8—42no—
60C22×C8—44no—
61C22×C8—42no—
62C22×C8C22⋊C412yes—
63C22×C8C2×D412yes—
64C4×D4—44no—
65C4×D4—42no—
66C4×D4—42no—
67C4×D4—44no—
68C4×D4—42no—
69C4×D4—42no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—42no—
72C2×C4⋊C4C2×C812yes—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—44no—
76C2×C42—22no—
77C2×C42—22no—
78C2×C42C22⋊C411yescharacteristic
79C2×C42C2×C811yescharacteristic
80C2×C42M1611yescharacteristic
81C2×C42C22×C411yescharacteristic
82C2×C16—44no—
83C2×C16—44no—
84C2×C16—44no—
85C4⋊C8—42no—
86C4⋊C8—42no—
87C22⋊C8—48no—
88C22⋊C8—48no—
89C4×C8—41no—
90C4×C8—42no—
91C4×C8—41no—
92C24—24no—
93C24—22no—
94C24—44no—
95C24—44no—
96C24—42no—
97C24—42no—
98C24C22⋊C811yescharacteristic
99C2×D4—24no—
100C2×D4—22no—
101C2×D4—42no—
102C2×D4—48no—
103C2×D4—42no—
104C2×D4—42no—
105C2×D4—48no—
106C2×D4—42no—
107C2×D4C2×C1614yes—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—42no—
115C22×C4—48no—
116C22×C4—44no—
117C22×C4—42no—
118C22×C4—44no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—44no—
122C22×C4—44no—
123C22×C4—84no—
124C22×C4—88no—
125C22×C4—84no—
126C22×C4C22⋊C811yescharacteristic
127C22×C4C22⋊C811yescharacteristic
128C22×C4C22⋊C811yescharacteristic
129C22×C4#9912yes—
130C22×C4#10012yes—
131C22×C4C2×C1612yes—
132C22×C4M3212yes—
133C22×C4C2×C22⋊C411yescharacteristic
134C22×C4C22×C811yescharacteristic
135C22×C4C2×M1611yescharacteristic
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—48no—
142C2×C8—42no—
143C2×C8—44no—
144C2×C8—84no—
145C2×C8AGL(1,ℤ/8)14yes—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C22⋊C4—44no—
149C22⋊C4—44no—
150C42—22no—
151C42—22no—
152C42—21no—
153C42—21no—
154C42—21no—
155C42—21no—
156C16—84no—
157C23—22no—
158C23—22no—
159C23—24no—
160C23—416no—
161C23—48no—
162C23—42no—
163C23—44no—
164C23—42no—
165C23—44no—
166C23—84no—
167C23—88no—
168C23—84no—
169C23#32312yes—
170C23C22⋊C1612yes—
171C23#34912yes—
172C23C2×C22⋊C811yescharacteristic
173D4—44no—
174D4—48no—
175D4—44no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—44no—
191C2×C4—42no—
192C2×C4—44no—
193C2×C4—88no—
194C2×C4—84no—
195C2×C4—82no—
196C2×C4—84no—
197C2×C4—82no—
198C2×C4#32312yes—
199C2×C4C22⋊C1612yes—
200C2×C4C2×#9911yescharacteristic
201C2×C4C2×#10011yescharacteristic
202C2×C4#41812yes—
203C2×C4#42112yes—
204C2×C4C22×C1611yescharacteristic
205C2×C4C2×M3211yescharacteristic
206C8—24no—
207C8—42no—
208C8—82no—
209C22—24no—
210C22—24no—
211C22—22no—
212C22—22no—
213C22—48no—
214C22—44no—
215C22—816no—
216C22—84no—
217C22—82no—
218C22—82no—
219C22#131414yes—
220C22C2×#32311yescharacteristic
221C22#147812yes—
222C22C2×C22⋊C1611yescharacteristic
223C22C2×#34911yescharacteristic
224C22#211712yes—
225C4—22no—
226C4—21no—
227C4—21no—
228C4—22no—
229C4—41no—
230C4—41no—
231C4—82no—
232C4#151012yes—
233C4#153311yescharacteristic
234C4#155511yescharacteristic
235C2—24no—
236C2—84no—
237C2#759211yescharacteristic
238C2C2×#131411yescharacteristic
239C2#1142111yescharacteristic
240C2#1162912yes—
241C2#1163612yes—
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C22×C16
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC23×D4C8
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12844C2semidirect
3. #7102C2semidirect
13. #1374C22semidirect
44. C22×D4C16semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
#128442
#71022