Finite Group Databasebeta

← #122244

Subgroups — #122244

Order 512.

537 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 100 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#21786C211yesmaximal, characteristic
2#11459C212yesmaximal, complemented
3#7056C214yesmaximal, complemented
4C22×C4×D4C411yescharacteristic
5C2×#431C411yescharacteristic
6C2×#431C2211yescharacteristic
7#2162—22no—
8#1954—22no—
9#1954C412yes—
10#1954C2212yes—
11#1779—22no—
12#1322C2214yescomplemented
13#1314—22no—
14C23×D4C811yescharacteristic
15C24×C4—21no—
16C2×C4×D4—21no—
17C2×C4×D4—21no—
18C2×C4×D4—42no—
19C2×C4×D4—42no—
20C2×C4×D4C812yes—
21C2×C4×D4C2×C412yes—
22C22×C4⋊C4C811yescharacteristic
23C22×C22⋊C4—21no—
24C22×C42C2×C411yescharacteristic
25#431—42no—
26#431—42no—
27C2×C4⋊C8—21no—
28C2×C4⋊C8C2×C411yescharacteristic
29C2×C4⋊C8C2311yescharacteristic
30C2×C22⋊C8—21no—
31C2×C22⋊C8—24no—
32C2×C22⋊C8—24no—
33C2×C22⋊C8D412yes—
34C2×C4×C8C2×C412yes—
35C2×C4×C8D412yes—
36C4⋊C16—42no—
37C22⋊C16—24no—
38C4×C16—42no—
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—42no—
50C23×C4—42no—
51C23×C4—82no—
52C23×C4—82no—
53C23×C4C22⋊C411yescharacteristic
54C23×C4C2×C811yescharacteristic
55C23×C4M1611yescharacteristic
56C22×C8—22no—
57C22×C8C22⋊C412yes—
58C22×C8C2×D412yes—
59C4×D4—44no—
60C4×D4—42no—
61C4×D4—42no—
62C4×D4—44no—
63C4×D4—42no—
64C4×D4—42no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4C2×C812yes—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—44no—
71C2×C42—22no—
72C2×C42—22no—
73C2×C42C22⋊C411yescharacteristic
74C2×C42C2×C811yescharacteristic
75C2×C42M1611yescharacteristic
76C2×C42C22×C411yescharacteristic
77C2×C16—44no—
78C4⋊C8—41no—
79C4⋊C8—41no—
80C4⋊C8—42no—
81C22⋊C8—42no—
82C22⋊C8—42no—
83C22⋊C8—84no—
84C22⋊C8—84no—
85C4×C8—22no—
86C4×C8—22no—
87C4×C8—42no—
88C24—22no—
89C24—24no—
90C24—42no—
91C24—42no—
92C24—82no—
93C24—82no—
94C24C22⋊C811yescharacteristic
95C2×D4—24no—
96C2×D4—22no—
97C2×D4—42no—
98C2×D4—42no—
99C2×D4—48no—
100C2×D4—48no—
101C2×D4—42no—
102C2×D4—42no—
103C2×D4C2×C1614yes—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4—44no—
113C22×C4—44no—
114C22×C4—44no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—84no—
119C22×C4—82no—
120C22×C4—81no—
121C22×C4—88no—
122C22×C4—84no—
123C22×C4—84no—
124C22×C4C22⋊C811yescharacteristic
125C22×C4C22⋊C811yescharacteristic
126C22×C4C22⋊C811yescharacteristic
127C22×C4C2×C1612yes—
128C22×C4M3212yes—
129C22×C4C2×C22⋊C411yescharacteristic
130C22×C4C22×C811yescharacteristic
131C22×C4C2×M1611yescharacteristic
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—42no—
135C2×C8—42no—
136C2×C8—42no—
137C2×C8—42no—
138C2×C8—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C22⋊C4—44no—
142C22⋊C4—44no—
143C42—22no—
144C42—22no—
145C42—21no—
146C42—21no—
147C42#9912yes—
148C42#10012yes—
149C16—84no—
150C23—21no—
151C23—21no—
152C23—24no—
153C23—41no—
154C23—41no—
155C23—44no—
156C23—48no—
157C23—44no—
158C23—42no—
159C23—42no—
160C23—88no—
161C23—84no—
162C23—88no—
163C23—84no—
164C23C22⋊C1612yes—
165C23C2×C22⋊C811yescharacteristic
166D4—44no—
167D4—44no—
168D4—48no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—24no—
172C2×C4—24no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—44no—
184C2×C4—42no—
185C2×C4—82no—
186C2×C4—84no—
187C2×C4—88no—
188C2×C4—84no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4#32312yes—
192C2×C4#32312yes—
193C2×C4C22⋊C1612yes—
194C2×C4#34912yes—
195C2×C4C2×#9911yescharacteristic
196C2×C4C2×#10011yescharacteristic
197C2×C4C22×C1611yescharacteristic
198C2×C4C2×M3211yescharacteristic
199C8—42no—
200C8—42no—
201C8—82no—
202C22—21no—
203C22—21no—
204C22—22no—
205C22—42no—
206C22—42no—
207C22—48no—
208C22—41no—
209C22—41no—
210C22—44no—
211C22—84no—
212C22—816no—
213C22—82no—
214C22—82no—
215C22C2×#32311yescharacteristic
216C22C2×C22⋊C1611yescharacteristic
217C22C2×#34911yescharacteristic
218C4—21no—
219C4—21no—
220C4—21no—
221C4—22no—
222C4—41no—
223C4—41no—
224C4—82no—
225C4#131414yes—
226C4#212811yescharacteristic
227C4#213411yescharacteristic
228C2—21no—
229C2—21no—
230C2—42no—
231C2—84no—
232C2C2×#131411yescharacteristic
233C2#1143311yescharacteristic
234C2#1143811yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC2×C4C22×C16
Frattini, cosocleC2×C4⋊C8C23
socleC22C2×#323
prime radicalC23×D4C8
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11459C2semidirect
3. #7056C2semidirect
12. #1322C22semidirect
44. C22×D4C16semidirect

Maximal subgroups

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

subgroupindex
#217862
#114592
#70562