Finite Group Databasebeta

← #122247

Subgroups — #122247

Order 512.

460 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 94 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×#1984C211yesmaximal, characteristic
2#11440C212yesmaximal
3#7057C214yesmaximal, complemented
4C22×C4×Q8C411yescharacteristic
5C22×C4⋊C8C2211yescharacteristic
6C22×C4×C8C411yescharacteristic
7#2112—22no—
8#1984—22no—
9#1984C412yes—
10#1984C2212yes—
11C2×#336—22no—
12#1322—22no—
13#1322C2214yes—
14C23×Q8C811yescharacteristic
15C23×C8—21no—
16C23×C8D412yes—
17C2×C4×Q8—21no—
18C2×C4×Q8—21no—
19C2×C4×Q8—41no—
20C2×C4×Q8—41no—
21C2×C4×Q8—41no—
22C2×C4×Q8—41no—
23C2×C4×Q8C812yes—
24C2×C4×Q8C2×C412yes—
25C22×C4⋊C4—21no—
26C22×C4⋊C4C811yescharacteristic
27C22×C42—21no—
28C22×C42C2×C411yescharacteristic
29C2×M32—42no—
30C2×C4⋊C8—41no—
31C2×C4⋊C8—41no—
32C2×C4⋊C8C2×C411yescharacteristic
33C2×C4⋊C8D412yes—
34C2×C4⋊C8C2311yescharacteristic
35C2×C4×C8—21no—
36C2×C4×C8—41no—
37C2×C4×C8—41no—
38C2×C4×C8C2×C412yes—
39C22⋊C16—24no—
40C22⋊C16—42no—
41#336—24no—
42#336—24no—
43C22×Q8—24no—
44C22×Q8—21no—
45C22×Q8—42no—
46C22×Q8—41no—
47C22×Q8—41no—
48C22×Q8C2×C812yes—
49C23×C4—21no—
50C23×C4—21no—
51C23×C4C22⋊C411yescharacteristic
52C23×C4C2×C811yescharacteristic
53C23×C4M1611yescharacteristic
54C22×C8—24no—
55C22×C8—22no—
56C22×C8—22no—
57C22×C8—22no—
58C22×C8—42no—
59C22×C8—41no—
60C22×C8—41no—
61C22×C8—44no—
62C22×C8C22⋊C412yes—
63C22×C8C2×D412yes—
64C4×Q8—42no—
65C4×Q8—42no—
66C4×Q8—42no—
67C4×Q8—42no—
68C4×Q8—42no—
69C4×Q8—42no—
70C4×Q8—42no—
71C4×Q8—42no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4C2×C812yes—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42—22no—
85C2×C42—22no—
86C2×C42—41no—
87C2×C42—41no—
88C2×C42—41no—
89C2×C42—41no—
90C2×C42C22⋊C411yescharacteristic
91C2×C42C2×C811yescharacteristic
92C2×C42M1611yescharacteristic
93C2×C42C22×C411yescharacteristic
94M32—84no—
95C2×C16—44no—
96C4⋊C8—41no—
97C4⋊C8—42no—
98C4⋊C8—41no—
99C4×C8—42no—
100C4×C8—42no—
101C24C22⋊C811yescharacteristic
102C2×Q8—24no—
103C2×Q8—22no—
104C2×Q8—42no—
105C2×Q8—48no—
106C2×Q8—48no—
107C2×Q8—42no—
108C2×Q8—42no—
109C2×Q8—42no—
110C2×Q8M3214yescomplemented
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—24no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—42no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—42no—
126C22×C4—44no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4C22⋊C811yescharacteristic
130C22×C4C22⋊C811yescharacteristic
131C22×C4C22⋊C811yescharacteristic
132C22×C4#9911yescharacteristic
133C22×C4#9911yescharacteristic
134C22×C4#10011yescharacteristic
135C22×C4#10011yescharacteristic
136C22×C4C2×C22⋊C411yescharacteristic
137C22×C4C22×C811yescharacteristic
138C22×C4C2×M1611yescharacteristic
139C2×C8—24no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—44no—
143C2×C8—44no—
144C2×C8—42no—
145C2×C8—44no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—42no—
149C2×C8—82no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4—44no—
153C4⋊C4—44no—
154C42—21no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42—21no—
161C42—21no—
162C42—42no—
163C42—42no—
164C42—42no—
165C42—42no—
166C16—84no—
167C23—22no—
168C23—22no—
169C23—21no—
170C23—21no—
171C23#32311yescharacteristic
172C23#32311yescharacteristic
173C23C2×C22⋊C811yescharacteristic
174Q8—44no—
175Q8—48no—
176Q8—44no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—24no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—42no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—44no—
202C2×C4—42no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4#32311yescharacteristic
206C2×C4#32311yescharacteristic
207C2×C4#34912yes—
208C2×C4#35412yes—
209C2×C4#35612yes—
210C2×C4C2×#9911yescharacteristic
211C2×C4C2×#10011yescharacteristic
212C2×C4C2×M3212yes—
213C8—44no—
214C8—42no—
215C8—42no—
216C22—24no—
217C22—24no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22#131912yes—
225C22#132112yes—
226C22C2×#32311yescharacteristic
227C22#211211yescharacteristic
228C22C2×#34911yescharacteristic
229C4—22no—
230C4—22no—
231C4—21no—
232C4—21no—
233C4—44no—
234C4—41no—
235C4—41no—
236C4—82no—
237C4#131912yes—
238C4C2×#35411yescharacteristic
239C4C2×#35611yescharacteristic
240C2—22no—
241C2—22no—
242C2—21no—
243C2—21no—
244C2#1141811yescharacteristic
245C2C2×#131911yescharacteristic
246C2C2×#132111yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4⋊C8C23
socleC22C2×#323
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7057C2semidirect
110. C2×Q8M32semidirect

Maximal subgroups

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

subgroupindex
C2×#19842
#114402
#70572