Finite Group Databasebeta

← #124534

Subgroups — #124534

Order 512.

382 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 90 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×#1940C211yesmaximal, characteristic
2#12846C211yesmaximal, characteristic
3#12846C211yesmaximal, characteristic
4#7124C214yesmaximal, complemented
5C22×C4⋊C8—21no—
6C22×C4⋊C8C411yescharacteristic
7C22×#97C2211yescharacteristic
8C2×C4×C4⋊C4C411yescharacteristic
9#2112—21no—
10#2112—21no—
11#1940—21no—
12#1940—21no—
13#1940C412yes—
14#1940C2212yes—
15C2×#336—21no—
16#1377—21no—
17#1377—21no—
18#1377C2212yes—
19#1377C2212yes—
20C23×C8—21no—
21C23×C8D411yescharacteristic
22C23×C8D411yescharacteristic
23C22×C4⋊C4—21no—
24C22×C42—21no—
25C22×C42C2×C411yescharacteristic
26C2×M32—22no—
27C2×M32—22no—
28C2×C4⋊C8—22no—
29C2×C4⋊C8—22no—
30C2×C4⋊C8—21no—
31C2×C4⋊C8—41no—
32C2×C4⋊C8—41no—
33C2×C4⋊C8—44no—
34C2×C4⋊C8C2×C412yes—
35C2×#97—41no—
36C2×#97—41no—
37C2×#97C2×C411yescharacteristic
38C2×#97D411yescharacteristic
39C2×#97D411yescharacteristic
40C2×#97C2311yescharacteristic
41C4×C4⋊C4—21no—
42C4×C4⋊C4—21no—
43C4×C4⋊C4C812yes—
44C4×C4⋊C4C2×C412yes—
45C2×#95C811yescharacteristic
46C2×#95C811yescharacteristic
47C22⋊C16—41no—
48C22⋊C16—41no—
49C22⋊C16—41no—
50C22⋊C16—41no—
51#336—22no—
52#336—22no—
53C23×C4—21no—
54C23×C4—21no—
55C23×C4C22⋊C411yescharacteristic
56C23×C4C2×C811yescharacteristic
57C23×C4M1611yescharacteristic
58C22×C8—21no—
59C22×C8—21no—
60C22×C8—21no—
61C22×C8—21no—
62C22×C8—22no—
63C22×C8—22no—
64C22×C8—22no—
65C22×C8—41no—
66C22×C8—41no—
67C22×C8—42no—
68C22×C8—42no—
69C22×C8—41no—
70C22×C8—41no—
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C22×C8C2×D411yescharacteristic
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—44no—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42—22no—
83C2×C42—22no—
84C2×C42—42no—
85C2×C42—42no—
86C2×C42C22⋊C411yescharacteristic
87C2×C42C2×C811yescharacteristic
88C2×C42M1611yescharacteristic
89C2×C42C22×C411yescharacteristic
90M32—42no—
91M32—42no—
92M32—42no—
93M32—42no—
94C2×C16—42no—
95C2×C16—42no—
96C4⋊C8—42no—
97C4⋊C8—42no—
98C4⋊C8—44no—
99C4⋊C8—44no—
100#97—41no—
101#97—41no—
102#97—41no—
103#97—41no—
104#95—21no—
105#95—21no—
106#95C2×C812yes—
107#95C2×C812yes—
108C24C22⋊C811yescharacteristic
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—24no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4C22⋊C811yescharacteristic
128C22×C4C22⋊C811yescharacteristic
129C22×C4C22⋊C811yescharacteristic
130C22×C4#9911yescharacteristic
131C22×C4#9911yescharacteristic
132C22×C4#10111yescharacteristic
133C22×C4#10111yescharacteristic
134C22×C4C2×C22⋊C411yescharacteristic
135C22×C4C22×C811yescharacteristic
136C22×C4C2×M1611yescharacteristic
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—42no—
148C2×C8—42no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8—82no—
155C2×C8—82no—
156C2×C8AGL(1,ℤ/8)12yes—
157C2×C8#13712yes—
158C4⋊C4—44no—
159C4⋊C4—44no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42—21no—
167C42—21no—
168C42—44no—
169C42—42no—
170C42—42no—
171C16—82no—
172C16—82no—
173C23—22no—
174C23—22no—
175C23—21no—
176C23—21no—
177C23#32411yescharacteristic
178C23#32411yescharacteristic
179C23C2×C22⋊C811yescharacteristic
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—24no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—48no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—84no—
210C2×C4#32411yescharacteristic
211C2×C4#32411yescharacteristic
212C2×C4C2×#9911yescharacteristic
213C2×C4C2×#10111yescharacteristic
214C2×C4#41811yescharacteristic
215C2×C4#41911yescharacteristic
216C2×C4#42111yescharacteristic
217C2×C4#42111yescharacteristic
218C8—22no—
219C8—22no—
220C8—41no—
221C8—41no—
222C8—82no—
223C22—24no—
224C22—24no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22#132712yes—
232C22#132812yes—
233C22C2×#32411yescharacteristic
234C22#147811yescharacteristic
235C22#147911yescharacteristic
236C4—21no—
237C4—21no—
238C4—44no—
239C4—41no—
240C4—41no—
241C4—41no—
242C4—41no—
243C4—82no—
244C4#151011yescharacteristic
245C4#151111yescharacteristic
246C4#153411yescharacteristic
247C4#155611yescharacteristic
248C2—21no—
249C2—22no—
250C2—22no—
251C2—21no—
252C2#759311yescharacteristic
253C2C2×#132711yescharacteristic
254C2C2×#132811yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#97C23
socleC22C2×#324
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7124C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19402
#128462
#71242