Finite Group Databasebeta

← #122246

Subgroups — #122246

Order 512.

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

normal subgroups only

This group has 624 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×#1954C211yesmaximal, characteristic
2#11440C212yesmaximal, complemented
3#7056C214yesmaximal, complemented
4C22×C4×D4C411yescharacteristic
5C22×C4⋊C8C2211yescharacteristic
6C22×C22⋊C8—22no—
7C22×C4×C8C411yescharacteristic
8#2112—22no—
9#1954—22no—
10#1954C412yes—
11#1954C2212yes—
12#1322—22no—
13#1322C2214yescomplemented
14C23×D4C811yescharacteristic
15C24×C4—21no—
16C23×C8—21no—
17C23×C8D412yes—
18C2×C4×D4—21no—
19C2×C4×D4—21no—
20C2×C4×D4—41no—
21C2×C4×D4—41no—
22C2×C4×D4—41no—
23C2×C4×D4—41no—
24C2×C4×D4C812yes—
25C2×C4×D4C2×C412yes—
26C22×C4⋊C4C811yescharacteristic
27C22×C22⋊C4—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×C22⋊C8—24no—
36C2×C22⋊C8—24no—
37C2×C22⋊C8—48no—
38C2×C4×C8—21no—
39C2×C4×C8—41no—
40C2×C4×C8—41no—
41C2×C4×C8C2×C412yes—
42C22⋊C16—24no—
43C22⋊C16—42no—
44C25—21no—
45C22×D4—24no—
46C22×D4—21no—
47C22×D4—41no—
48C22×D4—41no—
49C22×D4—42no—
50C22×D4C2×C812yes—
51C23×C4—21no—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—44no—
55C23×C4—44no—
56C23×C4—41no—
57C23×C4—41no—
58C23×C4—41no—
59C23×C4—41no—
60C23×C4C22⋊C411yescharacteristic
61C23×C4C2×C811yescharacteristic
62C23×C4M1611yescharacteristic
63C22×C8—24no—
64C22×C8—22no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—42no—
68C22×C8—41no—
69C22×C8—41no—
70C22×C8—44no—
71C22×C8C22⋊C412yes—
72C22×C8C2×D412yes—
73C4×D4—42no—
74C4×D4—42no—
75C4×D4—42no—
76C4×D4—42no—
77C4×D4—42no—
78C4×D4—42no—
79C4×D4—42no—
80C4×D4—42no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4C2×C812yes—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—42no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42C22⋊C411yescharacteristic
94C2×C42C2×C811yescharacteristic
95C2×C42M1611yescharacteristic
96C2×C42C22×C411yescharacteristic
97M32—84no—
98C2×C16—44no—
99C4⋊C8—41no—
100C4⋊C8—42no—
101C4⋊C8—41no—
102C22⋊C8—48no—
103C22⋊C8—48no—
104C4×C8—42no—
105C4×C8—42no—
106C24—24no—
107C24—22no—
108C24—44no—
109C24—42no—
110C24—44no—
111C24—41no—
112C24—41no—
113C24C22⋊C811yescharacteristic
114C2×D4—24no—
115C2×D4—22no—
116C2×D4—42no—
117C2×D4—42no—
118C2×D4—42no—
119C2×D4—48no—
120C2×D4—48no—
121C2×D4—42no—
122C2×D4M3214yescomplemented
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—44no—
133C22×C4—48no—
134C22×C4—44no—
135C22×C4—42no—
136C22×C4—44no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—84no—
144C22×C4—84no—
145C22×C4—82no—
146C22×C4—82no—
147C22×C4—82no—
148C22×C4—82no—
149C22×C4C22⋊C811yescharacteristic
150C22×C4C22⋊C811yescharacteristic
151C22×C4C22⋊C811yescharacteristic
152C22×C4#9911yescharacteristic
153C22×C4#9911yescharacteristic
154C22×C4#10011yescharacteristic
155C22×C4#10011yescharacteristic
156C22×C4C2×C22⋊C411yescharacteristic
157C22×C4C22×C811yescharacteristic
158C22×C4C2×M1611yescharacteristic
159C2×C8—24no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—44no—
163C2×C8—44no—
164C2×C8—42no—
165C2×C8—44no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—82no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C22⋊C4—44no—
173C22⋊C4—44no—
174C42—21no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—21no—
182C16—84no—
183C23—22no—
184C23—22no—
185C23—21no—
186C23—21no—
187C23—24no—
188C23—44no—
189C23—44no—
190C23—44no—
191C23—416no—
192C23—44no—
193C23—42no—
194C23—42no—
195C23—82no—
196C23—88no—
197C23—82no—
198C23—82no—
199C23—82no—
200C23#32311yescharacteristic
201C23#32311yescharacteristic
202C23C2×C22⋊C811yescharacteristic
203D4—44no—
204D4—48no—
205D4—44no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—84no—
228C2×C4—84no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—84no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4#32311yescharacteristic
235C2×C4#32311yescharacteristic
236C2×C4#34912yes—
237C2×C4#35312yes—
238C2×C4#35512yes—
239C2×C4C2×#9911yescharacteristic
240C2×C4C2×#10011yescharacteristic
241C2×C4C2×M3212yes—
242C8—44no—
243C8—42no—
244C8—42no—
245C22—24no—
246C22—24no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—21no—
253C22—48no—
254C22—44no—
255C22—88no—
256C22—88no—
257C22—84no—
258C22—82no—
259C22—82no—
260C22#131812yes—
261C22#132112yes—
262C22C2×#32311yescharacteristic
263C22#211211yescharacteristic
264C22C2×#34911yescharacteristic
265C4—22no—
266C4—22no—
267C4—21no—
268C4—21no—
269C4—41no—
270C4—41no—
271C4—82no—
272C4#131812yes—
273C4C2×#35311yescharacteristic
274C4C2×#35511yescharacteristic
275C2—21no—
276C2—22no—
277C2—22no—
278C2—21no—
279C2—84no—
280C2#1141811yescharacteristic
281C2C2×#131811yescharacteristic
282C2C2×#132111yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4⋊C8C23
socleC22C2×#323
prime radicalC23×D4C8
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
2. #11440C2semidirect
3. #7056C2semidirect
13. #1322C22semidirect
122. C2×D4M32semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
#114402
#70562