Finite Group Databasebeta

← #122267

Subgroups — #122267

Order 512.

460 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 76 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#21741C211yesmaximal, characteristic
2#11418C212yesmaximal, complemented
3#7059C212yesmaximal, complemented
4#7058C212yesmaximal, complemented
5C2×#525C411yescharacteristic
6C2×#407C411yescharacteristic
7C22×C22⋊C8C2211yescharacteristic
8#2112—22no—
9#1870—22no—
10#1869—22no—
11#1818C411yescharacteristic
12#1818C2211yescharacteristic
13#1780C411yescharacteristic
14#1780C2211yescharacteristic
15#1314—22no—
16#1314C2214yes—
17C22×#41C2×C411yescharacteristic
18C24×C4C2×C411yescharacteristic
19C22×M16—21no—
20C23×C8D412yes—
21#525—42no—
22#525—42no—
23C2×#123—21no—
24C2×C22⋊Q8C811yescharacteristic
25C2×C22⋊Q8C811yescharacteristic
26C2×#121C811yescharacteristic
27C2×#121C811yescharacteristic
28C2×#120—21no—
29C2×C4×D4—21no—
30C2×C4×D4—21no—
31C2×#117C2×C411yescharacteristic
32C2×M32—42no—
33#407—42no—
34C2×C22⋊C8—21no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—42no—
38C2×C22⋊C8C2×C411yescharacteristic
39C2×C22⋊C8C2×C411yescharacteristic
40C2×C22⋊C8C2×C411yescharacteristic
41C2×C22⋊C8D412yes—
42C2×C22⋊C8C2311yescharacteristic
43C22⋊C16—24no—
44C22⋊C16—42no—
45#336—22no—
46#336—22no—
47C25M1611yescharacteristic
48C2×#41—41no—
49C2×#41—41no—
50C2×#41—82no—
51C22×Q8C2×C811yescharacteristic
52C22×D4—21no—
53C22×D4—21no—
54C22×D4—21no—
55C22×D4C2×C811yescharacteristic
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—21no—
59C23×C4—82no—
60C23×C4C22⋊C411yescharacteristic
61C23×C4C22⋊C411yescharacteristic
62C23×C4C2×C811yescharacteristic
63C23×C4C2×C811yescharacteristic
64C23×C4M1611yescharacteristic
65C23×C4C22×C411yescharacteristic
66C2×M16—42no—
67C2×M16—82no—
68C22×C8—22no—
69C22×C8—22no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—82no—
73C22×C8C22⋊C412yes—
74C22×C8C2×D412yes—
75#123—44no—
76C22⋊Q8—42no—
77C22⋊Q8—42no—
78#121—42no—
79#121—42no—
80#120—44no—
81C4×D4—42no—
82C4×D4—42no—
83C4×D4—42no—
84C4×D4—42no—
85#117—41no—
86#117—41no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4C2×C811yescharacteristic
90C2×C4⋊C4C2×C811yescharacteristic
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C42—21no—
97M32—84no—
98C2×C16—44no—
99C22⋊C8—41no—
100C22⋊C8—42no—
101C22⋊C8—41no—
102C22⋊C8—42no—
103C22⋊C8—41no—
104C22⋊C8—41no—
105C22⋊C8—82no—
106C22⋊C8—82no—
107C24—22no—
108C24—21no—
109C24—42no—
110C24—81no—
111C24—81no—
112C24C22⋊C811yescharacteristic
113C24C22⋊C811yescharacteristic
114C24C2×M1611yescharacteristic
115#41—84no—
116#41—84no—
117C2×Q8—41no—
118C2×Q8—81no—
119C2×D4—41no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4—44no—
123C2×D4—42no—
124C2×D4—82no—
125C2×D4—81no—
126C2×D4—81no—
127C2×D4—81no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—82no—
146C22×C4—82no—
147C22×C4—84no—
148C22×C4—81no—
149C22×C4—81no—
150C22×C4—81no—
151C22×C4C22⋊C811yescharacteristic
152C22×C4C22⋊C811yescharacteristic
153C22×C4#9912yes—
154C22×C4#10012yes—
155C22×C4C2×C22⋊C411yescharacteristic
156C22×C4C22×C811yescharacteristic
157M16—84no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—44no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—82no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C22⋊C4—42no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C42—41no—
178C42—41no—
179C16—84no—
180C23—22no—
181C23—21no—
182C23—21no—
183C23—22no—
184C23—22no—
185C23—41no—
186C23—41no—
187C23—44no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—42no—
195C23—42no—
196C23—41no—
197C23—82no—
198C23—82no—
199C23—88no—
200C23#32312yes—
201C23C2×C22⋊C811yescharacteristic
202Q8—82no—
203D4—82no—
204D4—84no—
205D4—84no—
206D4—84no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4#32312yes—
233C2×C4C2×#9911yescharacteristic
234C2×C4C2×#10011yescharacteristic
235C8—42no—
236C8—42no—
237C8—81no—
238C8—81no—
239C22—21no—
240C22—21no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—44no—
245C22—41no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—82no—
252C22—82no—
253C22—82no—
254C22—82no—
255C22—88no—
256C22—81no—
257C22—81no—
258C22—81no—
259C22C2×#32311yescharacteristic
260C22#211712yes—
261C4—21no—
262C4—22no—
263C4—41no—
264C4—41no—
265C4—81no—
266C4—81no—
267C4—81no—
268C4#212112yes—
269C2—22no—
270C2—42no—
271C2—81no—
272C2—81no—
273C2—161no—
274C2#1142111yescharacteristic
275C2#1142611yescharacteristic
276C2#1143811yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×C22⋊C8C23
socleC22C2×#323
prime radicalC2×#525C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11418C2semidirect
3. #7059C2semidirect
4. #7058C2semidirect

Maximal subgroups

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

subgroupindex
#217412
#114182
#70592
#70582