Finite Group Databasebeta

← #124708

Subgroups — #124708

Order 512.

396 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 72 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#21414C211yesmaximal, characteristic
2#12790C211yesmaximal, characteristic, complemented
3#12606C211yesmaximal, characteristic, complemented
4#7130C212yesmaximal, complemented
5#7116C212yesmaximal, complemented
6C2×#540C411yescharacteristic
7C2×#405C2211yescharacteristic
8#2314C411yescharacteristic
9C2×#364—21no—
10C2×#349—21no—
11#2112—21no—
12C2×#347—21no—
13#1992C2211yescharacteristic
14#1991C411yescharacteristic
15#1961C2211yescharacteristic
16#1957C411yescharacteristic
17#1895—21no—
18#1893—21no—
19#1762—21no—
20C4×#102—21no—
21#1375C2212yes—
22#1375C2212yes—
23C22×M16D411yescharacteristic
24C23×C8D411yescharacteristic
25#540—22no—
26#540—22no—
27#540—42no—
28C2×#127C2×C411yescharacteristic
29C2×#125C2×C411yescharacteristic
30C2×#123—41no—
31C2×#121—21no—
32C2×#121—41no—
33C2×C4×D4—21no—
34C2×#117C2×C411yescharacteristic
35C2×M32—22no—
36C2×M32—22no—
37C2×#102—21no—
38C2×#102—21no—
39C2×#102—21no—
40C2×#102—21no—
41#405—42no—
42C2×#97D411yescharacteristic
43C2×#97C2311yescharacteristic
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8D411yescharacteristic
46#388—41no—
47#387—41no—
48#384C2×C411yescharacteristic
49#382C2×C411yescharacteristic
50#380—21no—
51C4×C4⋊C4—21no—
52#364—42no—
53#349—42no—
54C22⋊C16—24no—
55#347—42no—
56#340—21no—
57#340—21no—
58#340—21no—
59#340—21no—
60C22×D4—21no—
61C22×D4—41no—
62C22×D4—41no—
63C23×C4—41no—
64C23×C4C22⋊C411yescharacteristic
65C2×M16—21no—
66C2×M16—21no—
67C2×M16—41no—
68C2×M16—41no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—41no—
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8C2×D411yescharacteristic
75C22×C8C2×D411yescharacteristic
76#127—21no—
77#127—41no—
78#127M1612yes—
79#125—21no—
80#125—41no—
81#125M1612yes—
82#123—44no—
83#123—44no—
84#121—24no—
85#121—44no—
86#121—44no—
87#121—42no—
88C4×D4—24no—
89C4×D4—42no—
90#117—22no—
91#117—21no—
92#117—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C42—41no—
104C2×C42C22⋊C411yescharacteristic
105C2×C42C22×C411yescharacteristic
106M32—42no—
107M32—42no—
108M32—42no—
109M32—42no—
110C2×C16—24no—
111C2×C16—42no—
112#102—42no—
113#102—42no—
114#102—42no—
115#102—41no—
116#102—41no—
117#97—22no—
118#97—41no—
119C4×C8—22no—
120C4×C8—41no—
121#95—41no—
122#95—41no—
123#95—41no—
124C24—41no—
125C24#10011yescharacteristic
126C2×D4—22no—
127C2×D4—42no—
128C2×D4—42no—
129C2×D4—44no—
130C2×D4—41no—
131C2×D4—41no—
132C2×D4—41no—
133C2×D4—41no—
134C2×D4—41no—
135C2×D4—161no—
136C2×D4—161no—
137C2×D4—161no—
138C2×D4—161no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—44no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—161no—
152C22×C4—161no—
153C22×C4#10011yescharacteristic
154C22×C4C2×C22⋊C411yescharacteristic
155M16—42no—
156M16—42no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—44no—
167C2×C8#10412yes—
168C2×C8#13512yes—
169C2×C8AGL(1,ℤ/8)11yescharacteristic
170C2×C8#13711yescharacteristic
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—161no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—44no—
185C22⋊C4—44no—
186C22⋊C4—161no—
187C42—21no—
188C42—21no—
189C42—161no—
190C42#9912yes—
191C42C2×M1611yescharacteristic
192C42C2×M1611yescharacteristic
193C16—44no—
194C16—82no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—41no—
199C23—41no—
200C23—42no—
201C23—81no—
202C23—81no—
203C23—81no—
204C23—81no—
205C23—81no—
206C23—81no—
207C23C2×#10011yescharacteristic
208D4—42no—
209D4—42no—
210D4—81no—
211D4—81no—
212D4—162no—
213D4—162no—
214D4—164no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—164no—
239C2×C4#40711yescharacteristic
240C2×C4C2×#9911yescharacteristic
241C2×C4#41611yescharacteristic
242C2×C4#41711yescharacteristic
243C2×C4C2×#10411yescharacteristic
244C2×C4#42111yescharacteristic
245C8—22no—
246C8—22no—
247C8—21no—
248C8—21no—
249C8—41no—
250C22—21no—
251C22—21no—
252C22—42no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—81no—
257C22—82no—
258C22—82no—
259C22—82no—
260C22—161no—
261C22—161no—
262C22—161no—
263C22—161no—
264C22#146211yescharacteristic
265C22#153011yescharacteristic
266C22#155111yescharacteristic
267C4—42no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—82no—
272C4—161no—
273C4—161no—
274C4#148811yescharacteristic
275C4#148911yescharacteristic
276C4#150111yescharacteristic
277C4#151311yescharacteristic
278C2—21no—
279C2—21no—
280C2—81no—
281C2—162no—
282C2#761211yescharacteristic
283C2#1191211yescharacteristic
284C2#1191311yescharacteristic
subgroupG/H
center, socleC22#1530
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
prime radicalC2×#540C4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12790C2semidirect
3. #12606C2semidirect
4. #7130C2semidirect
5. #7116C2semidirect

Maximal subgroups

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

subgroupindex
#214142
#127902
#126062
#71302
#71162