Finite Group Databasebeta

← #125381

Subgroups — #125381

Order 512.

360 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 74 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#33055C211yesmaximal, characteristic
2#12608C211yesmaximal, characteristic, complemented
3#12549C211yesmaximal, characteristic, complemented
4#7135C212yesmaximal, complemented
5#7107C212yesmaximal, complemented
6C22×M32—21no—
7C2×#490C411yescharacteristic
8C2×#405C2211yescharacteristic
9#2629C411yescharacteristic
10#2619C411yescharacteristic
11C2×#349—21no—
12C2×#347—21no—
13#1991C2211yescharacteristic
14#1957C2211yescharacteristic
15#1936—21no—
16#1934—21no—
17#1895—21no—
18#1893—21no—
19#1836C411yescharacteristic
20#1376—21no—
21#1376C2212yes—
22#1376C2212yes—
23C22×M16D411yescharacteristic
24C23×C8D411yescharacteristic
25C2×#127C2×C411yescharacteristic
26C2×#125C2×C411yescharacteristic
27C2×#121—21no—
28C2×#117C2×C411yescharacteristic
29C2×M32—24no—
30C2×M32—22no—
31C2×M32—22no—
32C2×M32—42no—
33C2×M32—41no—
34C22×C16—41no—
35#490—42no—
36C2×#102—21no—
37C2×#102—21no—
38C2×#102—21no—
39C2×#102—21no—
40#405—42no—
41C2×#97C2×C411yescharacteristic
42C2×#97D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8C2311yescharacteristic
45#398—41no—
46#396—41no—
47#394—41no—
48#392—41no—
49#390—21no—
50#389—21no—
51#384C2×C411yescharacteristic
52#382C2×C411yescharacteristic
53#380—21no—
54#349—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×C4C22⋊C411yescharacteristic
64C2×M16—22no—
65C2×M16—22no—
66C2×M16—41no—
67C22×C8—22no—
68C22×C8—22no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C22×C8C2×D411yescharacteristic
75#127—22no—
76#127—41no—
77#125—22no—
78#125—41no—
79#121—42no—
80#121—42no—
81#117—22no—
82#117—41no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C42C22⋊C411yescharacteristic
96C2×C42C22×C411yescharacteristic
97M32—24no—
98M32—24no—
99M32—42no—
100M32—44no—
101M32—42no—
102C2×C16—44no—
103C2×C16—42no—
104C2×C16—42no—
105#102—42no—
106#102—42no—
107#102—42no—
108#102—41no—
109#102—41no—
110#97—22no—
111#97—22no—
112C4×C8—41no—
113C4×C8—41no—
114#95—41no—
115#95—41no—
116#95—41no—
117C24—41no—
118C24#10011yescharacteristic
119C2×D4—41no—
120C2×D4—41no—
121C2×D4—41no—
122C2×D4—41no—
123C2×D4—41no—
124C2×D4—41no—
125C2×D4—82no—
126C2×D4—82no—
127C2×D4—161no—
128C2×D4—161no—
129C2×D4—161no—
130C2×D4—161no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4#10011yescharacteristic
141C22×C4#10412yes—
142C22×C4#10412yes—
143C22×C4C2×C22⋊C411yescharacteristic
144M16—44no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—24no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8#9912yes—
158C2×C8#13512yes—
159C2×C8AGL(1,ℤ/8)11yescharacteristic
160C2×C8#13711yescharacteristic
161C4⋊C4—42no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—82no—
169C4⋊C4—161no—
170C4⋊C4—161no—
171C22⋊C4—42no—
172C22⋊C4—82no—
173C22⋊C4—161no—
174C22⋊C4—161no—
175C22⋊C4—161no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42#13112yes—
180C16—44no—
181C16—44no—
182C23—21no—
183C23—21no—
184C23—41no—
185C23—41no—
186C23—81no—
187C23—81no—
188C23—81no—
189C23—81no—
190C23—81no—
191C23—81no—
192C23—81no—
193C23#32912yes—
194C23C2×#10011yescharacteristic
195D4—81no—
196D4—81no—
197D4—81no—
198D4—81no—
199D4—164no—
200D4—164no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4#32912yes—
223C2×C4#40811yescharacteristic
224C2×C4C2×#9911yescharacteristic
225C2×C4#41611yescharacteristic
226C2×C4#41711yescharacteristic
227C2×C4C2×#10411yescharacteristic
228C2×C4C2×#10411yescharacteristic
229C8—22no—
230C8—21no—
231C8—22no—
232C8—21no—
233C8—41no—
234C22—21no—
235C22—21no—
236C22—22no—
237C22—42no—
238C22—41no—
239C22—41no—
240C22—81no—
241C22—161no—
242C22—161no—
243C22—161no—
244C22—161no—
245C22—161no—
246C22—161no—
247C22—161no—
248C22#147111yescharacteristic
249C22C2×#32911yescharacteristic
250C22#155111yescharacteristic
251C4—22no—
252C4—42no—
253C4—81no—
254C4—161no—
255C4—161no—
256C4—161no—
257C4#148311yescharacteristic
258C4#149411yescharacteristic
259C4#150111yescharacteristic
260C4#150411yescharacteristic
261C2—21no—
262C2—21no—
263C2—42no—
264C2—321no—
265C2#769011yescharacteristic
266C2#1198911yescharacteristic
267C2#1199111yescharacteristic
subgroupG/H
center, socleC22#1551
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×C8C23
prime radical#2619C4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12608C2semidirect
3. #12549C2semidirect
4. #7135C2semidirect
5. #7107C2semidirect

Maximal subgroups

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

subgroupindex
#330552
#126082
#125492
#71352
#71072