Finite Group Databasebeta

← #125608

Subgroups — #125608

Order 512.

340 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 66 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#33058C211yesmaximal, characteristic
2#12849C211yesmaximal, characteristic, complemented
3#12791C211yesmaximal, characteristic, complemented
4#7130C212yesmaximal, complemented
5#7116C212yesmaximal, complemented
6C2×#490C411yescharacteristic
7C2×C4×M16C2211yescharacteristic
8#2856C411yescharacteristic
9#2851C411yescharacteristic
10C2×#364—21no—
11C2×#349—21no—
12C2×#349—21no—
13#1992C2211yescharacteristic
14#1961C2211yescharacteristic
15#1934—21no—
16#1934—21no—
17#1895—21no—
18#1895—21no—
19#1836C411yescharacteristic
20#1376—21no—
21#1375C2212yes—
22#1375C2212yes—
23C22×M16D411yescharacteristic
24C22×M16D411yescharacteristic
25C2×#127C2×C411yescharacteristic
26C2×#125C2×C411yescharacteristic
27C2×#121—21no—
28C2×#121—41no—
29C22×C42C2×C411yescharacteristic
30C2×M32—22no—
31C2×M32—22no—
32#490—42no—
33C2×#102—21no—
34C2×#102—21no—
35C2×#102—21no—
36C2×#102—21no—
37C4×M16—42no—
38C2×#97C2×C411yescharacteristic
39C2×#97C2311yescharacteristic
40C2×C4×C8D411yescharacteristic
41C2×C4×C8D411yescharacteristic
42#390—21no—
43#390—41no—
44#389—21no—
45#387—41no—
46#385—41no—
47#384C2×C411yescharacteristic
48#382C2×C411yescharacteristic
49#380—21no—
50#364—42no—
51#349—42no—
52#349—42no—
53#340—21no—
54#340—21no—
55#340—21no—
56#340—21no—
57C22×D4—21no—
58C22×D4—41no—
59C22×D4—41no—
60C23×C4—21no—
61C23×C4C22⋊C411yescharacteristic
62C2×M16—22no—
63C2×M16—22no—
64C2×M16—21no—
65C2×M16—21no—
66C2×M16—41no—
67C2×M16—41no—
68C2×M16—41no—
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22⋊C411yescharacteristic
71C22×C8C2×D411yescharacteristic
72C22×C8C2×D411yescharacteristic
73#127—22no—
74#127—41no—
75#125—22no—
76#125—41no—
77#121—42no—
78#121—42no—
79#121—84no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C42—22no—
90C2×C42—21no—
91C2×C42—41no—
92C2×C42C22⋊C411yescharacteristic
93C2×C42C22×C411yescharacteristic
94M32—42no—
95M32—42no—
96M32—42no—
97M32—42no—
98C2×C16—42no—
99C2×C16—42no—
100#102—42no—
101#102—42no—
102#102—42no—
103#102—41no—
104#102—41no—
105#97—22no—
106#97—41no—
107C4×C8—22no—
108C4×C8—41no—
109#95—41no—
110#95—41no—
111#95—41no—
112C24—41no—
113C24#10011yescharacteristic
114C2×D4—41no—
115C2×D4—41no—
116C2×D4—41no—
117C2×D4—41no—
118C2×D4—41no—
119C2×D4—41no—
120C2×D4—82no—
121C2×D4—82no—
122C2×D4—82no—
123C2×D4—82no—
124C2×D4—84no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—81no—
139C22×C4#10011yescharacteristic
140C22×C4C2×C22⋊C411yescharacteristic
141M16—44no—
142M16—42no—
143M16—42no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8#9912yes—
154C2×C8#13512yes—
155C2×C8AGL(1,ℤ/8)11yescharacteristic
156C2×C8#13711yescharacteristic
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—82no—
164C4⋊C4—82no—
165C4⋊C4—82no—
166C22⋊C4—82no—
167C22⋊C4—84no—
168C22⋊C4—82no—
169C42—21no—
170C42—21no—
171C42—21no—
172C42—42no—
173C42#13112yes—
174C16—82no—
175C16—82no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23—41no—
180C23—41no—
181C23—82no—
182C23—82no—
183C23—81no—
184C23—81no—
185C23—81no—
186C23C2×#10011yescharacteristic
187D4—81no—
188D4—81no—
189D4—81no—
190D4—81no—
191D4—164no—
192D4—164no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—84no—
207C2×C4—82no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4#40811yescharacteristic
216C2×C4C2×#9911yescharacteristic
217C2×C4#41611yescharacteristic
218C2×C4#41711yescharacteristic
219C2×C4#42111yescharacteristic
220C2×C4#42111yescharacteristic
221C8—21no—
222C8—21no—
223C8—22no—
224C8—22no—
225C8—41no—
226C22—21no—
227C22—21no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—81no—
232C22—81no—
233C22—162no—
234C22—162no—
235C22—161no—
236C22—161no—
237C22—161no—
238C22#147111yescharacteristic
239C22#153311yescharacteristic
240C22#155111yescharacteristic
241C4—42no—
242C4—42no—
243C4—41no—
244C4—161no—
245C4—161no—
246C4—161no—
247C4#148211yescharacteristic
248C4#149511yescharacteristic
249C4#151311yescharacteristic
250C4#151611yescharacteristic
251C2—21no—
252C2—21no—
253C2—81no—
254C2—321no—
255C2#773011yescharacteristic
256C2#1191511yescharacteristic
257C2#1191711yescharacteristic
subgroupG/H
center, socleC22#1533
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
prime radical#2856C4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12849C2semidirect
3. #12791C2semidirect
4. #7130C2semidirect
5. #7116C2semidirect

Maximal subgroups

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

subgroupindex
#330582
#128492
#127912
#71302
#71162