Finite Group Databasebeta

← #327977

Subgroups — #327977

Order 512.

878 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 878 conjugacy classes of subgroups — too many to draw. Only its 168 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×#2571C211yesmaximal, characteristic, complemented
2#33107C212yesmaximal
3#27909C211yesmaximal, characteristic
4#10576C214yesmaximal, complemented
5#10351C214yesmaximal, complemented
6#8723C211yesmaximal, characteristic, complemented
7#8707C212yesmaximal, complemented
8C2×#481—22no—
9C2×#480—22no—
10#2856C2211yescharacteristic
11#2571—22no—
12#2571—24no—
13#2571C2214yes—
14C2×#400—22no—
15C2×#389—22no—
16C2×#384C2211yescharacteristic
17C2×#382—22no—
18C2×C4×C4⋊C4C2212yes—
19#2093—22no—
20#1982C2214yes—
21#1957C2212yes—
22#1948C2214yes—
23#1926—24no—
24#1874—22no—
25#1845C2212yes—
26#1797C2212yes—
27C4×C22⋊C8C2211yescharacteristic
28#1296C2214yes—
29#1279C2218yes—
30C2×#127C2311yescharacteristic
31C2×#121—21no—
32C2×#121—22no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4D412yes—
36C22×C4⋊C4D412yes—
37C22×C4⋊C4D412yes—
38C22×C42—22no—
39C22×C42C2311yescharacteristic
40#481—48no—
41#480—48no—
42C2×#107—24no—
43C2×#106—24no—
44C2×C4⋊C8C2314yes—
45C2×#102—24no—
46C2×#102—24no—
47C2×#102—24no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—22no—
50C2×C4×C8C2312yes—
51#400—28no—
52#400—28no—
53#390—21no—
54#390—22no—
55#389—28no—
56#389—28no—
57#384—22no—
58#384D412yes—
59#384C2311yescharacteristic
60#384C2312yes—
61#382—28no—
62#382—28no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4D414yes—
66C4×C4⋊C4C2314yes—
67C2×#95—22no—
68C2×#95—24no—
69C2×#95—22no—
70#340—24no—
71#336—22no—
72#336—21no—
73C22×D4—21no—
74C22×D4—22no—
75C22×D4—41no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D412yes—
81C22×C8—22no—
82C22×C8—24no—
83#127—24no—
84#121—44no—
85#121—44no—
86#121—44no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—28no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—44no—
98C2×C4⋊C4—44no—
99C2×C4⋊C4—44no—
100C2×C4⋊C4—44no—
101C2×C4⋊C4—84no—
102C2×C4⋊C4C2×D414yes—
103C2×C4⋊C4C2×D414yes—
104C2×C4⋊C4C2×D414yes—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—42no—
107C2×C42—24no—
108C2×C42—24no—
109C2×C42—24no—
110C2×C42—48no—
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D412yes—
113C2×C42C2411yescharacteristic
114#107—48no—
115#106—48no—
116C4⋊C8—28no—
117#102—48no—
118#102—48no—
119#102—48no—
120C22⋊C8—24no—
121C22⋊C8—44no—
122C4×C8—24no—
123#95—28no—
124#95—216no—
125#95—24no—
126#95—24no—
127C24—41no—
128C24#12011yescharacteristic
129C2×D4—42no—
130C2×D4—42no—
131C2×D4—42no—
132C2×D4—44no—
133C2×D4—42no—
134C2×D4—42no—
135C2×D4—44no—
136C2×D4—84no—
137C2×D4—82no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—41no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—48no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—41no—
155C22×C4#12011yescharacteristic
156C22×C4#12012yes—
157C22×C4#13514yes—
158C22×C4AGL(1,ℤ/8)14yes—
159C22×C4C22×D411yescharacteristic
160C22×C4C22×D412yes—
161C2×C8—24no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—44no—
165C2×C8—44no—
166C2×C8—44no—
167C4⋊C4—24no—
168C4⋊C4—24no—
169C4⋊C4—24no—
170C4⋊C4—24no—
171C4⋊C4—44no—
172C4⋊C4—44no—
173C4⋊C4—42no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C4⋊C4—48no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—82no—
181C4⋊C4—88no—
182C22⋊C4—82no—
183C22⋊C4—84no—
184C42—22no—
185C42—24no—
186C42—48no—
187C42—48no—
188C422−1+414yes—
189C23—21no—
190C23—22no—
191C23—21no—
192C23—81no—
193C23—82no—
194C23—82no—
195C23—81no—
196C23—81no—
197C23#44914yes—
198C23#45312yes—
199C23C2×#12011yescharacteristic
200D4—84no—
201D4—84no—
202D4—84no—
203D4—164no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—28no—
209C2×C4—28no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—48no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—88no—
223C2×C4—88no—
224C2×C4—88no—
225C2×C4—82no—
226C2×C4—81no—
227C2×C4#44914yes—
228C2×C4#53612yes—
229C2×C4#53614yes—
230C2×C4C2×#13512yes—
231C2×C4C2×AGL(1,ℤ/8)12yes—
232C2×C4#57514yes—
233C8—44no—
234C8—84no—
235C22—24no—
236C22—22no—
237C22—24no—
238C22—22no—
239C22—161no—
240C22—162no—
241C22—161no—
242C22—162no—
243C22—161no—
244C22#162014yes—
245C22#261411yescharacteristic
246C22C2×#44912yes—
247C22#300512yes—
248C22C2×#45311yescharacteristic
249C22#302011yescharacteristic
250C4—24no—
251C4—22no—
252C4—41no—
253C4—84no—
254C4—84no—
255C4—84no—
256C4—161no—
257C4#323214yes—
258C4#323414yes—
259C2—24no—
260C2—321no—
261C2C2×#162011yescharacteristic
262C2#1854711yescharacteristic
263C2#1878012yes—
264C2#2964012yes—
265C2#2965011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2571C2semidirect
4. #10576C2semidirect
5. #10351C2semidirect
6. #8723C2semidirect
7. #8707C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
#331072
#279092
#105762
#103512
#87232
#87072