Finite Group Databasebeta

← #241293

Subgroups — #241293

Order 512.

1158 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1158 conjugacy classes of subgroups — too many to draw. Only its 234 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×#2551C211yesmaximal, characteristic
2#21365C212yesmaximal
3#20764C211yesmaximal, characteristic
4#10575C218yesmaximal, complemented
5#8040C211yesmaximal, characteristic
6#7904C212yesmaximal
7C22×C4⋊Q8—22no—
8C2×#432C2212yes—
9C2×C4×M16C2211yescharacteristic
10#2551—22no—
11#2551—24no—
12#2551C2214yes—
13C2×#395—22no—
14C2×#389—22no—
15#2285C2211yescharacteristic
16C2×#384C2211yescharacteristic
17C2×C4×C4⋊C4C2212yes—
18#1948C2214yes—
19#1947C2214yes—
20#1853—22no—
21#1852—22no—
22#1836—22no—
23#1834C2214yes—
24#1776—24no—
25#1736—22no—
26#1296C2214yes—
27#1279C2218yes—
28C23×Q8—22no—
29C22×M16—21no—
30C2×C4⋊Q8—28no—
31C2×C4⋊Q8—28no—
32C2×C4⋊Q8—416no—
33C2×#117—21no—
34C22×C4⋊C4—24no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4D412yes—
37C22×C4⋊C4D412yes—
38C22×C4⋊C4D412yes—
39C22×C42—22no—
40C22×C42C2311yescharacteristic
41#432—24no—
42#432Q818yescomplemented
43C2×#107—24no—
44C2×#106—24no—
45C2×C4⋊C8C2314yes—
46C2×C22⋊C8—22no—
47C4×M16—22no—
48C4×M16Q814yescomplemented
49C2×#97—21no—
50C2×#97—22no—
51C2×C4×C8C2312yes—
52#395—28no—
53#395—28no—
54#389—28no—
55#389—28no—
56#384—22no—
57#384D412yes—
58#384C2312yes—
59#384C2312yes—
60#382—21no—
61#380—22no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4D414yes—
66C4×C4⋊C4C2314yes—
67C2×#95—22no—
68C2×#95—22no—
69#340—28no—
70#340—24no—
71#340—24no—
72C22×Q8—28no—
73C22×Q8—416no—
74C22×Q8—84no—
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4C2×D412yes—
79C23×C4C2×D411yescharacteristic
80C2×M16—24no—
81C2×M16—42no—
82C2×M16—42no—
83C22×C8—22no—
84C22×C8—24no—
85C4⋊Q8—432no—
86C4⋊Q8—416no—
87C4⋊Q8—416no—
88#117—44no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—28no—
92C2×C4⋊C4—28no—
93C2×C4⋊C4—216no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—416no—
101C2×C4⋊C4—48no—
102C2×C4⋊C4—44no—
103C2×C4⋊C4C2×D414yes—
104C2×C4⋊C4C2×D414yes—
105C2×C4⋊C4C2×D414yes—
106C2×C22⋊C4—41no—
107C2×C42—24no—
108C2×C42—24no—
109C2×C42—48no—
110C2×C42—41no—
111C2×C42C2×D412yes—
112C2×C42C2×D411yescharacteristic
113C2×C42C2×Q812yes—
114C2×C42C2×Q814yes—
115C2×C42#4112yes—
116C2×C42C2411yescharacteristic
117#107—48no—
118#106—48no—
119C4⋊C8—24no—
120C4⋊C8C2×Q818yescomplemented
121C22⋊C8—28no—
122#97—24no—
123#97—28no—
124C4×C8—22no—
125C4×C8C2×Q814yescomplemented
126#95—28no—
127#95—24no—
128#95—24no—
129#95—42no—
130C24#12011yescharacteristic
131C2×Q8—432no—
132C2×Q8—88no—
133C2×Q8—832no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—28no—
139C22×C4—28no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4—48no—
148C22×C4—44no—
149C22×C4—42no—
150C22×C4#12011yescharacteristic
151C22×C4#12012yes—
152C22×C4C22⋊Q814yes—
153C22×C4C22⋊Q812yes—
154C22×C4C4⋊Q812yes—
155C22×C4C4⋊Q814yes—
156C22×C4C22×D411yescharacteristic
157C22×C4C22×D412yes—
158M16—48no—
159C2×C8—24no—
160C2×C8—28no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—44no—
164C2×C8—44no—
165C4⋊C4—28no—
166C4⋊C4—28no—
167C4⋊C4—24no—
168C4⋊C4—24no—
169C4⋊C4—44no—
170C4⋊C4—48no—
171C4⋊C4—432no—
172C4⋊C4—44no—
173C4⋊C4—48no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—82no—
177C4⋊C4AGL(1,ℤ/8)18yes—
178C4⋊C4#13718yes—
179C22⋊C4—82no—
180C42—48no—
181C42—48no—
182C42—82no—
183C42C22⋊Q812yes—
184C42C22⋊Q814yes—
185C42C4⋊Q812yes—
186C42C4⋊Q814yes—
187C42C22×Q811yescharacteristic
188C42C22×Q812yes—
189C42C2×#4111yescharacteristic
190C23—22no—
191C23—22no—
192C23#39512yes—
193C23#45412yes—
194C23#45512yes—
195C23C2×#12011yescharacteristic
196Q8—832no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—28no—
206C2×C4—28no—
207C2×C4—28no—
208C2×C4—28no—
209C2×C4—416no—
210C2×C4—416no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—48no—
216C2×C4—44no—
217C2×C4—88no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4#39514yes—
222C2×C4#39512yes—
223C2×C4C2×C22⋊Q811yescharacteristic
224C2×C4C2×C22⋊Q812yes—
225C2×C4C2×C4⋊Q811yescharacteristic
226C2×C4C2×C4⋊Q812yes—
227C2×C4C2×AGL(1,ℤ/8)14yes—
228C2×C4C2×#13714yes—
229C8—44no—
230C8—48no—
231C22—24no—
232C22—22no—
233C22—24no—
234C22—22no—
235C22#206114yes—
236C22C2×#39511yescharacteristic
237C22#300312yes—
238C22#300412yes—
239C22C2×#45511yescharacteristic
240C22C2×#45411yescharacteristic
241C4—24no—
242C4—21no—
243C4—21no—
244C4—22no—
245C4—48no—
246C4—48no—
247C4—84no—
248C4—162no—
249C4#307914yes—
250C4#316514yes—
251C2—24no—
252C2#1852911yescharacteristic
253C2#1853412yes—
254C2#1855111yescharacteristic
255C2#2355212yes—
256C2C2×#206111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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
4. #10575C2semidirect
42. #432Q8semidirect
48. C4×M16Q8semidirect
120. C4⋊C8C2×Q8semidirect
125. C4×C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
#213652
#207642
#105752
#80402
#79042