Finite Group Databasebeta

← #382731

Subgroups — #382731

Order 512.

1842 conjugacy classes of subgroups; only the 278 normal subgroups are shown.

normal subgroups only

This group has 1842 conjugacy classes of subgroups — too many to draw. Only its 278 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×#2315C211yesmaximal, characteristic
2C2×#1819C212yesmaximal
3#11351C218yesmaximal, complemented
4#10735C214yesmaximal
5C23×C22⋊C4C2211yescharacteristic
6C23×C22⋊C4C2212yes—
7#2315—24no—
8#2315C414yescomplemented
9#2315C2214yes—
10C2×#385—24no—
11C2×#380C414yescomplemented
12C2×#379C2214yes—
13#1819—24no—
14#1819—24no—
15C2×#342C2218yes—
16C2×#342C22116yes—
17C26C2311yescharacteristic
18C24×C4—22no—
19C24×C4C2×C412yes—
20C22×C4⋊C4—24no—
21C22×C22⋊C4—28no—
22C22×C22⋊C4—22no—
23C22×C22⋊C4—22no—
24C22×C22⋊C4—22no—
25C22×C22⋊C4—28no—
26C22×C22⋊C4—48no—
27C22×C22⋊C4C2×C412yes—
28C22×C22⋊C4C2×C414yes—
29C22×C22⋊C4C2×C414yes—
30C22×C22⋊C4D414yes—
31C22×C22⋊C4D414yes—
32C22×C22⋊C4C2318yes—
33C22×C22⋊C4C2314yes—
34C22×C22⋊C4C2312yes—
35#385—216no—
36#385—216no—
37#380—28no—
38#380C2×C4116yes—
39#379—24no—
40#379—24no—
41#379—28no—
42C2×#95—24no—
43#342—216no—
44#342—416no—
45C25—24no—
46C25—24no—
47C25—84no—
48C25C22×C411yescharacteristic
49C25C22×C411yescharacteristic
50C25C2×D412yes—
51C25C2×D412yes—
52C25#4118yes—
53C25C2411yescharacteristic
54C23×C4—24no—
55C23×C4—24no—
56C23×C4—24no—
57C23×C4—24no—
58C23×C4—24no—
59C23×C4—28no—
60C23×C4—48no—
61C23×C4—88no—
62C23×C4—84no—
63C23×C4C22⋊C414yes—
64C23×C4C22⋊C414yes—
65C23×C4C22×C414yes—
66C2×C4⋊C4—216no—
67C2×C4⋊C4—88no—
68C2×C22⋊C4—28no—
69C2×C22⋊C4—216no—
70C2×C22⋊C4—216no—
71C2×C22⋊C4—28no—
72C2×C22⋊C4—216no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—24no—
75C2×C22⋊C4—24no—
76C2×C22⋊C4—28no—
77C2×C22⋊C4—28no—
78C2×C22⋊C4—216no—
79C2×C22⋊C4—216no—
80C2×C22⋊C4—48no—
81C2×C22⋊C4—48no—
82C2×C22⋊C4—48no—
83C2×C22⋊C4—48no—
84C2×C22⋊C4—48no—
85C2×C22⋊C4—48no—
86C2×C22⋊C4—48no—
87C2×C22⋊C4—816no—
88C2×C22⋊C4C22⋊C418yes—
89C2×C22⋊C4C22×C418yes—
90C2×C22⋊C4C2×D418yes—
91#95—216no—
92C24—22no—
93C24—22no—
94C24—24no—
95C24—24no—
96C24—22no—
97C24—22no—
98C24—24no—
99C24—24no—
100C24—232no—
101C24—44no—
102C24—48no—
103C24—416no—
104C24—82no—
105C24—84no—
106C24—88no—
107C24—82no—
108C24—816no—
109C24—816no—
110C24C2×C22⋊C411yescharacteristic
111C24C2×C22⋊C411yescharacteristic
112C24C2×C22⋊C412yes—
113C24#11714yes—
114C24#11714yes—
115C24#12318yes—
116C24#12318yes—
117C24C23×C411yescharacteristic
118C24C22×D412yes—
119C24C2×#4114yes—
120C24H(5,𝔽2)14yes—
121C242−1+414yes—
122C22×C4—24no—
123C22×C4—24no—
124C22×C4—28no—
125C22×C4—28no—
126C22×C4—28no—
127C22×C4—48no—
128C22×C4—48no—
129C22×C4—48no—
130C22×C4—416no—
131C22×C4—48no—
132C22×C4—48no—
133C22×C4—416no—
134C22×C4—88no—
135C22×C4—88no—
136C22×C4—88no—
137C22×C4—88no—
138C22×C4—84no—
139C22×C4—84no—
140C22×C4—832no—
141C22×C4—816no—
142C22×C4C2×C22⋊C418yes—
143C4⋊C4—816no—
144C22⋊C4—216no—
145C22⋊C4—216no—
146C22⋊C4—216no—
147C22⋊C4—48no—
148C22⋊C4—48no—
149C22⋊C4—48no—
150C22⋊C4—48no—
151C22⋊C4—48no—
152C22⋊C4—416no—
153C22⋊C4—816no—
154C22⋊C4—816no—
155C23—21no—
156C23—21no—
157C23—24no—
158C23—21no—
159C23—22no—
160C23—21no—
161C23—22no—
162C23—28no—
163C23—216no—
164C23—28no—
165C23—216no—
166C23—28no—
167C23—216no—
168C23—42no—
169C23—44no—
170C23—44no—
171C23—42no—
172C23—44no—
173C23—48no—
174C23—48no—
175C23—48no—
176C23—48no—
177C23—48no—
178C23—416no—
179C23—48no—
180C23—48no—
181C23—416no—
182C23—84no—
183C23—84no—
184C23—864no—
185C23—88no—
186C23—84no—
187C23—84no—
188C23—88no—
189C23#38114yes—
190C23#38112yes—
191C23#38112yes—
192C23C22×C22⋊C411yescharacteristic
193C23C2×#11712yes—
194C23#51812yes—
195C23#51912yes—
196C23C2×#12314yes—
197C23#53514yes—
198C23#53614yes—
199C23#539116yes—
200C2×C4—28no—
201C2×C4—48no—
202C2×C4—48no—
203C2×C4—48no—
204C2×C4—416no—
205C2×C4—48no—
206C2×C4—88no—
207C2×C4—88no—
208C2×C4—816no—
209C2×C4—816no—
210C2×C4—816no—
211C2×C4—816no—
212C2×C4—88no—
213C2×C4—1616no—
214C22—24no—
215C22—24no—
216C22—22no—
217C22—22no—
218C22—22no—
219C22—24no—
220C22—28no—
221C22—22no—
222C22—24no—
223C22—24no—
224C22—28no—
225C22—44no—
226C22—416no—
227C22—42no—
228C22—42no—
229C22—44no—
230C22—432no—
231C22—416no—
232C22—44no—
233C22—42no—
234C22—42no—
235C22—44no—
236C22—816no—
237C22—88no—
238C22#182814yes—
239C22C2×#38111yescharacteristic
240C22#230911yescharacteristic
241C22#231011yescharacteristic
242C22#231314yes—
243C22#262318yes—
244C4—48no—
245C4—88no—
246C4—168no—
247C2—21no—
248C2—28no—
249C2—21no—
250C2—24no—
251C2—24no—
252C2—42no—
253C2—44no—
254C2#1434211yescharacteristic
255C2C2×#182812yes—
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC26C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C26 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11351C2semidirect
8. #2315C4semidirect
11. C2×#380C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23152
C2×#18192
#113512
#107352