Finite Group Databasebeta

← #327934

Subgroups — #327934

Order 512.

721 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 721 conjugacy classes of subgroups — too many to draw. Only its 140 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×#2845C211yesmaximal, characteristic
2#27983C211yesmaximal, characteristic
3#27956C212yesmaximal
4#10585C214yesmaximal, complemented
5#10361C214yesmaximal, complemented
6#8719C211yesmaximal, characteristic
7#8719C212yesmaximal
8#2857C2211yescharacteristic
9#2845—22no—
10#2845—24no—
11#2845C2214yes—
12C2×#398—22no—
13C2×#398—22no—
14C2×#398—22no—
15C2×#382C2212yes—
16C2×#382C2211yescharacteristic
17#2085—24no—
18#2074—22no—
19#1993C2214yes—
20#1960C2212yes—
21#1872—22no—
22#1853—22no—
23#1852—22no—
24#1847C2214yes—
25#1836C2212yes—
26#1749C2212yes—
27#1749C2211yescharacteristic
28#1295C2214yes—
29#1295C2218yes—
30C2×C4⋊Q8C2311yescharacteristic
31C2×C22⋊Q8—21no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4D412yes—
36C22×C4⋊C4D412yes—
37C22×C4⋊C4D412yes—
38C22×C42C2311yescharacteristic
39C2×#107—24no—
40C2×#106—24no—
41C2×#103—24no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—21no—
44C2×#97C2312yes—
45C2×#97C2314yes—
46#398—28no—
47#398—28no—
48#398—28no—
49#398—28no—
50#398—28no—
51#398—28no—
52#391—21no—
53#389—22no—
54#384C2311yescharacteristic
55#382—22no—
56#382—22no—
57#382—22no—
58#382D412yes—
59#382D414yes—
60#382C2312yes—
61#382C2314yes—
62#380—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—22no—
67#340—24no—
68#340—24no—
69#340—24no—
70#336—21no—
71#336—22no—
72C22×Q8—21no—
73C23×C4—22no—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D412yes—
78C22×C8—24no—
79C22×C8—22no—
80C4⋊Q8—24no—
81C22⋊Q8—44no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—28no—
86C2×C4⋊C4—28no—
87C2×C4⋊C4—28no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—82no—
93C2×C4⋊C4—82no—
94C2×C4⋊C4—82no—
95C2×C4⋊C4—84no—
96C2×C4⋊C4—84no—
97C2×C4⋊C4—84no—
98C2×C4⋊C4C2×D414yes—
99C2×C4⋊C4C2×D414yes—
100C2×C4⋊C4C2×D414yes—
101C2×C22⋊C4—41no—
102C2×C42—24no—
103C2×C42C2×D412yes—
104C2×C42C2×D411yescharacteristic
105C2×C42C2411yescharacteristic
106#107—84no—
107#106—84no—
108#103—84no—
109C22⋊C8—28no—
110C22⋊C8—42no—
111#97—28no—
112#97—24no—
113#95—28no—
114#95—24no—
115#95—24no—
116#95—24no—
117#95—24no—
118#95—24no—
119#95—24no—
120#95—42no—
121C24#12011yescharacteristic
122C2×Q8—42no—
123C2×Q8—42no—
124C2×Q8—42no—
125C22×C4—22no—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—22no—
129C22×C4—24no—
130C22×C4—24no—
131C22×C4—24no—
132C22×C4—42no—
133C22×C4—41no—
134C22×C4—44no—
135C22×C4—44no—
136C22×C4—44no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—44no—
141C22×C4—44no—
142C22×C4—44no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4#12012yes—
146C22×C4#12011yescharacteristic
147C22×C4C22×D412yes—
148C22×C4C22×D411yescharacteristic
149C2×C8—28no—
150C2×C8—44no—
151C2×C8—44no—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8—42no—
155C4⋊C4—44no—
156C4⋊C4—44no—
157C4⋊C4—44no—
158C4⋊C4—44no—
159C4⋊C4—42no—
160C4⋊C4—44no—
161C4⋊C4—82no—
162C4⋊C4—82no—
163C4⋊C4—82no—
164C4⋊C4—82no—
165C4⋊C4—88no—
166C4⋊C4—88no—
167C4⋊C4—88no—
168C22⋊C4—82no—
169C42—24no—
170C42—22no—
171C422−1+414yes—
172C23—21no—
173C23—22no—
174C23—21no—
175C23#45412yes—
176C23#45614yes—
177C23C2×#12011yescharacteristic
178Q8—84no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—44no—
190C2×C4—48no—
191C2×C4—44no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—88no—
201C2×C4—88no—
202C2×C4—88no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—82no—
207C2×C4—81no—
208C2×C4#53614yes—
209C2×C4#53612yes—
210C8—48no—
211C8—82no—
212C22—22no—
213C22—24no—
214C22—22no—
215C22—24no—
216C22#166314yes—
217C22#261411yescharacteristic
218C22C2×#45411yescharacteristic
219C22C2×#45612yes—
220C22#301912yes—
221C22#301911yescharacteristic
222C4—22no—
223C4—24no—
224C4—41no—
225C4—42no—
226C4—84no—
227C4—84no—
228C4—84no—
229C4—161no—
230C4—161no—
231C2—24no—
232C2C2×#166311yescharacteristic
233C2#1900111yescharacteristic
234C2#1900612yes—
235C2#2965111yescharacteristic
236C2#2965312yes—
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. #10585C2semidirect
5. #10361C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28452
#279832
#279562
#105852
#103612
#87192