Finite Group Databasebeta

← #327924

Subgroups — #327924

Order 512.

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

normal subgroups only

This group has 812 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, complemented
2#27982C211yesmaximal, characteristic
3#27918C212yesmaximal
4#10585C214yesmaximal, complemented
5#10360C214yesmaximal, complemented
6#8719C211yesmaximal, characteristic, complemented
7#8718C212yesmaximal, complemented
8C2×#481—22no—
9C2×#480—22no—
10#2856C2211yescharacteristic
11#2845—22no—
12#2845—24no—
13#2845C2214yes—
14C2×#398—22no—
15C2×#398—22no—
16C2×#398—22no—
17C2×#382C2212yes—
18C2×#382C2211yescharacteristic
19#2092—24no—
20#2073—22no—
21#1961C2214yes—
22#1961C2212yes—
23#1874—22no—
24#1847C2214yes—
25#1836C2212yes—
26#1749C2211yescharacteristic
27#1749C2212yes—
28#1295C2218yes—
29#1295C2214yes—
30C2×#127C2311yescharacteristic
31C2×#121—22no—
32C2×#121—21no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4D412yes—
37C22×C4⋊C4D412yes—
38C22×C4⋊C4D412yes—
39C22×C42C2311yescharacteristic
40#481—48no—
41#480—48no—
42C2×#107—24no—
43C2×#106—24no—
44C2×#102—24no—
45C2×#102—24no—
46C2×#102—24no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—21no—
49C2×#97C2312yes—
50C2×#97C2314yes—
51#398—28no—
52#398—28no—
53#398—28no—
54#398—28no—
55#398—28no—
56#398—28no—
57#390—21no—
58#390—22no—
59#384C2311yescharacteristic
60#382—22no—
61#382—22no—
62#382—22no—
63#382D412yes—
64#382D414yes—
65#382C2312yes—
66#382C2314yes—
67C2×#95—22no—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—22no—
71#340—24no—
72#336—21no—
73#336—22no—
74C22×D4—22no—
75C22×D4—21no—
76C22×D4—41no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4C2×D412yes—
81C23×C4C2×D411yescharacteristic
82C22×C8—24no—
83C22×C8—22no—
84#127—24no—
85#121—44no—
86#121—44no—
87#121—44no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—28no—
91C2×C4⋊C4—28no—
92C2×C4⋊C4—28no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—44no—
97C2×C4⋊C4—44no—
98C2×C4⋊C4—82no—
99C2×C4⋊C4—84no—
100C2×C4⋊C4—84no—
101C2×C4⋊C4—84no—
102C2×C4⋊C4C2×D414yes—
103C2×C4⋊C4C2×D414yes—
104C2×C4⋊C4C2×D414yes—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—41no—
107C2×C42—24no—
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D412yes—
110C2×C42C2411yescharacteristic
111#107—48no—
112#106—48no—
113#102—48no—
114#102—48no—
115#102—84no—
116C22⋊C8—28no—
117C22⋊C8—42no—
118#97—28no—
119#97—24no—
120#95—28no—
121#95—24no—
122#95—24no—
123#95—24no—
124#95—24no—
125#95—24no—
126#95—24no—
127C24—41no—
128C24#12011yescharacteristic
129C2×D4—42no—
130C2×D4—42no—
131C2×D4—42no—
132C2×D4—42no—
133C2×D4—44no—
134C2×D4—44no—
135C2×D4—42no—
136C2×D4—84no—
137C2×D4—82no—
138C22×C4—24no—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—41no—
146C22×C4—42no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4—44no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—41no—
157C22×C4#12012yes—
158C22×C4#12011yescharacteristic
159C22×C4C22×D412yes—
160C22×C4C22×D411yescharacteristic
161C2×C8—28no—
162C2×C8—44no—
163C2×C8—44no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—82no—
174C4⋊C4—82no—
175C4⋊C4—82no—
176C4⋊C4—88no—
177C4⋊C4—88no—
178C4⋊C4—88no—
179C22⋊C4—84no—
180C22⋊C4—82no—
181C42—22no—
182C42—24no—
183C422−1+414yes—
184C23—21no—
185C23—21no—
186C23—22no—
187C23—81no—
188C23—82no—
189C23—81no—
190C23—81no—
191C23—82no—
192C23#45314yes—
193C23#45412yes—
194C23C2×#12011yescharacteristic
195D4—84no—
196D4—84no—
197D4—84no—
198D4—164no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—48no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—44no—
220C2×C4—88no—
221C2×C4—88no—
222C2×C4—88no—
223C2×C4—82no—
224C2×C4—81no—
225C2×C4#53614yes—
226C2×C4#53612yes—
227C8—48no—
228C8—82no—
229C22—24no—
230C22—22no—
231C22—24no—
232C22—22no—
233C22—161no—
234C22—162no—
235C22—161no—
236C22—162no—
237C22—161no—
238C22#165614yes—
239C22#261411yescharacteristic
240C22C2×#45312yes—
241C22C2×#45411yescharacteristic
242C22#301911yescharacteristic
243C22#302012yes—
244C4—24no—
245C4—22no—
246C4—42no—
247C4—41no—
248C4—84no—
249C4—84no—
250C4—84no—
251C4—161no—
252C2—24no—
253C2—321no—
254C2C2×#165611yescharacteristic
255C2#1899312yes—
256C2#1900311yescharacteristic
257C2#2965012yes—
258C2#2965111yescharacteristic
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×#2845C2semidirect
4. #10585C2semidirect
5. #10360C2semidirect
6. #8719C2semidirect
7. #8718C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28452
#279822
#279182
#105852
#103602
#87192
#87182