Finite Group Databasebeta

← #327004

Subgroups — #327004

Order 512.

1047 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1047 conjugacy classes of subgroups — too many to draw. Only its 184 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×#2582C211yesmaximal, characteristic
2#32957C212yesmaximal
3#27910C211yesmaximal, characteristic
4#10439C214yesmaximal, complemented
5#10237C214yesmaximal, complemented
6#8725C211yesmaximal, characteristic
7#8311C212yesmaximal
8C22×C4×Q8C2212yes—
9C2×#484—22no—
10C2×#477C2214yes—
11#2857C2211yescharacteristic
12#2582—22no—
13#2582—24no—
14#2582C2214yes—
15C2×#398—22no—
16C2×#391—22no—
17C2×#389—22no—
18C2×#384C2211yescharacteristic
19#2095—22no—
20#1956C2212yes—
21#1863—24no—
22#1852—22no—
23#1845C2212yes—
24#1803C2214yes—
25#1797C2212yes—
26#1787—22no—
27C4×C22⋊C8C2211yescharacteristic
28C2×Q8⋊C8C2218yes—
29#1296C2214yes—
30C23×Q8D412yes—
31C2×C4⋊Q8C2311yescharacteristic
32C2×C22⋊Q8—21no—
33C2×C4×Q8—22no—
34C2×C4×Q8—22no—
35C2×C4×Q8—48no—
36C2×C4×Q8D414yes—
37C2×C4×Q8C2314yes—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4D412yes—
42C22×C4⋊C4D412yes—
43C22×C42—22no—
44C22×C42C2311yescharacteristic
45#484—48no—
46#477—48no—
47C2×#107—24no—
48C2×#107—24no—
49C2×C4⋊C8C2314yes—
50C2×#103—24no—
51C2×#103—24no—
52C2×#103—24no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—22no—
55C2×C4×C8C2312yes—
56#398—28no—
57#398—28no—
58#391—28no—
59#391—28no—
60#391—21no—
61#389—28no—
62#389—28no—
63#389—22no—
64#384—22no—
65#384D412yes—
66#384C2311yescharacteristic
67#384C2312yes—
68#380—22no—
69C2×#95—24no—
70C2×#95—22no—
71#340—24no—
72#336—22no—
73#336—21no—
74Q8⋊C8—48no—
75C22×Q8—22no—
76C22×Q8—24no—
77C22×Q8—21no—
78C22×Q8—44no—
79C22×Q8—44no—
80C22×Q8Q1618yes—
81C22×Q8C2×D414yes—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D412yes—
87C22×C8—22no—
88C22×C8—24no—
89C4⋊Q8—24no—
90C22⋊Q8—44no—
91C4×Q8—28no—
92C4×Q8—28no—
93C4×Q8—44no—
94C4×Q8—44no—
95C4×Q8—48no—
96C4×Q8—48no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—28no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—44no—
108C2×C4⋊C4—48no—
109C2×C4⋊C4—48no—
110C2×C4⋊C4—44no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—84no—
113C2×C4⋊C4C2×D414yes—
114C2×C4⋊C4C2×D414yes—
115C2×C22⋊C4—41no—
116C2×C42—24no—
117C2×C42—24no—
118C2×C42—24no—
119C2×C42—48no—
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D412yes—
122C2×C42C2411yescharacteristic
123#107—48no—
124#107—48no—
125C4⋊C8—28no—
126#103—48no—
127#103—48no—
128#103—48no—
129C22⋊C8—24no—
130C22⋊C8—44no—
131C4×C8—24no—
132#95—216no—
133#95—28no—
134#95—42no—
135C24#12011yescharacteristic
136C2×Q8—24no—
137C2×Q8—28no—
138C2×Q8—24no—
139C2×Q8—24no—
140C2×Q8—24no—
141C2×Q8—24no—
142C2×Q8—416no—
143C2×Q8—44no—
144C2×Q8—44no—
145C2×Q8—416no—
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8C2×Q1618yes—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—28no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—41no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—48no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4#12011yescharacteristic
169C22×C4#12012yes—
170C22×C4C2×Q1614yes—
171C22×C4#13714yes—
172C22×C4C22×D411yescharacteristic
173C22×C4C22×D412yes—
174C2×C8—24no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—44no—
178C2×C8—44no—
179C2×C8—44no—
180C4⋊C4—24no—
181C4⋊C4—24no—
182C4⋊C4—44no—
183C4⋊C4—48no—
184C4⋊C4—44no—
185C4⋊C4—48no—
186C4⋊C4—42no—
187C4⋊C4—48no—
188C4⋊C4—48no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C4⋊C4—88no—
194C22⋊C4—82no—
195C42—24no—
196C42—22no—
197C42—48no—
198C42—48no—
199C42H(5,𝔽2)14yes—
200C23—21no—
201C23—22no—
202C23—21no—
203C23#45114yes—
204C23#45312yes—
205C23C2×#12011yescharacteristic
206Q8—28no—
207Q8—28no—
208Q8—416no—
209Q8—44no—
210Q8—44no—
211Q8—84no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—28no—
217C2×C4—28no—
218C2×C4—21no—
219C2×C4—28no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—44no—
223C2×C4—416no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—48no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—88no—
232C2×C4—88no—
233C2×C4—81no—
234C2×C4—82no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4#45114yes—
239C2×C4#53414yes—
240C2×C4#53512yes—
241C2×C4C22×Q1612yes—
242C2×C4C2×#13712yes—
243C2×C4#57514yes—
244C8—44no—
245C8—84no—
246C22—24no—
247C22—22no—
248C22—24no—
249C22—22no—
250C22#163314yes—
251C22#261311yescharacteristic
252C22C2×#45112yes—
253C22#300512yes—
254C22C2×#45311yescharacteristic
255C22#302011yescharacteristic
256C4—24no—
257C4—22no—
258C4—48no—
259C4—41no—
260C4—84no—
261C4—84no—
262C4—161no—
263C4—161no—
264C4#318914yes—
265C4#319614yes—
266C2—24no—
267C2C2×#163311yescharacteristic
268C2#1852611yescharacteristic
269C2#1877912yes—
270C2#2959712yes—
271C2#2961611yescharacteristic
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. #10439C2semidirect
5. #10237C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
#329572
#279102
#104392
#102372
#87252
#83112