Finite Group Databasebeta

← #327008

Subgroups — #327008

Order 512.

1047 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1047 conjugacy classes of subgroups — too many to draw. Only its 176 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#27905C211yesmaximal, characteristic
4#10437C214yesmaximal, complemented
5#10237C214yesmaximal, complemented
6#8723C211yesmaximal, characteristic
7#8311C212yesmaximal
8C22×C4×Q8C2212yes—
9C2×#483—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#2096—22no—
20#1956C2212yes—
21#1863—24no—
22#1852—22no—
23#1844C2212yes—
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#483—48no—
46#477—48no—
47C2×#107—24no—
48C2×#106—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—28no—
76C22×Q8—22no—
77C22×Q8—24no—
78C22×Q8—24no—
79C22×Q8—21no—
80C22×Q8—44no—
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—44no—
110C2×C4⋊C4—44no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—84no—
114C2×C4⋊C4C2×D414yes—
115C2×C4⋊C4C2×D414yes—
116C2×C22⋊C4—41no—
117C2×C42—24no—
118C2×C42—24no—
119C2×C42—24no—
120C2×C42—48no—
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D412yes—
123C2×C42C2411yescharacteristic
124#107—48no—
125#106—48no—
126C4⋊C8—28no—
127#103—48no—
128#103—48no—
129#103—48no—
130C22⋊C8—24no—
131C22⋊C8—44no—
132C4×C8—24no—
133#95—216no—
134#95—28no—
135#95—42no—
136C24#12011yescharacteristic
137C2×Q8—28no—
138C2×Q8—24no—
139C2×Q8—24no—
140C2×Q8—24no—
141C2×Q8—24no—
142C2×Q8—24no—
143C2×Q8—416no—
144C2×Q8—416no—
145C2×Q8—44no—
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8—44no—
149C2×Q8—42no—
150C2×Q8#13718yes—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—28no—
154C22×C4—24no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—41no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—48no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4#12011yescharacteristic
170C22×C4#12012yes—
171C22×C4#13514yes—
172C22×C4#13714yes—
173C22×C4C22×D411yescharacteristic
174C22×C4C22×D412yes—
175C2×C8—24no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—44no—
179C2×C8—44no—
180C2×C8—44no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C4⋊C4—48no—
186C4⋊C4—42no—
187C4⋊C4—48no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C4⋊C4—48no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—82no—
194C4⋊C4—82no—
195C4⋊C4—88no—
196C22⋊C4—82no—
197C42—24no—
198C42—22no—
199C42—48no—
200C42—48no—
201C42H(5,𝔽2)14yes—
202C23—21no—
203C23—22no—
204C23—21no—
205C23#45214yes—
206C23#45312yes—
207C23C2×#12011yescharacteristic
208Q8—28no—
209Q8—28no—
210Q8—416no—
211Q8—44no—
212Q8—44no—
213Q8—84no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—28no—
219C2×C4—28no—
220C2×C4—21no—
221C2×C4—28no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—44no—
225C2×C4—416no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—48no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—88no—
234C2×C4—88no—
235C2×C4—81no—
236C2×C4—82no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4#45214yes—
241C2×C4#53414yes—
242C2×C4#53512yes—
243C2×C4C2×#13512yes—
244C2×C4C2×#13714yes—
245C2×C4C2×#13712yes—
246C8—44no—
247C8—84no—
248C22—24no—
249C22—22no—
250C22—24no—
251C22—22no—
252C22#163014yes—
253C22#261311yescharacteristic
254C22C2×#45212yes—
255C22#300412yes—
256C22C2×#45311yescharacteristic
257C22#302011yescharacteristic
258C4—24no—
259C4—22no—
260C4—48no—
261C4—41no—
262C4—84no—
263C4—84no—
264C4—161no—
265C4—161no—
266C4#319014yes—
267C4#319514yes—
268C2—24no—
269C2C2×#163011yescharacteristic
270C2#1854611yescharacteristic
271C2#1876112yes—
272C2#2959912yes—
273C2#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. #10437C2semidirect
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
#279052
#104372
#102372
#87232
#83112