Finite Group Databasebeta

← #349974

Subgroups — #349974

Order 512.

828 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 164 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
1#34374C211yesmaximal, characteristic
2#29939C212yesmaximal
3#25904C211yesmaximal, characteristic
4#9065C214yesmaximal, complemented
5#8922C214yesmaximal, complemented
6#8709C212yesmaximal
7#8345C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#487C2212yes—
10#2857C2211yescharacteristic
11#2382—22no—
12#2365C2214yes—
13#2355—22no—
14Q8×C4⋊C4—22no—
15#2323—24no—
16#2314—22no—
17#2285C2212yes—
18#2100—22no—
19#2083—22no—
20#2076—24no—
21#2026—22no—
22#1990C2212yes—
23#1987C2214yes—
24#1947C2214yes—
25#1873—22no—
26#1797C2212yes—
27C4×C22⋊C8C2211yescharacteristic
28C2×#332C2214yes—
29#1279C2218yes—
30#533—28no—
31C2×C4⋊Q8C2311yescharacteristic
32C2×C4⋊Q8C2312yes—
33C2×#125D412yes—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8—21no—
36C2×C4×Q8—24no—
37C2×C4×Q8—22no—
38C2×#117—22no—
39C2×#117D412yes—
40C22×C42C2311yescharacteristic
41#487—28no—
42C2×#107—24no—
43C2×C4⋊C8C2314yes—
44C2×#103—24no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—22no—
47C2×C4×C8C2312yes—
48#391—24no—
49#391—21no—
50#389—28no—
51#389—24no—
52#389—22no—
53#388—28no—
54#387—28no—
55#387—28no—
56#384—22no—
57#384D412yes—
58#384D412yes—
59#384C2311yescharacteristic
60#382—22no—
61#382—22no—
62#380—22no—
63#380—22no—
64#380D414yes—
65C4×C4⋊C4—24no—
66C4×C4⋊C4—24no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4C2314yes—
69#340—24no—
70#340—24no—
71#340—24no—
72#340—24no—
73#336—22no—
74#336—21no—
75#332—28no—
76C22×Q8—22no—
77C22×Q8—21no—
78C23×C4C2×D412yes—
79C23×C4C2×D411yescharacteristic
80C22×C8—24no—
81C22×C8—22no—
82C4⋊Q8—24no—
83C4⋊Q8—24no—
84C4⋊Q8#4118yes—
85#125—28no—
86C22⋊Q8—216no—
87C22⋊Q8—44no—
88C4×Q8—216no—
89C4×Q8—84no—
90C4×Q8—84no—
91#117—28no—
92#117—82no—
93#117—82no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—28no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4C2×D414yes—
106C2×C4⋊C4C2×D414yes—
107C2×C4⋊C4C2×D414yes—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—41no—
111C2×C42—22no—
112C2×C42—24no—
113C2×C42—24no—
114C2×C42—28no—
115C2×C42C2×D412yes—
116C2×C42C2×D411yescharacteristic
117C2×C42C2411yescharacteristic
118#107—84no—
119C4⋊C8—28no—
120#103—48no—
121C22⋊C8—42no—
122C22⋊C8—82no—
123C4×C8—24no—
124#95—28no—
125#95—28no—
126#95—24no—
127#95—24no—
128#95—42no—
129C24#12011yescharacteristic
130C2×Q8—28no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—44no—
136C2×Q8—42no—
137C22×C4—24no—
138C22×C4—22no—
139C22×C4—24no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4#12012yes—
150C22×C4#12011yescharacteristic
151C22×C4C22×D411yescharacteristic
152C22×C4C22×D412yes—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—44no—
157C2×C8—44no—
158C2×C8—44no—
159C4⋊C4—216no—
160C4⋊C4—24no—
161C4⋊C4—28no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—28no—
165C4⋊C4—44no—
166C4⋊C4—42no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—82no—
172C4⋊C4—82no—
173C4⋊C4—88no—
174C4⋊C4—82no—
175C4⋊C4#12318yes—
176C22⋊C4—28no—
177C22⋊C4—84no—
178C22⋊C4—82no—
179C42—28no—
180C42—22no—
181C42—24no—
182C42—44no—
183C42—44no—
184C42—88no—
185C42C2×#4114yes—
186C23—21no—
187C23—22no—
188C23—22no—
189C23—21no—
190C23—21no—
191C23C2×#12011yescharacteristic
192Q8—48no—
193Q8—48no—
194Q8—84no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—28no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—81no—
215C2×C4—82no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4#45412yes—
220C2×C4C2×#12312yes—
221C2×C4#52514yes—
222C2×C4#57612yes—
223C2×C4#57712yes—
224C2×C4#57814yes—
225C8—82no—
226C8—84no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22#284911yescharacteristic
233C22#301112yes—
234C22#301212yes—
235C22C2×#45411yescharacteristic
236C22#302011yescharacteristic
237C4—22no—
238C4—22no—
239C4—42no—
240C4—48no—
241C4—84no—
242C4—84no—
243C4—161no—
244C4—161no—
245C4#312014yes—
246C4#312414yes—
247C2—81no—
248C2#1869611yescharacteristic
249C2#1873211yescharacteristic
250C2#1895312yes—
251C2#3299111yescharacteristic
252C2#3301712yes—
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. #9065C2semidirect
5. #8922C2semidirect

Maximal subgroups

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

subgroupindex
#343742
#299392
#259042
#90652
#89222
#87092
#83452