Finite Group Databasebeta

← #343559

Subgroups — #343559

Order 512.

496 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 138 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#26022C212yesmaximal
2#24995C212yesmaximal
3#10657C212yesmaximal
4#10626C212yesmaximal
5#10573C211yesmaximal, characteristic
6#10439C211yesmaximal, characteristic
7#9553C212yesmaximal
8#8883C212yesmaximal
9#8816C211yesmaximal, characteristic
10#2582C2211yescharacteristic
11#2551C2211yescharacteristic
12#2355C2212yes—
13#2101—22no—
14#2074—22no—
15#2066—22no—
16#2024—22no—
17#1988—22no—
18#1984C2212yes—
19#1948C2212yes—
20#1947C2212yes—
21#1939C2212yes—
22#1928—22no—
23#1919—22no—
24#1918—22no—
25#1845C2211yescharacteristic
26#1845C2211yescharacteristic
27#1834C2212yes—
28#1834C2212yes—
29#1803C2212yes—
30#1803C2212yes—
31C4×#103C2212yes—
32C2×Q8⋊C8C2212yes—
33C2×C82C2211yescharacteristic
34#1296C2211yescharacteristic
35#1296C2212yes—
36#1296C2211yescharacteristic
37#1279C2212yes—
38#1277C2212yes—
39C2×C4⋊Q8—22no—
40C2×C4×Q8C2312yes—
41C2×#107—22no—
42C2×#107—41no—
43C2×#107—41no—
44C2×#106—22no—
45C2×C4⋊C8—22no—
46C2×C4⋊C8C2312yes—
47C2×C4⋊C8C2312yes—
48C2×#103—22no—
49C2×#103—22no—
50C2×#103D412yes—
51C2×#103D412yes—
52C2×C4×C8C2312yes—
53C2×C4×C8C2311yescharacteristic
54#398—41no—
55#395—41no—
56#391—22no—
57#391—22no—
58#389—22no—
59#389—22no—
60#387—42no—
61#384C2311yescharacteristic
62#384C2312yes—
63#384C2311yescharacteristic
64#382—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4C2312yes—
68#340—22no—
69#340—22no—
70#340—22no—
71#340—22no—
72#340—22no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340D412yes—
77#340D412yes—
78#336—22no—
79#336—22no—
80#336—22no—
81Q8⋊C8—24no—
82C82—22no—
83C22×Q8—41no—
84C22×Q8C2×D412yes—
85C22×C8—21no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8C2×D412yes—
89C22×C8C2×D412yes—
90C4⋊Q8—44no—
91C4⋊Q8—82no—
92C4×Q8—22no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×Q8—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4C2×D412yes—
107C2×C4⋊C4C2×D412yes—
108C2×C4⋊C4C2×D412yes—
109C2×C42—22no—
110C2×C42—22no—
111C2×C42—42no—
112C2×C42C2411yescharacteristic
113#107—44no—
114#107—82no—
115#107—82no—
116#106—44no—
117C4⋊C8—22no—
118C4⋊C8—22no—
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—82no—
122#103—24no—
123#103—42no—
124#103—44no—
125#103—82no—
126C4×C8—22no—
127C4×C8—21no—
128C4×C8—22no—
129C4×C8#4112yes—
130#95—22no—
131#95—42no—
132#95—42no—
133#95—42no—
134#95—41no—
135#95—42no—
136#95—41no—
137C2×Q8—22no—
138C2×Q8—22no—
139C2×Q8—22no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—22no—
143C2×Q8—84no—
144C2×Q8—161no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—42no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4C22×D412yes—
153C22×C4H(5,𝔽2)11yescharacteristic
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—42no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—84no—
185C4⋊C4—82no—
186C4⋊C4—81no—
187C4⋊C4—81no—
188C4⋊C4—161no—
189C4⋊C4#13514yes—
190C42—42no—
191C42—42no—
192C42—42no—
193C42—42no—
194C42—162no—
195C42C2×#4111yescharacteristic
196C422−1+411yescharacteristic
197C422−1+412yes—
198C23D4211yescharacteristic
199Q8—42no—
200Q8—42no—
201Q8—42no—
202Q8—42no—
203Q8—164no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—24no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4#53612yes—
226C2×C4#53911yescharacteristic
227C2×C4#54712yes—
228C2×C4#56311yescharacteristic
229C2×C4C2×#13512yes—
230C2×C4#57512yes—
231C2×C4#57712yes—
232C2×C4#57812yes—
233C8—24no—
234C8—42no—
235C8—42no—
236C8—82no—
237C8—82no—
238C22#268011yescharacteristic
239C22#327712yes—
240C22#328212yes—
241C22#329211yescharacteristic
242C22#329311yescharacteristic
243C4—21no—
244C4—21no—
245C4—44no—
246C4—82no—
247C4—82no—
248C4—82no—
249C4—161no—
250C4—161no—
251C4—162no—
252C4#323312yes—
253C4#324212yes—
254C4#334212yes—
255C4#334812yes—
256C2#1992012yes—
257C2#1999611yescharacteristic
258C2#2031911yescharacteristic
259C2#3109312yes—
260C2#3111011yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#260222
#249952
#106572
#106262
#105732
#104392
#95532
#88832
#88162