Finite Group Databasebeta

← #312158

Subgroups — #312158

Order 512.

546 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 142 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#26513C212yesmaximal
2#23217C212yesmaximal
3#10541C212yesmaximal
4#10352C211yesmaximal, characteristic
5#10317C211yesmaximal, characteristic
6#9565C212yesmaximal
7#9536C212yesmaximal
8#9059C212yesmaximal
9#8928C211yesmaximal, characteristic
10C2×#498C2211yescharacteristic
11C2×#487C2212yes—
12#2571C2211yescharacteristic
13#2369C2211yescharacteristic
14#2365C2212yes—
15#2101—22no—
16#2087—22no—
17#1990C2212yes—
18#1988C2212yes—
19#1987C2212yes—
20#1981C2212yes—
21#1956C2211yescharacteristic
22#1949—22no—
23#1948C2212yes—
24#1939C2212yes—
25#1933—22no—
26#1928—22no—
27#1919—22no—
28#1863—22no—
29C4×#107C2212yes—
30C8×C4⋊C4C2212yes—
31C4×#103—22no—
32C2×#333C2211yescharacteristic
33C2×#332C2212yes—
34#1296C2211yescharacteristic
35#1279C2212yes—
36#1279C2212yes—
37#1277C2212yes—
38#1273C2211yescharacteristic
39C2×C4⋊Q8C2312yes—
40C2×C4⋊Q8C2311yescharacteristic
41C2×C4×Q8—22no—
42#498—24no—
43#487—24no—
44#487—24no—
45C2×#107D412yes—
46C2×#107D412yes—
47C2×#106—22no—
48C2×#106—41no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8C2312yes—
51C2×C4⋊C8C2312yes—
52C2×#103—22no—
53C2×#103—22no—
54C2×#103—22no—
55C2×#103—22no—
56C2×#103—41no—
57C2×C4×C8—22no—
58C2×C4×C8C2311yescharacteristic
59C2×C4×C8C2312yes—
60#400—41no—
61#391—22no—
62#391—41no—
63#389—22no—
64#389—22no—
65#389—22no—
66#387—42no—
67#384C2311yescharacteristic
68#382—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4C2312yes—
72C4×C4⋊C4C2312yes—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—22no—
79#340—22no—
80#340D412yes—
81#340D412yes—
82#336—22no—
83#336—22no—
84#333—22no—
85#332—24no—
86C22×Q8—22no—
87C22×Q8—21no—
88C22×C8—21no—
89C22×C8—22no—
90C22×C8—22no—
91C22×C8C2×D412yes—
92C22×C8C2×D412yes—
93C4⋊Q8—22no—
94C4⋊Q8—22no—
95C4⋊Q8—22no—
96C4⋊Q8—22no—
97C4⋊Q8—21no—
98C4⋊Q8#4114yes—
99C4⋊Q8#4112yes—
100C4×Q8—44no—
101C4×Q8—82no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4C2×D412yes—
111C2×C4⋊C4C2×D412yes—
112C2×C4⋊C4C2×D412yes—
113C2×C4⋊C4C2×D412yes—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—42no—
117C2×C42—41no—
118C2×C42C2411yescharacteristic
119#107—24no—
120#107—42no—
121#106—44no—
122#106—44no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—44no—
128#103—28no—
129#103—28no—
130#103—44no—
131#103—82no—
132#103—82no—
133C4×C8—22no—
134C4×C8—22no—
135C4×C8—22no—
136C4×C8—82no—
137#95—22no—
138#95—22no—
139#95—42no—
140#95—42no—
141#95—41no—
142#95—42no—
143#95—42no—
144C2×Q8—24no—
145C2×Q8—22no—
146C2×Q8—42no—
147C2×Q8—41no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—81no—
153C2×Q8—81no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—42no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4C22×D412yes—
162C22×C4H(5,𝔽2)11yescharacteristic
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—42no—
178C4⋊C4—22no—
179C4⋊C4—24no—
180C4⋊C4—24no—
181C4⋊C4—24no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—81no—
194C4⋊C4—84no—
195C4⋊C4—161no—
196C4⋊C4—161no—
197C4⋊C4#13514yes—
198C42—42no—
199C42—42no—
200C42—42no—
201C42—42no—
202C42—84no—
203C42—161no—
204C42C2×#4111yescharacteristic
205C42C2×#4112yes—
206C422−1+411yescharacteristic
207C23D4211yescharacteristic
208Q8—44no—
209Q8—44no—
210Q8—82no—
211Q8—82no—
212Q8—82no—
213Q8—161no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—24no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—82no—
235C2×C4—81no—
236C2×C4#52512yes—
237C2×C4#53911yescharacteristic
238C2×C4#54812yes—
239C2×C4#55111yescharacteristic
240C2×C4C2×#13512yes—
241C2×C4#57512yes—
242C2×C4#57712yes—
243C2×C4#57812yes—
244C8—42no—
245C8—42no—
246C8—42no—
247C8—82no—
248C8—82no—
249C22#246311yescharacteristic
250C22#326712yes—
251C22#328512yes—
252C22#328911yescharacteristic
253C22#329111yescharacteristic
254C4—21no—
255C4—21no—
256C4—44no—
257C4—84no—
258C4—82no—
259C4—82no—
260C4—82no—
261C4—82no—
262C4—161no—
263C4#310212yes—
264C4#312012yes—
265C4#336512yes—
266C4#337712yes—
267C2#1983912yes—
268C2#2001311yescharacteristic
269C2#2014311yescharacteristic
270C2#2689212yes—
271C2#2692411yescharacteristic
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
#265132
#232172
#105412
#103522
#103172
#95652
#95362
#90592
#89282