Finite Group Databasebeta

← #312160

Subgroups — #312160

Order 512.

568 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 146 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#26440C212yesmaximal
2#23217C212yesmaximal
3#10541C212yesmaximal
4#10349C211yesmaximal, characteristic
5#10317C211yesmaximal, characteristic
6#9566C212yesmaximal
7#9533C212yesmaximal
8#9059C212yesmaximal
9#8929C211yesmaximal, characteristic
10C2×#498C2211yescharacteristic
11C2×#487C2212yes—
12#2551C2211yescharacteristic
13#2369C2211yescharacteristic
14#2365C2212yes—
15#2086—22no—
16#2057—22no—
17#1990C2212yes—
18#1988—22no—
19#1988C2212yes—
20#1987C2212yes—
21#1981C2212yes—
22#1956C2211yescharacteristic
23#1947C2212yes—
24#1939C2212yes—
25#1933—22no—
26#1928—22no—
27#1918—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⋊Q8—22no—
40C2×C4⋊Q8C2312yes—
41C2×C4⋊Q8C2311yescharacteristic
42C2×C4×Q8—22no—
43#498—24no—
44#487—24no—
45#487—24no—
46C2×#107—22no—
47C2×#107D412yes—
48C2×#107D412yes—
49C2×#106—41no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×#103—22no—
54C2×#103—22no—
55C2×#103—22no—
56C2×#103—22no—
57C2×#103—41no—
58C2×C4×C8—22no—
59C2×C4×C8C2311yescharacteristic
60C2×C4×C8C2312yes—
61#395—41no—
62#391—22no—
63#391—41no—
64#389—22no—
65#389—22no—
66#389—22no—
67#387—42no—
68#384C2311yescharacteristic
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×Q8—41no—
89C22×C8—21no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8C2×D412yes—
93C22×C8C2×D412yes—
94C4⋊Q8—22no—
95C4⋊Q8—22no—
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98C4⋊Q8—21no—
99C4⋊Q8—44no—
100C4⋊Q8—82no—
101C4⋊Q8#4114yes—
102C4⋊Q8#4112yes—
103C4×Q8—44no—
104C4×Q8—82no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4C2×D412yes—
114C2×C4⋊C4C2×D412yes—
115C2×C4⋊C4C2×D412yes—
116C2×C4⋊C4C2×D412yes—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42—42no—
120C2×C42—41no—
121C2×C42C2411yescharacteristic
122#107—24no—
123#107—24no—
124#107—82no—
125#106—44no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—82no—
131#103—28no—
132#103—28no—
133#103—44no—
134#103—82no—
135#103—82no—
136C4×C8—22no—
137C4×C8—22no—
138C4×C8—22no—
139C4×C8—82no—
140#95—22no—
141#95—22no—
142#95—42no—
143#95—42no—
144#95—42no—
145#95—41no—
146C2×Q8—24no—
147C2×Q8—22no—
148C2×Q8—42no—
149C2×Q8—41no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—84no—
155C2×Q8—81no—
156C2×Q8—81no—
157C2×Q8—161no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—42no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4C22×D412yes—
166C22×C4H(5,𝔽2)11yescharacteristic
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—42no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—24no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—84no—
197C4⋊C4—84no—
198C4⋊C4—81no—
199C4⋊C4—161no—
200C4⋊C4#13514yes—
201C4⋊C4#13714yes—
202C42—42no—
203C42—42no—
204C42—42no—
205C42—42no—
206C42—84no—
207C42—161no—
208C42C2×#4111yescharacteristic
209C42C2×#4112yes—
210C422−1+411yescharacteristic
211C23D4211yescharacteristic
212Q8—44no—
213Q8—44no—
214Q8—82no—
215Q8—82no—
216Q8—82no—
217Q8—161no—
218Q8—164no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—82no—
240C2×C4—81no—
241C2×C4#52512yes—
242C2×C4#53911yescharacteristic
243C2×C4#54812yes—
244C2×C4#55111yescharacteristic
245C2×C4C2×#13512yes—
246C2×C4C2×#13712yes—
247C2×C4#57612yes—
248C2×C4#57812yes—
249C8—42no—
250C8—42no—
251C8—42no—
252C8—82no—
253C8—82no—
254C22#246311yescharacteristic
255C22#327012yes—
256C22#328012yes—
257C22#328911yescharacteristic
258C22#329111yescharacteristic
259C4—21no—
260C4—21no—
261C4—44no—
262C4—44no—
263C4—84no—
264C4—82no—
265C4—82no—
266C4—82no—
267C4—161no—
268C4#310712yes—
269C4#312412yes—
270C4#336412yes—
271C4#337512yes—
272C2#1978212yes—
273C2#1998711yescharacteristic
274C2#2009711yescharacteristic
275C2#2689312yes—
276C2#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
#264402
#232172
#105412
#103492
#103172
#95662
#95332
#90592
#89292