Finite Group Databasebeta

← #312157

Subgroups — #312157

Order 512.

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

normal subgroups only

This group has 540 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#26507C212yesmaximal
2#23158C212yesmaximal
3#10543C212yesmaximal
4#10352C211yesmaximal, characteristic
5#10309C211yesmaximal, characteristic
6#9567C212yesmaximal
7#9533C212yesmaximal
8#9066C212yesmaximal
9#8928C211yesmaximal, characteristic
10C2×#500C2212yes—
11C2×#494C2211yescharacteristic
12#2571C2211yescharacteristic
13#2369C2212yes—
14#2369C2211yescharacteristic
15#2101—22no—
16#2086—22no—
17#1990C2212yes—
18#1987C2212yes—
19#1987C2212yes—
20#1987C2212yes—
21#1956C2211yescharacteristic
22#1949—22no—
23#1948C2212yes—
24#1942—22no—
25#1939C2212yes—
26#1921—22no—
27#1921—22no—
28#1920—22no—
29C4×#107—22no—
30C4×#107C2212yes—
31C8×C4⋊C4C2212yes—
32C2×#333C2211yescharacteristic
33C2×#333C2212yes—
34#1296C2211yescharacteristic
35#1279C2212yes—
36#1279C2212yes—
37#1277C2212yes—
38#1273C2211yescharacteristic
39C22×Q16—41no—
40C2×C4⋊Q8C2312yes—
41C2×C4⋊Q8C2311yescharacteristic
42#500—24no—
43#500—24no—
44#494—24no—
45C2×#107—22no—
46C2×#107—22no—
47C2×#107D412yes—
48C2×#107D412yes—
49C2×#106—22no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×#103—41no—
54C2×C4×C8—22no—
55C2×C4×C8C2311yescharacteristic
56C2×C4×C8C2312yes—
57#400—41no—
58#391—41no—
59#391—42no—
60#389—22no—
61#389—22no—
62#389—22no—
63#389—22no—
64#384C2311yescharacteristic
65#382—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4C2312yes—
70C4×C4⋊C4C2312yes—
71#340—22no—
72#340—22no—
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—24no—
85#333—22no—
86C22×Q8—22no—
87C22×Q8—21no—
88C2×Q16—44no—
89C2×Q16—82no—
90C2×Q16—82no—
91C22×C8—21no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8C2×D412yes—
95C22×C8C2×D412yes—
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98C4⋊Q8—22no—
99C4⋊Q8—22no—
100C4⋊Q8—21no—
101C4⋊Q8#4114yes—
102C4⋊Q8#4112yes—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4C2×D412yes—
112C2×C4⋊C4C2×D412yes—
113C2×C4⋊C4C2×D412yes—
114C2×C4⋊C4C2×D412yes—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42—41no—
118C2×C42—42no—
119C2×C42C2411yescharacteristic
120#107—28no—
121#107—28no—
122#107—24no—
123#107—24no—
124#106—44no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—44no—
130#103—82no—
131C4×C8—22no—
132C4×C8—22no—
133C4×C8—22no—
134C4×C8—44no—
135#95—22no—
136#95—22no—
137#95—42no—
138#95—42no—
139#95—42no—
140#95—41no—
141#95—42no—
142C2×Q8—22no—
143C2×Q8—24no—
144C2×Q8—42no—
145C2×Q8—41no—
146C2×Q8—81no—
147C2×Q8—82no—
148C2×Q8—82no—
149C2×Q8—81no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—42no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4C22×D412yes—
158C22×C4H(5,𝔽2)11yescharacteristic
159Q16—88no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8#13514yes—
175C4⋊C4—22no—
176C4⋊C4—24no—
177C4⋊C4—24no—
178C4⋊C4—24no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—81no—
191C4⋊C4—161no—
192C4⋊C4—162no—
193C4⋊C4—161no—
194C4⋊C4#13714yes—
195C42—42no—
196C42—42no—
197C42—42no—
198C42—42no—
199C42—161no—
200C42—162no—
201C42C2×#4111yescharacteristic
202C42C2×#4112yes—
203C422−1+411yescharacteristic
204C23D4211yescharacteristic
205Q8—84no—
206Q8—82no—
207Q8—161no—
208Q8—162no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—24no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—82no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—81no—
231C2×C4#52512yes—
232C2×C4#53911yescharacteristic
233C2×C4#54812yes—
234C2×C4#55111yescharacteristic
235C2×C4C2×#13512yes—
236C2×C4C2×#13712yes—
237C2×C4#57812yes—
238C2×C4#57812yes—
239C8—24no—
240C8—42no—
241C8—42no—
242C8—82no—
243C8—82no—
244C22#246311yescharacteristic
245C22#327012yes—
246C22#328312yes—
247C22#328811yescharacteristic
248C22#329211yescharacteristic
249C4—21no—
250C4—21no—
251C4—44no—
252C4—84no—
253C4—82no—
254C4—82no—
255C4—82no—
256C4—82no—
257C4—161no—
258C4#310812yes—
259C4#312712yes—
260C4#336412yes—
261C4#337412yes—
262C2#1986512yes—
263C2#2001911yescharacteristic
264C2#2011711yescharacteristic
265C2#2689512yes—
266C2#2692011yescharacteristic
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
#265072
#231582
#105432
#103522
#103092
#95672
#95332
#90662
#89282