Finite Group Databasebeta

← #305216

Subgroups — #305216

Order 512.

709 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 709 conjugacy classes of subgroups — too many to draw. Only its 156 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#59869C211yesmaximal, characteristic
2#29355C212yesmaximal
3#28719C211yesmaximal, characteristic
4#10353C214yesmaximal
5#9868C212yesmaximal
6#9060C214yesmaximal
7#8232C211yesmaximal, characteristic
8C2×#487C2212yes—
9#2862C2211yescharacteristic
10#2809—22no—
11#2571C2212yes—
12#2568—22no—
13#2546—22no—
14#2508—24no—
15#2369C2212yes—
16#2355—22no—
17C4×C4⋊Q8C2212yes—
18C4×C4⋊Q8C2211yescharacteristic
19#2074—22no—
20#2072—22no—
21#2024—22no—
22#1990C2212yes—
23#1988C2214yes—
24#1981C2214yes—
25#1935—24no—
26#1805—22no—
27#1767C2212yes—
28C4×C4⋊C8C2211yescharacteristic
29C2×#333C2212yes—
30#1296C2212yes—
31#1279C2218yes—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8D412yes—
34C2×C4⋊Q8C2312yes—
35C2×C4⋊Q8C2311yescharacteristic
36C2×C4×Q8—22no—
37C2×C4×Q8D412yes—
38#487—28no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8C2314yes—
41C2×#103—24no—
42C2×#103—24no—
43C2×#103—24no—
44C2×#97—22no—
45C2×C4×C8C2312yes—
46#400—24no—
47#400—24no—
48#398—28no—
49#391—24no—
50#391—24no—
51#391—22no—
52#391—22no—
53#391—22no—
54#391D412yes—
55#389—24no—
56#389—24no—
57#389—22no—
58#389—22no—
59#389D414yes—
60#387—28no—
61#384D412yes—
62#384C2311yescharacteristic
63#382—24no—
64#382—24no—
65#382—21no—
66#382—21no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—24no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4C2314yes—
72C43C2311yescharacteristic
73#340—24no—
74#340—24no—
75#340—24no—
76#336—22no—
77#336—21no—
78#333—24no—
79C22×Q8—22no—
80C22×Q8—22no—
81C22×Q8C2×D412yes—
82C22×C8—22no—
83C22×C8—24no—
84C4⋊Q8—24no—
85C4⋊Q8—22no—
86C4⋊Q8—24no—
87C4⋊Q8—44no—
88C4⋊Q8—84no—
89C4⋊Q8#4114yes—
90C4×Q8—48no—
91C4×Q8—44no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4C2×D414yes—
104C2×C4⋊C4C2×D414yes—
105C2×C4⋊C4C2×D412yes—
106C2×C42—22no—
107C2×C42—24no—
108C2×C42—24no—
109C2×C42C2×D412yes—
110C2×C42C2×D411yescharacteristic
111C2×C42C2×D412yes—
112C2×C42C2×D411yescharacteristic
113C2×C42C2411yescharacteristic
114C4⋊C8—24no—
115C4⋊C8—24no—
116C4⋊C8—42no—
117#103—48no—
118#103—48no—
119#103—84no—
120#97—82no—
121C4×C8—24no—
122#95—24no—
123#95—28no—
124#95—28no—
125#95—24no—
126#95—24no—
127#95—24no—
128#95—41no—
129#95—42no—
130#95—41no—
131C2×Q8—22no—
132C2×Q8—44no—
133C2×Q8—44no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—44no—
137C2×Q8—44no—
138C2×Q8—44no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8#12414yes—
142C22×C4—22no—
143C22×C4—24no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—42no—
147C22×C4#12011yescharacteristic
148C22×C4#12012yes—
149C22×C4#12011yescharacteristic
150C22×C4C22×D411yescharacteristic
151C22×C4C22×D412yes—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—44no—
156C2×C8—44no—
157C2×C8—44no—
158C4⋊C4—28no—
159C4⋊C4—24no—
160C4⋊C4—24no—
161C4⋊C4—24no—
162C4⋊C4—24no—
163C4⋊C4—42no—
164C4⋊C4—44no—
165C4⋊C4—48no—
166C4⋊C4—44no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—44no—
172C4⋊C4—82no—
173C4⋊C4—82no—
174C4⋊C4—84no—
175C4⋊C4—84no—
176C42—24no—
177C42—22no—
178C42—22no—
179C42—22no—
180C42—41no—
181C42—48no—
182C42—44no—
183C42—84no—
184C42C2×#4112yes—
185C422−1+412yes—
186C23C2×#12011yescharacteristic
187Q8—44no—
188Q8—82no—
189Q8—88no—
190Q8—84no—
191Q8—84no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—24no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4#45512yes—
215C2×C4C2×#12411yescharacteristic
216C2×C4#53611yescharacteristic
217C2×C4#54814yes—
218C2×C4#57514yes—
219C2×C4#57712yes—
220C2×C4#57812yes—
221C8—82no—
222C8—84no—
223C22#243611yescharacteristic
224C22#300512yes—
225C22#301312yes—
226C22C2×#45511yescharacteristic
227C22#302211yescharacteristic
228C4—21no—
229C4—21no—
230C4—22no—
231C4—41no—
232C4—44no—
233C4—42no—
234C4—82no—
235C4—84no—
236C4—84no—
237C4—162no—
238C4#337114yes—
239C4#337714yes—
240C2#1855211yescharacteristic
241C2#1869311yescharacteristic
242C2#1884912yes—
243C2#2591711yescharacteristic
244C2#2594112yes—
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ 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
#598692
#293552
#287192
#103532
#98682
#90602
#82322