Finite Group Databasebeta

← #325224

Subgroups — #325224

Order 512.

482 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 162 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#33291C211yesmaximal, characteristic
2#29321C212yesmaximal
3#21673C211yesmaximal, characteristic
4#10658C212yesmaximal
5#10657C212yesmaximal
6#9907C212yesmaximal
7#9622C212yesmaximal
8#9622C212yesmaximal
9#8273C211yesmaximal, characteristic
10#2859C2212yes—
11C4×#125C2211yescharacteristic
12#2085—22no—
13#2059—22no—
14#1991—22no—
15#1991—22no—
16#1948C2212yes—
17#1948C2212yes—
18#1947C2212yes—
19#1947C2212yes—
20#1944—22no—
21#1943—22no—
22#1939C2212yes—
23#1916—22no—
24#1834C2211yescharacteristic
25#1834C2211yescharacteristic
26#1834C2212yes—
27#1831C2211yescharacteristic
28#1771—22no—
29#1770C2212yes—
30#1767C2212yes—
31C42×C8C2211yescharacteristic
32C2×#334C2212yes—
33#1296C2212yes—
34#1296C2212yes—
35#1296C2212yes—
36#1296C2212yes—
37#1277C2211yescharacteristic
38#1277C2211yescharacteristic
39C2×#125—22no—
40C2×#125D411yescharacteristic
41C2×#107—22no—
42C2×#107—22no—
43C2×#106—22no—
44C2×#106—22no—
45C2×C4⋊C8—22no—
46C2×C4⋊C8C2312yes—
47C2×C4⋊C8C2312yes—
48C2×C4⋊C8C2312yes—
49C2×#97—22no—
50C2×#97—22no—
51C2×C4×C8D412yes—
52C2×C4×C8D412yes—
53C2×C4×C8C2311yescharacteristic
54C2×C4×C8C2311yescharacteristic
55#389—22no—
56#389—22no—
57#389—22no—
58#389—22no—
59#389D412yes—
60#387—22no—
61#384C2312yes—
62#384C2312yes—
63#382—22no—
64#382D411yescharacteristic
65C4×C4⋊C4—22no—
66C4×C4⋊C4C2312yes—
67C43C2311yescharacteristic
68#340—24no—
69#340—21no—
70#340—21no—
71#340—21no—
72#340—21no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#336—22no—
78#336—22no—
79#336—22no—
80#336—22no—
81#334Q818yes—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8C2×D412yes—
86C22×C8C2×D411yescharacteristic
87C22×C8C2×D411yescharacteristic
88#125—41no—
89#125—41no—
90#125—44no—
91#125—44no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4C2×D412yes—
103C2×C4⋊C4C2×D412yes—
104C2×C42—22no—
105C2×C42C2×D412yes—
106C2×C42C2×D411yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C42#4112yes—
109C2×C42C2411yescharacteristic
110#107—44no—
111#107—82no—
112#106—44no—
113#106—82no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—24no—
117C4⋊C8—44no—
118C4⋊C8C2×Q814yes—
119C4⋊C8C2×Q814yes—
120#97—44no—
121#97—82no—
122C4×C8—22no—
123C4×C8—42no—
124C4×C8—42no—
125C4×C8C2×Q814yes—
126#95—22no—
127#95—22no—
128#95—22no—
129#95—42no—
130#95—42no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4#12112yes—
136C22×C4#12112yes—
137C22×C4C22×D411yescharacteristic
138C22×C4C22×D411yescharacteristic
139C22×C4C2×#4111yescharacteristic
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—24no—
144C2×C8—24no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—42no—
148C2×C8—42no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8C4⋊Q812yes—
155C2×C8C4⋊Q812yes—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—82no—
171C4⋊C4—82no—
172C4⋊C4—84no—
173C4⋊C4—84no—
174C42—22no—
175C42—22no—
176C42—21no—
177C42—21no—
178C42—22no—
179C42—42no—
180C42—42no—
181C42—42no—
182C42C22×Q812yes—
183C422−1+412yes—
184C23C2×#12111yescharacteristic
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4C2×C4⋊Q811yescharacteristic
205C2×C4#53611yescharacteristic
206C2×C4D4×Q812yes—
207C2×C4#55712yes—
208C2×C4#57511yescharacteristic
209C2×C4#57511yescharacteristic
210C2×C4#57512yes—
211C2×C4#57512yes—
212C2×C4#57712yes—
213C8—22no—
214C8—22no—
215C8—44no—
216C8—44no—
217C8—42no—
218C8—82no—
219C22#254611yescharacteristic
220C22#304012yes—
221C22#305411yescharacteristic
222C22#306011yescharacteristic
223C22#306712yes—
224C4—22no—
225C4—42no—
226C4—42no—
227C4—82no—
228C4—82no—
229C4—162no—
230C4—162no—
231C4#316211yescharacteristic
232C4#316611yescharacteristic
233C4#324012yes—
234C4#338812yes—
235C4#338812yes—
236C2#1930312yes—
237C2#1936612yes—
238C2#2868111yescharacteristic
239C2#2869511yescharacteristic
240C2#2870111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
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
#332912
#293212
#216732
#106582
#106572
#99072
#96222
#82732