Finite Group Databasebeta

← #325197

Subgroups — #325197

Order 512.

498 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 170 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#33256C211yesmaximal, characteristic
2#29314C212yesmaximal
3#21673C211yesmaximal, characteristic
4#10654C212yesmaximal
5#10653C212yesmaximal
6#9901C212yesmaximal
7#9622C212yesmaximal
8#9622C212yesmaximal
9C4×#334C211yesmaximal, characteristic
10#2859C2212yes—
11C4×#125C2211yescharacteristic
12#2085—22no—
13#2059—22no—
14#1948C2212yes—
15#1948C2212yes—
16#1947C2212yes—
17#1947C2212yes—
18#1942—22no—
19#1941—22no—
20#1941—22no—
21#1941—22no—
22#1939C2212yes—
23#1916—22no—
24#1844C2212yes—
25#1834C2211yescharacteristic
26#1834C2211yescharacteristic
27#1831C2211yescharacteristic
28#1771—22no—
29#1770C2212yes—
30C4×C4⋊C8C2212yes—
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×#106—22no—
46C2×#106—22no—
47C2×C4⋊C8—22no—
48C2×C4⋊C8C2312yes—
49C2×C4⋊C8C2312yes—
50C2×C4⋊C8C2312yes—
51C2×#97—22no—
52C2×C4×C8—22no—
53C2×C4×C8D412yes—
54C2×C4×C8D412yes—
55C2×C4×C8C2311yescharacteristic
56C2×C4×C8C2311yescharacteristic
57#389—22no—
58#389—22no—
59#389—22no—
60#389—22no—
61#389D412yes—
62#387—22no—
63#384C2312yes—
64#384C2312yes—
65#382—22no—
66#382D411yescharacteristic
67C4×C4⋊C4—22no—
68C4×C4⋊C4C2312yes—
69C43C2311yescharacteristic
70#340—21no—
71#340—21no—
72#340—21no—
73#340—21no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#336—22no—
79#336—22no—
80#336—22no—
81#336—22no—
82#334Q818yes—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8C2×D411yescharacteristic
87C22×C8C2×D411yescharacteristic
88C22×C8C2×D412yes—
89#125—41no—
90#125—41no—
91#125—84no—
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×D411yescharacteristic
106C2×C42C2×D412yes—
107C2×C42C2×D411yescharacteristic
108C2×C42#4112yes—
109C2×C42C2411yescharacteristic
110#107—44no—
111#107—82no—
112#106—44no—
113#106—44no—
114#106—44no—
115#106—82no—
116C4⋊C8—22no—
117C4⋊C8—22no—
118C4⋊C8—24no—
119C4⋊C8—44no—
120C4⋊C8C2×Q814yes—
121C4⋊C8C2×Q814yes—
122#97—44no—
123C4×C8—24no—
124C4×C8—22no—
125C4×C8—42no—
126C4×C8—82no—
127C4×C8C2×Q814yes—
128#95—22no—
129#95—22no—
130#95—22no—
131#95—42no—
132#95—42no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4#12112yes—
138C22×C4#12112yes—
139C22×C4C22×D411yescharacteristic
140C22×C4C22×D411yescharacteristic
141C22×C4C2×#4111yescharacteristic
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—24no—
147C2×C8—24no—
148C2×C8—42no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8C4⋊Q812yes—
156C2×C8C4⋊Q812yes—
157C2×C8#13514yes—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—82no—
173C4⋊C4—82no—
174C4⋊C4—84no—
175C4⋊C4—84no—
176C42—22no—
177C42—22no—
178C42—21no—
179C42—21no—
180C42—22no—
181C42—24no—
182C42—42no—
183C42—42no—
184C42C22×Q812yes—
185C422−1+412yes—
186C23C2×#12111yescharacteristic
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4#47114yes—
206C2×C4C2×C4⋊Q811yescharacteristic
207C2×C4#53611yescharacteristic
208C2×C4D4×Q812yes—
209C2×C4#55712yes—
210C2×C4C2×#13512yes—
211C2×C4#57511yescharacteristic
212C2×C4#57511yescharacteristic
213C2×C4#57512yes—
214C2×C4#57712yes—
215C8—22no—
216C8—22no—
217C8—24no—
218C8—44no—
219C8—44no—
220C8—82no—
221C22#254611yescharacteristic
222C22#303512yes—
223C22#305411yescharacteristic
224C22#306011yescharacteristic
225C22C2×#47112yes—
226C4—24no—
227C4—22no—
228C4—42no—
229C4—82no—
230C4—82no—
231C4—162no—
232C4—162no—
233C4#316011yescharacteristic
234C4#316611yescharacteristic
235C4#323212yes—
236C4#338812yes—
237C4#338812yes—
238C2#1915912yes—
239C2#1922112yes—
240C2#2868111yescharacteristic
241C2#2869011yescharacteristic
242C2#2870011yescharacteristic
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
#332562
#293142
#216732
#106542
#106532
#99012
#96222
C4×#3342