Finite Group Databasebeta

← #102500

Subgroups — #102500

Order 512.

568 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 76 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#32954C212yesmaximal
2#32790C211yesmaximal, characteristic
3#8303C211yesmaximal, characteristic
4#7727C212yesmaximal
5#7044C211yesmaximal, characteristic
6C22×C4×Q8C2211yescharacteristic
7C2×#475—22no—
8#2857C2212yes—
9#2100—22no—
10#2076—22no—
11#2057—22no—
12#1865—24no—
13#1853—22no—
14#1817C2211yescharacteristic
15#1817C2211yescharacteristic
16#1803—22no—
17#1797C2212yes—
18#1787—22no—
19C2×#324—22no—
20#1272—24no—
21C23×Q8D411yescharacteristic
22C2×C4⋊Q8D412yes—
23C2×C22⋊Q8—22no—
24C2×C4×Q8—22no—
25C2×C4×Q8—22no—
26C2×C4×Q8—44no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4D411yescharacteristic
29C22×C42—21no—
30C22×C42C2311yescharacteristic
31#475—48no—
32C2×#107—22no—
33C2×#106—24no—
34C2×C4⋊C8—22no—
35C2×#103—24no—
36C2×#103—24no—
37C2×C22⋊C8—22no—
38C2×C22⋊C8—21no—
39C2×C22⋊C8D412yes—
40#391—22no—
41#389—22no—
42#389—22no—
43#384D412yes—
44#380—22no—
45#380—22no—
46#340—24no—
47#340—24no—
48#340—22no—
49#336—22no—
50#336—22no—
51#336D414yes—
52#324—44no—
53C22×Q8—21no—
54C22×Q8—22no—
55C22×Q8—22no—
56C22×Q8—22no—
57C22×Q8—42no—
58C22×Q8—42no—
59C22×Q8Q1612yes—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4C2×D411yescharacteristic
63C23×C4C2×D411yescharacteristic
64C23×C4#4111yescharacteristic
65C22×C8—22no—
66C22×C8—24no—
67C22×C8—41no—
68C4⋊Q8—44no—
69C22⋊Q8—84no—
70C4×Q8—28no—
71C4×Q8—44no—
72C4×Q8—44no—
73C4×Q8—44no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—42no—
86C2×C22⋊C4—42no—
87C2×C42—42no—
88C2×C42—42no—
89C2×C42—42no—
90C2×C42—42no—
91C2×C42C2×D412yes—
92C2×C42C2×D412yes—
93#107—82no—
94#106—48no—
95C4⋊C8—44no—
96#103—48no—
97#103—48no—
98C22⋊C8—42no—
99C22⋊C8—42no—
100C22⋊C8—42no—
101C22⋊C8—42no—
102#95—42no—
103#95—42no—
104C24#12111yescharacteristic
105C2×Q8—24no—
106C2×Q8—24no—
107C2×Q8—24no—
108C2×Q8—44no—
109C2×Q8—42no—
110C2×Q8—42no—
111C2×Q8—44no—
112C2×Q8—48no—
113C2×Q8—42no—
114C2×Q8—44no—
115C2×Q8—82no—
116C2×Q8—82no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—24no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—42no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—82no—
136C22×C4#12012yes—
137C22×C4#12011yescharacteristic
138C22×C4#12112yes—
139C22×C4#12711yescharacteristic
140C22×C4C2×Q1611yescharacteristic
141C22×C4#13511yescharacteristic
142C22×C4#13712yes—
143C2×C8—44no—
144C2×C8—44no—
145C2×C8—44no—
146C2×C8—44no—
147C2×C8—42no—
148C2×C8—81no—
149C2×C8—82no—
150C4⋊C4—24no—
151C4⋊C4—42no—
152C4⋊C4—44no—
153C4⋊C4—44no—
154C4⋊C4—44no—
155C4⋊C4—44no—
156C4⋊C4—44no—
157C4⋊C4—44no—
158C4⋊C4—82no—
159C4⋊C4—84no—
160C22⋊C4—84no—
161C42—24no—
162C42—42no—
163C42—48no—
164C23—22no—
165C23—21no—
166C23—22no—
167C23—21no—
168C23#39211yescharacteristic
169C23#46711yescharacteristic
170C23#47011yescharacteristic
171Q8—28no—
172Q8—44no—
173Q8—44no—
174Q8—44no—
175Q8—164no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—82no—
196C2×C4—82no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4#45112yes—
206C2×C4#45212yes—
207C2×C4B(4,𝔽2)12yes—
208C2×C4#47114yes—
209C8—84no—
210C8—82no—
211C8—161no—
212C22—21no—
213C22—21no—
214C22—22no—
215C22—22no—
216C22—41no—
217C22—42no—
218C22—41no—
219C22#200111yescharacteristic
220C22#200912yes—
221C22#201011yescharacteristic
222C22#201912yes—
223C22#202111yescharacteristic
224C4—24no—
225C4—22no—
226C4—22no—
227C4—22no—
228C4—44no—
229C4—41no—
230C4—82no—
231C4—162no—
232C4—162no—
233C2—41no—
234C2—41no—
235C2#998211yescharacteristic
236C2#998811yescharacteristic
237C2#1004412yes—
238C2#1019112yes—
239C2#1032411yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C22×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#329542
#327902
#83032
#77272
#70442