Finite Group Databasebeta

← #103000

Subgroups — #103000

Order 512.

537 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 84 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#27611C212yesmaximal
2#15256C211yesmaximal, characteristic
3#11370C211yesmaximal, characteristic
4#7607C212yesmaximal
5#7017C211yesmaximal, characteristic
6C2×#389C2212yes—
7#2285C2211yescharacteristic
8#2099—22no—
9#2084—22no—
10#1920—24no—
11#1853—22no—
12#1824—22no—
13#1817C2212yes—
14C2×#339—22no—
15#1728C2211yescharacteristic
16#1441—22no—
17#1441—22no—
18#1434—22no—
19C4×#95C2211yescharacteristic
20#1271—24no—
21C22×M16—21no—
22C2×#117—21no—
23C22×C4⋊C4—22no—
24C22×C4⋊C4D412yes—
25C22×C4⋊C4D412yes—
26C22×C42D412yes—
27C22×C42C2311yescharacteristic
28C2×#106—24no—
29C2×C22⋊C8—22no—
30#389—22no—
31#389—22no—
32#389—22no—
33#389—22no—
34#384D411yescharacteristic
35#384D411yescharacteristic
36#382—21no—
37#380—22no—
38C4×C4⋊C4—22no—
39C2×#95—24no—
40C2×#95—22no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—22no—
44#340—24no—
45#340—24no—
46#340—24no—
47#339—44no—
48#336—22no—
49#336—22no—
50#336D414yes—
51C23×C4—22no—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4C2×D412yes—
55C23×C4C2×D411yescharacteristic
56C2×M16—22no—
57C2×M16—41no—
58C2×M16—42no—
59C2×M16—42no—
60C22×C8—24no—
61C22×C8—41no—
62#117—42no—
63#117—42no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—44no—
73C2×C4⋊C4—44no—
74C2×C4⋊C4—44no—
75C2×C4⋊C4—44no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—42no—
78C2×C22⋊C4—41no—
79C2×C42—22no—
80C2×C42—42no—
81C2×C42—42no—
82C2×C42—42no—
83C2×C42—41no—
84C2×C42SD1614yes—
85C2×C42C2×D411yescharacteristic
86C2×C42C2×D412yes—
87C2×C42#4111yescharacteristic
88#106—48no—
89C22⋊C8—44no—
90#95—44no—
91#95—48no—
92#95—42no—
93#95—42no—
94#95—42no—
95#95—82no—
96#95—82no—
97C24#12011yescharacteristic
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—24no—
102C22×C4—24no—
103C22×C4—24no—
104C22×C4—24no—
105C22×C4—42no—
106C22×C4—44no—
107C22×C4—44no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4—44no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—42no—
122C22×C4—44no—
123C22×C4#12012yes—
124C22×C4#12112yes—
125C22×C4#12111yescharacteristic
126C22×C4#12711yescharacteristic
127C22×C4C2×SD1612yes—
128C22×C4#13712yes—
129M16—44no—
130M16—82no—
131C2×C8—44no—
132C2×C8—44no—
133C2×C8—42no—
134C2×C8—44no—
135C2×C8—81no—
136C2×C8—81no—
137C4⋊C4—24no—
138C4⋊C4—24no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—44no—
142C4⋊C4—44no—
143C4⋊C4—44no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—82no—
147C4⋊C4—84no—
148C4⋊C4—84no—
149C22⋊C4—82no—
150C42—41no—
151C42—42no—
152C42—41no—
153C42—44no—
154C42—82no—
155C42—81no—
156C42—81no—
157C23—22no—
158C23—22no—
159C23#39211yescharacteristic
160C23#44812yes—
161C23#45511yescharacteristic
162C23#45611yescharacteristic
163C23#45812yes—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—48no—
178C2×C4—44no—
179C2×C4—44no—
180C2×C4—44no—
181C2×C4—44no—
182C2×C4—44no—
183C2×C4—44no—
184C2×C4—84no—
185C2×C4—84no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—84no—
191C2×C4—84no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—82no—
199C2×C4#45414yes—
200C2×C4#46112yes—
201C2×C4#49211yescharacteristic
202C2×C4#49711yescharacteristic
203C8—84no—
204C8—82no—
205C22—22no—
206C22—22no—
207C22—22no—
208C22—22no—
209C22—24no—
210C22#160812yes—
211C22#161112yes—
212C22#200312yes—
213C22#200711yescharacteristic
214C22#201012yes—
215C22#201311yescharacteristic
216C22#202211yescharacteristic
217C4—24no—
218C4—21no—
219C4—22no—
220C4—21no—
221C4—42no—
222C4—84no—
223C4—84no—
224C4—161no—
225C4—161no—
226C4—162no—
227C2—22no—
228C2—22no—
229C2#994112yes—
230C2#997311yescharacteristic
231C2#1000912yes—
232C2#1001411yescharacteristic
233C2#1028511yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#120
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
#276112
#152562
#113702
#76072
#70172