Finite Group Databasebeta

← #142165

Subgroups — #142165

Order 512.

396 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 96 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#33211C211yesmaximal, characteristic
2#12785C211yesmaximal, characteristic
3C4×#338C211yesmaximal, characteristic
4#7255C214yesmaximal
5C2×#499C411yescharacteristic
6C2×C4×M16—21no—
7C2×C4×M16C2211yescharacteristic
8#2851—21no—
9#2150—21no—
10C2×#349C2212yes—
11#1991—21no—
12#1944C412yes—
13#1943—21no—
14#1846C411yescharacteristic
15#1809—21no—
16#1809C2212yes—
17C2×#338—21no—
18C2×#338C2212yes—
19#1377—21no—
20C22×M16—22no—
21C22×M16D411yescharacteristic
22C22×M16C2311yescharacteristic
23C2×#125—21no—
24C22×C42D411yescharacteristic
25C2×M32—22no—
26#499—42no—
27C2×#107—21no—
28C2×#107C2×C412yes—
29C2×#106—21no—
30C2×#106C2×C412yes—
31C4×M16—42no—
32C4×M16—42no—
33C4×M16—42no—
34C2×#97—22no—
35C2×#97C2×C411yescharacteristic
36C2×#97D411yescharacteristic
37C2×C4×C8—22no—
38C2×C4×C8C2×C411yescharacteristic
39C2×C4×C8D411yescharacteristic
40#389—22no—
41#387—42no—
42#385—42no—
43#382—21no—
44#380—22no—
45C4⋊C16—42no—
46#349—24no—
47#340—22no—
48#340—22no—
49#338—24no—
50#338—22no—
51#338Q814yes—
52C23×C4SD1612yes—
53C23×C4C2×D411yescharacteristic
54C2×M16—24no—
55C2×M16—24no—
56C2×M16—22no—
57C2×M16—22no—
58C2×M16—41no—
59C2×M16—41no—
60C2×M16—41no—
61C2×M16—42no—
62C2×M16—42no—
63C2×M16—42no—
64C2×M16—42no—
65C2×M16C2×Q812yes—
66C2×M16#4112yes—
67C22×C8—24no—
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22×C411yescharacteristic
71C22×C8C2×D411yescharacteristic
72#125—42no—
73#125—41no—
74#125—41no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C22⋊C4—42no—
80C2×C42—22no—
81C2×C42—21no—
82C2×C42—21no—
83C2×C42C22⋊C411yescharacteristic
84C2×C42C22⋊C411yescharacteristic
85C2×C42D812yes—
86M32—42no—
87M32—42no—
88C2×C16—42no—
89#107—42no—
90#107—42no—
91#106—42no—
92#106—42no—
93#97—41no—
94#97—41no—
95#97—44no—
96C4×C8—22no—
97C4×C8—22no—
98C4×C8—44no—
99#95—42no—
100#95—42no—
101#95—42no—
102C24C2×SD1611yescharacteristic
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—82no—
112C22×C4#10212yes—
113C22×C4#10212yes—
114C22×C4C2×C22⋊C411yescharacteristic
115C22×C4C22⋊Q812yes—
116C22×C4C2×D811yescharacteristic
117C22×C4#13712yes—
118M16—42no—
119M16—42no—
120M16—44no—
121M16—44no—
122M16—44no—
123M16—44no—
124M16—44no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—28no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—44no—
136C2×C8—44no—
137C2×C8#10412yes—
138C2×C8C4×Q812yes—
139C2×C8#12411yescharacteristic
140C2×C8C4⋊Q811yescharacteristic
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—84no—
146C4⋊C4—82no—
147C4⋊C4—82no—
148C22⋊C4—84no—
149C42—22no—
150C42—22no—
151C42—21no—
152C42—21no—
153C42#9912yes—
154C42#10411yescharacteristic
155C42#10411yescharacteristic
156C16—82no—
157C23—21no—
158C23—21no—
159C23—42no—
160C23C2×#10211yescharacteristic
161C23#47512yes—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—42no—
170C2×C4—44no—
171C2×C4—42no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—82no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4#32712yes—
180C2×C4#32712yes—
181C2×C4#39111yescharacteristic
182C2×C4C2×#9911yescharacteristic
183C2×C4#41011yescharacteristic
184C2×C4#41911yescharacteristic
185C2×C4C2×#10411yescharacteristic
186C2×C4C2×#10411yescharacteristic
187C2×C4#47412yes—
188C8—21no—
189C8—22no—
190C8—21no—
191C8—22no—
192C8—48no—
193C8—41no—
194C22—22no—
195C22—22no—
196C22—42no—
197C22—42no—
198C22—81no—
199C22C2×#32711yescharacteristic
200C22#150211yescharacteristic
201C22#198811yescharacteristic
202C4—22no—
203C4—22no—
204C4—22no—
205C4—42no—
206C4—162no—
207C4—162no—
208C4#150611yescharacteristic
209C4#150811yescharacteristic
210C4#199411yescharacteristic
211C4#199611yescharacteristic
212C2—21no—
213C2—21no—
214C2—42no—
215C2#990911yescharacteristic
216C2#1168812yes—
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC22×M16C23
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332112
#127852
C4×#3382
#72552