Finite Group Databasebeta

← #141978

Subgroups — #141978

Order 512.

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

normal subgroups only

This group has 328 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#21022C211yesmaximal, characteristic
2#12791C211yesmaximal, characteristic
3#12607C211yesmaximal, characteristic
4#7194C212yesmaximal
5#7192C212yesmaximal
6C2×#424C411yescharacteristic
7C2×C4×M16C2211yescharacteristic
8#2296C411yescharacteristic
9C2×#364—21no—
10C2×#349C2212yes—
11C2×#349C2212yes—
12C2×#347—21no—
13#1917—21no—
14#1917—21no—
15#1916—21no—
16#1916—21no—
17#1729C411yescharacteristic
18#1729C2211yescharacteristic
19#1729C2211yescharacteristic
20C2×#336C411yescharacteristic
21#1376—21no—
22#1375—21no—
23C22×M16C2×C411yescharacteristic
24C22×M16C2311yescharacteristic
25C23×C8C2×C411yescharacteristic
26C23×C8C2×C411yescharacteristic
27C2×#117C2×C411yescharacteristic
28C2×#117C2×C411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C2×M32—22no—
31C2×M32—22no—
32#424—42no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8—21no—
35C4×M16—42no—
36C2×#97D411yescharacteristic
37C2×#97D411yescharacteristic
38C2×C4×C8D411yescharacteristic
39C2×C4×C8D411yescharacteristic
40#389—21no—
41#389—21no—
42#385—22no—
43C4×C4⋊C4—21no—
44C4×C4⋊C4—21no—
45#364—42no—
46#349—22no—
47#349—24no—
48#349Q814yes—
49#347—42no—
50#336—21no—
51#336—21no—
52#336—21no—
53#336—21no—
54#336—21no—
55#336—21no—
56#336—21no—
57#336—21no—
58C23×C4—21no—
59C23×C4C22×C411yescharacteristic
60C2×M16—22no—
61C2×M16—22no—
62C2×M16—41no—
63C2×M16—41no—
64C2×M16—41no—
65C2×M16C2×Q812yes—
66C2×M16#4112yes—
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—41no—
72C22×C8—41no—
73C22×C8—41no—
74C22×C8—41no—
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C2×D411yescharacteristic
78C22×C8C2×D411yescharacteristic
79#117—41no—
80#117—41no—
81#117—41no—
82#117—41no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4M1611yescharacteristic
87C2×C4⋊C4M1611yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4M1612yes—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—41no—
94C2×C42—41no—
95C2×C42C22⋊C411yescharacteristic
96C2×C42C22⋊C411yescharacteristic
97M32—24no—
98M32—42no—
99M32—42no—
100M32—42no—
101C2×C16—42no—
102C2×C16—42no—
103C4⋊C8—42no—
104C4⋊C8—42no—
105#97—22no—
106#97—41no—
107C4×C8—22no—
108C4×C8—41no—
109#95—22no—
110#95—22no—
111C24C2×M1611yescharacteristic
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—81no—
124C22×C4C2×C22⋊C411yescharacteristic
125C22×C4C4×Q812yes—
126C22×C4C2×M1611yescharacteristic
127C22×C4#13112yes—
128M16—42no—
129M16—42no—
130M16—44no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—82no—
147C2×C8—82no—
148C2×C8#9912yes—
149C2×C8C22⋊Q812yes—
150C2×C8#12411yescharacteristic
151C2×C8C4⋊Q811yescharacteristic
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—44no—
155C4⋊C4—42no—
156C4⋊C4—82no—
157C22⋊C4—42no—
158C22⋊C4—42no—
159C42—21no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—42no—
164C42—41no—
165C42—41no—
166C42—41no—
167C42—41no—
168C16—44no—
169C16—82no—
170C23—21no—
171C23—21no—
172C23—41no—
173C23—41no—
174C23#40711yescharacteristic
175C23#44612yes—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—84no—
196C2×C4#39111yescharacteristic
197C2×C4#40811yescharacteristic
198C2×C4C2×#9911yescharacteristic
199C2×C4#41011yescharacteristic
200C2×C4#41311yescharacteristic
201C2×C4#41311yescharacteristic
202C2×C4#44612yes—
203C8—22no—
204C8—22no—
205C8—21no—
206C8—21no—
207C8—41no—
208C8—81no—
209C8—81no—
210C22—21no—
211C22—21no—
212C22—22no—
213C22—42no—
214C22—42no—
215C22—41no—
216C22—41no—
217C22#146411yescharacteristic
218C22#146511yescharacteristic
219C22#198511yescharacteristic
220C4—22no—
221C4—42no—
222C4—82no—
223C4—81no—
224C4—81no—
225C4—81no—
226C4#146811yescharacteristic
227C4#147311yescharacteristic
228C4#199611yescharacteristic
229C4#199811yescharacteristic
230C2—21no—
231C2—21no—
232C2—42no—
233C2#988111yescharacteristic
234C2#1152311yescharacteristic
235C2#1152411yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC23×C4C22×C4
Frattini, cosocleC22×M16C23
socleC22#1464
prime radicalC24C2×M16
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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
#210222
#127912
#126072
#71942
#71922