Finite Group Databasebeta

← #135197

Subgroups — #135197

Order 512.

354 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 94 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#20365C211yesmaximal, characteristic
2#12849C212yesmaximal
3#7193C212yesmaximal
4#7191C212yesmaximal
5C2×C4×M16C411yescharacteristic
6C2×C4×M16C2211yescharacteristic
7C2×#376C411yescharacteristic
8C2×#349C2214yes—
9#1746—22no—
10#1746—22no—
11#1728C411yescharacteristic
12#1728C411yescharacteristic
13#1728C2211yescharacteristic
14#1728C2211yescharacteristic
15#1376—22no—
16#1375—22no—
17C22×M16C2×C411yescharacteristic
18C22×M16C2×C411yescharacteristic
19C22×M16C2×C411yescharacteristic
20C22×M16C2311yescharacteristic
21C22×C42C2×C411yescharacteristic
22C22×C42C2×C411yescharacteristic
23C22×C42C2×C411yescharacteristic
24C2×M32—24no—
25C4×M16—22no—
26C4×M16—22no—
27C4×M16—22no—
28C4×M16C814yes—
29C2×#97—21no—
30C2×#97D412yes—
31C2×C4×C8—21no—
32C2×C4×C8D412yes—
33#376—21no—
34#376—21no—
35#376—22no—
36C2×#95—22no—
37#349—24no—
38#349—24no—
39#336—22no—
40#336—22no—
41#336—22no—
42#336—22no—
43C23×C4—21no—
44C23×C4—21no—
45C23×C4M1612yes—
46C23×C4C22×C411yescharacteristic
47C2×M16—21no—
48C2×M16—21no—
49C2×M16—21no—
50C2×M16—21no—
51C2×M16—41no—
52C2×M16—41no—
53C2×M16—41no—
54C2×M16—41no—
55C2×M16—41no—
56C2×M16—41no—
57C2×M16—41no—
58C2×M16—41no—
59C2×M16C2×C812yes—
60C2×M16C2×C812yes—
61C2×M16#4112yes—
62C2×M16#4112yes—
63C22×C8—21no—
64C22×C8—21no—
65C22×C8C22⋊C412yes—
66C22×C8C2×D412yes—
67C2×C42—21no—
68C2×C42—21no—
69C2×C42—21no—
70C2×C42—21no—
71C2×C42—21no—
72C2×C42—21no—
73C2×C42—41no—
74C2×C42—41no—
75C2×C42—41no—
76C2×C42—41no—
77C2×C42C22⋊C411yescharacteristic
78C2×C42C22⋊C411yescharacteristic
79C2×C42C2×C812yes—
80C2×C42M1612yes—
81C2×C42M1612yes—
82M32—44no—
83M32—44no—
84C2×C16—44no—
85#97—24no—
86#97—24no—
87C4×C8—24no—
88C4×C8—24no—
89#95—24no—
90#95—24no—
91C24C2×M1611yescharacteristic
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—42no—
103C22×C4—42no—
104C22×C4—82no—
105C22×C4—82no—
106C22×C4—82no—
107C22×C4C2×C22⋊C411yescharacteristic
108C22×C4#11711yescharacteristic
109C22×C4#11711yescharacteristic
110C22×C4C22×C811yescharacteristic
111C22×C4C2×M1611yescharacteristic
112C22×C4C2×M1611yescharacteristic
113M16—42no—
114M16—42no—
115M16—42no—
116M16—42no—
117M16—42no—
118M16—42no—
119M16—42no—
120M16—42no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—41no—
132C2×C8#12314yes—
133C42—21no—
134C42—21no—
135C42—22no—
136C42—22no—
137C42—42no—
138C42—42no—
139C42—41no—
140C42—41no—
141C42—41no—
142C42—41no—
143C42C22⋊C812yes—
144C42C22⋊C812yes—
145C16—84no—
146C23—21no—
147C23—21no—
148C23—42no—
149C23#40711yescharacteristic
150C23#43111yescharacteristic
151C23#43211yescharacteristic
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—44no—
167C2×C4—44no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—82no—
171C2×C4—82no—
172C2×C4—84no—
173C2×C4—84no—
174C2×C4#38111yescharacteristic
175C2×C4C2×C22⋊C811yescharacteristic
176C2×C4#41011yescharacteristic
177C2×C4#41011yescharacteristic
178C2×C4#41311yescharacteristic
179C2×C4#41311yescharacteristic
180C2×C4#43111yescharacteristic
181C2×C4#43211yescharacteristic
182C8—24no—
183C8—42no—
184C8—42no—
185C8—42no—
186C22—22no—
187C22—21no—
188C22—21no—
189C22—41no—
190C22—41no—
191C22—42no—
192C22—42no—
193C22#146411yescharacteristic
194C22#146411yescharacteristic
195C22#181711yescharacteristic
196C4—21no—
197C4—21no—
198C4—42no—
199C4—42no—
200C4—82no—
201C4—82no—
202C4#145811yescharacteristic
203C4#145811yescharacteristic
204C4#182011yescharacteristic
205C4#182211yescharacteristic
206C2—21no—
207C2—21no—
208C2—41no—
209C2—41no—
210C2#869911yescharacteristic
211C2#1152212yes—
subgroupG/H
centerC2×C4#410
commutatorC22×C4C22×C8
Frattini, cosocleC22×M16C23
socleC22#1464
prime radicalC24C2×M16
LCSG ▷ C22×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
#203652
#128492
#71932
#71912