Finite Group Databasebeta

← #135244

Subgroups — #135244

Order 512.

299 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 74 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#20774C211yesmaximal, characteristic
2#8731C212yesmaximal
3#7190C212yesmaximal
4#7188C211yesmaximal, characteristic
5#7188C211yesmaximal, characteristic
6C4×#117C2211yescharacteristic
7#1840—22no—
8#1839—22no—
9#1729C411yescharacteristic
10#1729C411yescharacteristic
11#1729C2211yescharacteristic
12#1729C2211yescharacteristic
13#1728C411yescharacteristic
14C2×#336C411yescharacteristic
15#1458C2212yes—
16#1458C2212yes—
17#1276—22no—
18#1274—22no—
19C22×M16C2×C411yescharacteristic
20C22×M16C2×C411yescharacteristic
21C23×C8C2×C411yescharacteristic
22C23×C8C2×C411yescharacteristic
23C2×#117C2×C411yescharacteristic
24C2×#117C2311yescharacteristic
25C22×C42C2×C411yescharacteristic
26#431—22no—
27#431—22no—
28C4×C4⋊C4—21no—
29C4×C4⋊C4—21no—
30C4×C22⋊C4—22no—
31#376D412yes—
32C43D412yes—
33#336—21no—
34#336—21no—
35#336—22no—
36#336—22no—
37#336—21no—
38#336—21no—
39#336—22no—
40#336—22no—
41#324—22no—
42#324—22no—
43#323—22no—
44#323—22no—
45C23×C4—21no—
46C23×C4C22×C411yescharacteristic
47C2×M16—22no—
48C2×M16—22no—
49C2×M16—41no—
50C2×M16—41no—
51C2×M16—41no—
52C2×M16—41no—
53C22×C8—21no—
54C22×C8—21no—
55C22×C8—21no—
56C22×C8—21no—
57C22×C8—22no—
58C22×C8—22no—
59C22×C8—41no—
60C22×C8—41no—
61C22×C8—41no—
62C22×C8—41no—
63#117—22no—
64#117—22no—
65#117—41no—
66#117—41no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4#4111yescharacteristic
69C2×C4⋊C4#4111yescharacteristic
70C2×C22⋊C4—21no—
71C2×C22⋊C4#4112yes—
72C2×C42—21no—
73C2×C42—22no—
74C2×C42—22no—
75C2×C42—41no—
76C2×C42—41no—
77C2×C42C22⋊C412yes—
78C2×C42C22⋊C411yescharacteristic
79C2×C42C22⋊C411yescharacteristic
80C2×C42C2×D412yes—
81C4⋊C8—42no—
82C4⋊C8—42no—
83C22⋊C8—42no—
84C22⋊C8—42no—
85C4×C8—42no—
86C4×C8—42no—
87#95—22no—
88#95—22no—
89C24#11711yescharacteristic
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—22no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4—81no—
100C22×C4C2×C22⋊C411yescharacteristic
101C22×C4#11711yescharacteristic
102C22×C4#12312yes—
103C22×C4#12312yes—
104C22×C4#13112yes—
105C22×C4#13112yes—
106M16—82no—
107M16—82no—
108C2×C8—41no—
109C2×C8—41no—
110C2×C8—41no—
111C2×C8—41no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—41no—
117C2×C8—41no—
118C2×C8—41no—
119C2×C8—41no—
120C2×C8—82no—
121C2×C8—82no—
122C2×C8—82no—
123C2×C8—82no—
124C4⋊C4—41no—
125C4⋊C4—41no—
126C4⋊C4—42no—
127C22⋊C4—42no—
128C22⋊C4—42no—
129C42—22no—
130C42—22no—
131C42—22no—
132C42—22no—
133C42—21no—
134C42—21no—
135C42—21no—
136C42—21no—
137C42—44no—
138C42—42no—
139C42—82no—
140C23—21no—
141C23—21no—
142C23—41no—
143C23—41no—
144C23#38111yescharacteristic
145C23#43312yes—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—84no—
166C2×C4#40811yescharacteristic
167C2×C4#40811yescharacteristic
168C2×C4#41011yescharacteristic
169C2×C4#41011yescharacteristic
170C2×C4#41311yescharacteristic
171C2×C4#41311yescharacteristic
172C2×C4#43312yes—
173C8—81no—
174C8—81no—
175C8—81no—
176C8—81no—
177C8—162no—
178C8—162no—
179C22—21no—
180C22—21no—
181C22—22no—
182C22—42no—
183C22—42no—
184C22—41no—
185C22—41no—
186C22#181811yescharacteristic
187C22#182011yescharacteristic
188C22#182211yescharacteristic
189C4—22no—
190C4—42no—
191C4—42no—
192C4—42no—
193C4—82no—
194C4—81no—
195C4#147311yescharacteristic
196C4#147311yescharacteristic
197C4#147311yescharacteristic
198C4#147311yescharacteristic
199C2—21no—
200C2—21no—
201C2—42no—
202C2#870511yescharacteristic
203C2#1286411yescharacteristic
204C2#1286511yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC23×C4C22×C4
Frattini, cosocleC2×#117C23
socleC22#1820
prime radicalC24#117
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#117 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#207742
#87312
#71902
#71882