Finite Group Databasebeta

← #122349

Subgroups — #122349

Order 512.

311 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 78 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#21350C211yesmaximal, characteristic
2#11460C212yesmaximal
3#7067C214yesmaximal
4C2×#431C2211yescharacteristic
5C2×C4×M16C411yescharacteristic
6C4×#117C411yescharacteristic
7#2163—22no—
8#1940—22no—
9#1851C412yes—
10#1851C2212yes—
11C4×C4⋊C8—22no—
12#1322—22no—
13#1314C2214yes—
14C22×M16C2×C412yes—
15C2×#117—21no—
16C22×C42C2×C411yescharacteristic
17#431—41no—
18#431—41no—
19C2×C4⋊C8—22no—
20C2×C4⋊C8—22no—
21C2×C4⋊C8D412yes—
22C2×C22⋊C8C2×C411yescharacteristic
23C2×C22⋊C8C2311yescharacteristic
24C4×M16—41no—
25C4×M16—41no—
26C2×#97—21no—
27C2×C4×C8—21no—
28C2×C4×C8D412yes—
29C4×C4⋊C4—21no—
30C4×C4⋊C4C812yes—
31C4×C22⋊C4—21no—
32C4×C22⋊C4C2×C412yes—
33#376C811yescharacteristic
34#376C811yescharacteristic
35C43—21no—
36C4⋊C16—42no—
37C22⋊C16—24no—
38#346—42no—
39#336—22no—
40#336—22no—
41C23×C4M1611yescharacteristic
42C23×C4M1611yescharacteristic
43C23×C4C22×C411yescharacteristic
44C2×M16—44no—
45C2×M16—42no—
46C22×C8—22no—
47C22×C8C22⋊C412yes—
48C22×C8C2×D412yes—
49#117—44no—
50C2×C4⋊C4—22no—
51C2×C22⋊C4—22no—
52C2×C42—21no—
53C2×C42—21no—
54C2×C42—21no—
55C2×C42—21no—
56C2×C42—21no—
57C2×C42—21no—
58C2×C42—22no—
59C2×C42—22no—
60C2×C42C22⋊C411yescharacteristic
61C2×C42C22⋊C411yescharacteristic
62C2×C42C2×C811yescharacteristic
63C2×C42C2×C811yescharacteristic
64C2×C16—44no—
65C4⋊C8—42no—
66C4⋊C8—44no—
67C4⋊C8—44no—
68C22⋊C8—41no—
69C22⋊C8—41no—
70#97—42no—
71C4×C8—22no—
72C4×C8—22no—
73C4×C8—41no—
74C4×C8—41no—
75#95—21no—
76#95—21no—
77#95C2×C812yes—
78#95C2×C812yes—
79C24C2×M1611yescharacteristic
80C22×C4—22no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—22no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4C22⋊C811yescharacteristic
92C22×C4C22⋊C811yescharacteristic
93C22×C4C22⋊C811yescharacteristic
94C22×C4C22⋊C811yescharacteristic
95C22×C4C2×C22⋊C411yescharacteristic
96C22×C4C22×C811yescharacteristic
97M16—44no—
98M16—44no—
99C2×C8—24no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—42no—
103C2×C8—42no—
104C2×C8—42no—
105C2×C8—42no—
106C4⋊C4—44no—
107C22⋊C4—44no—
108C42—24no—
109C42—21no—
110C42—21no—
111C42—21no—
112C42—21no—
113C42—21no—
114C42—21no—
115C42—42no—
116C42—42no—
117C42—44no—
118C42—42no—
119C42#9911yescharacteristic
120C42#9911yescharacteristic
121C42#10111yescharacteristic
122C42#10111yescharacteristic
123C16—84no—
124C23—21no—
125C23—22no—
126C23—41no—
127C23—41no—
128C23C2×C22⋊C811yescharacteristic
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—24no—
133C2×C4—22no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4#32411yescharacteristic
151C2×C4#32411yescharacteristic
152C2×C4#32411yescharacteristic
153C2×C4#32411yescharacteristic
154C2×C4C2×#9911yescharacteristic
155C2×C4C2×#10111yescharacteristic
156C8—42no—
157C8—42no—
158C8—44no—
159C22—21no—
160C22—21no—
161C22—41no—
162C22—41no—
163C22—41no—
164C22—82no—
165C22C2×#32411yescharacteristic
166C22#211712yes—
167C4—21no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—44no—
172C4—41no—
173C4—41no—
174C4—41no—
175C4—82no—
176C4#132712yes—
177C4#212012yes—
178C2—22no—
179C2—81no—
180C2#1144311yescharacteristic
181C2#1145211yescharacteristic
182C2C2×#132711yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×C22⋊C8C23
socleC22C2×#324
prime radicalC24C2×M16
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#213502
#114602
#70672