Finite Group Databasebeta

← #94137

Subgroups — #94137

Order 512.

207 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 52 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#12982C211yesmaximal, characteristic
2#11485C211yesmaximal, characteristic
3#8241C211yesmaximal, characteristic
4#3407C411yescharacteristic
5#2931C2211yescharacteristic
6#2117—21no—
7#2115—21no—
8#2115C411yescharacteristic
9#1543C411yescharacteristic
10#1525C411yescharacteristic
11C8×M16—21no—
12#1376—21no—
13#1327—21no—
14#1321C412yes—
15#1299—21no—
16C2×#131C2×C411yescharacteristic
17C2×#131D411yescharacteristic
18#532C2×C411yescharacteristic
19#504—42no—
20C2×M32—21no—
21C2×M32—21no—
22C2×M32—41no—
23C2×M32—41no—
24#436—41no—
25C8×D4—41no—
26#431—21no—
27#423D411yescharacteristic
28#408C2×C411yescharacteristic
29#408Q811yescharacteristic
30C4×M16—21no—
31C2×C4×C8D411yescharacteristic
32#350—42no—
33#349—22no—
34#349—41no—
35#349—41no—
36#333—21no—
37#331C812yes—
38#329—21no—
39#329C812yes—
40#322—41no—
41C82—41no—
42C2×#41C4211yescharacteristic
43C2×#41C4⋊C411yescharacteristic
44#131—41no—
45#131—41no—
46#131—41no—
47#131—41no—
48C2×M16—21no—
49C2×M16—21no—
50C2×M16—41no—
51C2×M16C22⋊C411yescharacteristic
52C2×M16C4⋊C411yescharacteristic
53C22×C8—21no—
54C22×C8—21no—
55C22×C8C22⋊C411yescharacteristic
56C22×C8C4⋊C411yescharacteristic
57C4⋊Q8M1611yescharacteristic
58#127C2×C811yescharacteristic
59#124C2×C811yescharacteristic
60#124M1611yescharacteristic
61#121—41no—
62C4×D4—41no—
63C2×C42C22⋊C411yescharacteristic
64M32—22no—
65M32—42no—
66M32—41no—
67M32—42no—
68M32—41no—
69M32—41no—
70M32—81no—
71C2×C16—41no—
72C2×C16—42no—
73C2×C16—41no—
74C2×C16—81no—
75C4⋊C8—22no—
76C4⋊C8—41no—
77C22⋊C8—41no—
78C22⋊C8—41no—
79C22⋊C8—41no—
80#97—22no—
81C4×C8—21no—
82C4×C8—21no—
83C4×C8—41no—
84C4×C8—41no—
85#41—41no—
86#41—41no—
87#41—41no—
88#41—41no—
89C2×Q8#9711yescharacteristic
90C2×Q8C4⋊C811yescharacteristic
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—41no—
94C2×D4C4×C811yescharacteristic
95C2×D4C4⋊C811yescharacteristic
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4#9511yescharacteristic
100M16—42no—
101M16—44no—
102M16—41no—
103M16—41no—
104M16—81no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—42no—
111C2×C8—41no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—42no—
117C2×C8#9911yescharacteristic
118C2×C8#9911yescharacteristic
119C4⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C42—21no—
123C42C22⋊C811yescharacteristic
124C42C22⋊C811yescharacteristic
125C16—42no—
126C16—81no—
127C16—82no—
128C23—21no—
129C23—21no—
130C23#34411yescharacteristic
131Q8—41no—
132Q8—41no—
133D4—41no—
134D4—41no—
135D4—42no—
136D4—82no—
137D4—81no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—42no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4#33611yescharacteristic
150C2×C4#34211yescharacteristic
151C8—21no—
152C8—44no—
153C8—41no—
154C8—41no—
155C8—41no—
156C8—82no—
157C8AΣL(1,𝔽16)11yescharacteristic
158C8#35211yescharacteristic
159C22—21no—
160C22—41no—
161C22—41no—
162C22—41no—
163C22—81no—
164C22—81no—
165C22#128211yescharacteristic
166C4—22no—
167C4—21no—
168C4—41no—
169C4—41no—
170C4—41no—
171C4#139011yescharacteristic
172C4#139211yescharacteristic
173C2—21no—
174C2—41no—
175C2—81no—
176C2—81no—
177C2#720311yescharacteristic
subgroupG/H
centerC8AΣL(1,𝔽16)
commutatorC2×D4C4×C8
Frattini, cosocle#2931C22
socleC2#7203
prime radical#532C2×C4
LCSG ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ #2931 ◁ G
LECSG ▷ #2931 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#129822
#114852
#82412