Finite Group Databasebeta

← #93200

Subgroups — #93200

Order 512.

266 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 28 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#8459C211yesmaximal, characteristic
2#7133C211yesmaximal, characteristic, complemented
3#7061C211yesmaximal, characteristic, complemented
4#3497C411yescharacteristic
5#1985C2211yescharacteristic
6#1551—21no—
7#1530—21no—
8#1467C411yescharacteristic
9#1377—21no—
10#1322—21no—
11C2×2−1+4—21no—
12#550—41no—
13D4×Q8—41no—
14#548—42no—
15#537—41no—
16#537C812yes—
17#532—21no—
18#532—41no—
19C2×C4×Q8C2×C411yescharacteristic
20C2×M32—41no—
21#432—21no—
22C2×C4⋊C8D411yescharacteristic
23C2×#97D411yescharacteristic
24C22⋊C16—41no—
25#336—41no—
26#333—41no—
27#331—41no—
28#329—42no—
29#324—41no—
302−1+4—84no—
31C2×#41—22no—
32C2×#41—42no—
33C2×#41—42no—
34C22×Q8—42no—
35C22×Q8M1611yescharacteristic
36C22×C8—21no—
37C22×C8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—41no—
40#127—21no—
41#127—41no—
42#124—22no—
43#124—42no—
44C22⋊Q8—44no—
45C22⋊Q8—42no—
46#121—42no—
47#121—42no—
48#121—44no—
49#121—41no—
50#121—41no—
51C4×Q8—41no—
52C4×Q8—41no—
53C4×D4—41no—
54C4×D4—41no—
55C4×D4—42no—
56C4×D4—42no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4C2×C811yescharacteristic
59C2×C42—21no—
60C2×C42C22⋊C411yescharacteristic
61M32—42no—
62M32—81no—
63C2×C16—81no—
64C2×C16—81no—
65C4⋊C8—41no—
66C4⋊C8—41no—
67C22⋊C8—41no—
68#97—41no—
69#97—41no—
70#41—84no—
71#41—84no—
72#41—84no—
73#41—84no—
74#41—84no—
75C2×Q8—22no—
76C2×Q8—42no—
77C2×Q8—41no—
78C2×Q8—42no—
79C2×Q8—84no—
80C2×Q8—84no—
81C2×D4—22no—
82C2×D4—41no—
83C2×D4—42no—
84C2×D4—42no—
85C2×D4—42no—
86C2×D4—41no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—42no—
90C22×C4—44no—
91C22×C4C22⋊C811yescharacteristic
92C22×C4#9911yescharacteristic
93C22×C4#10011yescharacteristic
94C2×C8—21no—
95C2×C8—21no—
96C2×C8—41no—
97C2×C8—41no—
98C2×C8—41no—
99C2×C8—81no—
100C2×C8—81no—
101C4⋊C4—41no—
102C4⋊C4—42no—
103C4⋊C4—41no—
104C4⋊C4—42no—
105C4⋊C4—41no—
106C22⋊C4—42no—
107C22⋊C4—44no—
108C22⋊C4—42no—
109C42—21no—
110C42—21no—
111C42—41no—
112C42—41no—
113C16—82no—
114C16—161no—
115C23—42no—
116C23#32311yescharacteristic
117Q8—42no—
118Q8—42no—
119Q8—82no—
120Q8—84no—
121D4—82no—
122D4—84no—
123D4—84no—
124D4—81no—
125D4—81no—
126D4—161no—
127D4—162no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—42no—
139C2×C4—88no—
140C2×C4—82no—
141C2×C4—82no—
142C2×C4—81no—
143C2×C4AΣL(1,𝔽16)11yescharacteristic
144C2×C4AΣL(1,𝔽16)11yescharacteristic
145C8—41no—
146C8—41no—
147C8—81no—
148C8—161no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—82no—
153C22—82no—
154C22—82no—
155C22#131611yescharacteristic
156C4—21no—
157C4—42no—
158C4—41no—
159C4—41no—
160C4—82no—
161C4—81no—
162C4#140411yescharacteristic
163C4#140511yescharacteristic
164C2—41no—
165C2—162no—
166C2#734211yescharacteristic
167C2#734511yescharacteristic
168C2#734611yescharacteristic
subgroupG/H
center, socleC22#1316
commutatorC2×C4⋊C4C2×C8
Frattini, cosocle#1985C22
prime radical#3497C4
LCSG ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C23 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #1985 ◁ G
LECSG ▷ #1985 ▷ C2×C42 ▷ C23 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #7133C2semidirect
3. #7061C2semidirect

Maximal subgroups

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

subgroupindex
#84592
#71332
#70612