Finite Group Databasebeta

← #93227

Subgroups — #93227

Order 512.

196 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 26 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#10026C211yesmaximal, characteristic
2#7287C211yesmaximal, characteristic, complemented
3#7105C211yesmaximal, characteristic, complemented
4#3245C411yescharacteristic
5#3109—21no—
6#1955C2211yescharacteristic
7#1530—21no—
8#1459—21no—
9#1377—21no—
10#1290C411yescharacteristic
11#532—21no—
12C2×C4×D4D411yescharacteristic
13C2×M32—41no—
14#481—41no—
15#476—41no—
16#474—41no—
17#466—41no—
18#459—41no—
19#459—41no—
20#447—41no—
21#442—42no—
22#432—21no—
23C2×C4⋊C8C2×C411yescharacteristic
24C2×C22⋊C8—41no—
25C2×C22⋊C8—41no—
26C2×#97D411yescharacteristic
27#329—42no—
28#324—41no—
29#323—22no—
30C2×#41—41no—
31C22×D4SD1611yescharacteristic
32C23×C4—21no—
33C2×D8—42no—
34C2×D8—41no—
35C22×C8—21no—
36C22×C8—21no—
37C22×C8—81no—
38C4⋊Q8—21no—
39#127—21no—
40#124—41no—
41#121—41no—
42#121—41no—
43C4×D4—41no—
44C4×D4—41no—
45C4×D4—41no—
46C4×D4—41no—
47C2×C4⋊C4D811yescharacteristic
48C2×C22⋊C4—21no—
49C2×C42C22⋊C411yescharacteristic
50M32—42no—
51M32—81no—
52C2×C16—81no—
53#107—41no—
54#106—42no—
55C4⋊C8—41no—
56C4⋊C8—41no—
57#102—42no—
58#102—42no—
59#102—42no—
60C22⋊C8—41no—
61C22⋊C8—42no—
62C22⋊C8—81no—
63C22⋊C8—81no—
64C22⋊C8—81no—
65#97—22no—
66#97—41no—
67C24—21no—
68#41—162no—
69C2×Q8—41no—
70C2×D4—41no—
71C2×D4—41no—
72C2×D4—41no—
73C2×D4—42no—
74C2×D4—41no—
75C2×D4—81no—
76C22×C4—21no—
77C22×C4—41no—
78C22×C4—41no—
79C22×C4—81no—
80C22×C4—81no—
81C22×C4—81no—
82C22×C4#9911yescharacteristic
83C22×C4#10211yescharacteristic
84C22×C4#10411yescharacteristic
85D8—84no—
86D8—161no—
87C2×C8—21no—
88C2×C8—21no—
89C2×C8—41no—
90C2×C8—41no—
91C2×C8—42no—
92C2×C8—41no—
93C2×C8—81no—
94C2×C8—82no—
95C2×C8—161no—
96C2×C8—161no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C22⋊C4—81no—
101C22⋊C4—81no—
102C22⋊C4—81no—
103C42—21no—
104C42—21no—
105C16—82no—
106C23—22no—
107C23—41no—
108C23—41no—
109C23—81no—
110C23—81no—
111C23—81no—
112C23#32711yescharacteristic
113Q8—161no—
114D4—42no—
115D4—81no—
116D4—82no—
117D4—81no—
118D4—81no—
119D4—161no—
120D4—162no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—81no—
129C2×C4—81no—
130C2×C4—81no—
131C2×C4—81no—
132C2×C4—161no—
133C2×C4—162no—
134C2×C4AΣL(1,𝔽16)11yescharacteristic
135C2×C4#35311yescharacteristic
136C8—41no—
137C8—41no—
138C8—82no—
139C8—81no—
140C8—162no—
141C22—21no—
142C22—21no—
143C22—21no—
144C22—42no—
145C22—82no—
146C22—81no—
147C22—81no—
148C22—161no—
149C22—161no—
150C22—161no—
151C22—161no—
152C22#134311yescharacteristic
153C4—21no—
154C4—41no—
155C4—41no—
156C4—81no—
157C4—161no—
158C4#140211yescharacteristic
159C4#140411yescharacteristic
160C2—41no—
161C2—82no—
162C2—321no—
163C2#739711yescharacteristic
164C2#740011yescharacteristic
165C2#740811yescharacteristic
subgroupG/H
center, socleC22#1343
commutatorC2×C4⋊C8C2×C4
Frattini, cosocle#1955C22
prime radical#10026C2
LCSG ▷ C2×C4⋊C8 ▷ C2×C42 ▷ C23 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #1955 ◁ G
LECSG ▷ #1955 ▷ C2×C42 ▷ C23 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #7287C2semidirect
3. #7105C2semidirect

Maximal subgroups

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

subgroupindex
#100262
#72872
#71052