Finite Group Databasebeta

← #94459

Subgroups — #94459

Order 512.

231 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 42 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#12798C211yesmaximal, characteristic
2#8964C211yesmaximal, characteristic
3#7649C211yesmaximal, characteristic
4#2983C411yescharacteristic, complemented
5C4×#124C2211yescharacteristic
6C2×#354C412yes—
7#1839—21no—
8#1799—21no—
9#1478—21no—
10#1461C411yescharacteristic, complemented
11#1393—21no—
12#1276—21no—
13#1272C412yescomplemented
14#1270—21no—
15C2×#124C2×C411yescharacteristic
16C2×C4×Q8C2×C411yescharacteristic
17C2×C4×D4D411yescharacteristic
18#446—42no—
19#432—41no—
20#431—21no—
21#431—21no—
22C2×#99—22no—
23#391C2×C411yescharacteristic
24#390D411yescharacteristic
25#388—41no—
26#382Q811yescharacteristic
27#380—41no—
28C4×C22⋊C4—41no—
29C43D411yescharacteristic
30#354—42no—
31#342—41no—
32#336—22no—
33#336—41no—
34#336—41no—
35D4⋊C8—42no—
36#324—41no—
37C22×Q8C4211yescharacteristic
38C22×D4C22⋊C411yescharacteristic
39C23×C4—21no—
40C22×C8—42no—
41#124—21no—
42#124—21no—
43#124—41no—
44C4×Q8—41no—
45C4×Q8—41no—
46C4×D4—41no—
47C4×D4—41no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—81no—
50C2×C4⋊C4C4⋊C411yescharacteristic
51C2×C4⋊C4C4⋊C411yescharacteristic
52C2×C22⋊C4—21no—
53C2×C22⋊C4—41no—
54C2×C22⋊C4—41no—
55C2×C22⋊C4—42no—
56C2×C42—21no—
57C2×C42—21no—
58C2×C42—21no—
59C2×C42C22⋊C411yescharacteristic
60C2×C42C22⋊C411yescharacteristic
61C2×C42C4⋊C411yescharacteristic
62C4⋊C8—22no—
63C4⋊C8—41no—
64C4⋊C8—42no—
65#99—42no—
66#99—42no—
67C22⋊C8—41no—
68C22⋊C8—41no—
69#97—42no—
70C4×C8—41no—
71C4×C8—41no—
72#95—41no—
73#95—41no—
74#95—81no—
75C24—21no—
76C2×Q8—21no—
77C2×Q8—21no—
78C2×Q8—41no—
79C2×Q8—41no—
80C2×D4—21no—
81C2×D4—41no—
82C2×D4—41no—
83C2×D4#10011yescharacteristic
84C2×D4#10111yescharacteristic
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4—82no—
93C22×C4—82no—
94C22×C4#9511yescharacteristic
95C22×C4#9911yescharacteristic
96C22×C4#9911yescharacteristic
97C2×C8—42no—
98C2×C8—81no—
99C2×C8—82no—
100C2×C8—82no—
101C2×C8—82no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—42no—
105C4⋊C4—162no—
106C22⋊C4—42no—
107C22⋊C4—42no—
108C22⋊C4—81no—
109C22⋊C4—82no—
110C22⋊C4—82no—
111C22⋊C4—82no—
112C42—21no—
113C42—21no—
114C42—21no—
115C42—21no—
116C42—41no—
117C42—42no—
118C42—41no—
119C42—41no—
120C42—81no—
121C23—21no—
122C23—21no—
123C23—41no—
124C23—41no—
125C23—81no—
126C23—81no—
127C23#34211yescharacteristic
128Q8—81no—
129Q8—81no—
130D4—41no—
131D4—41no—
132D4—81no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—82no—
150C2×C4—81no—
151C2×C4—81no—
152C2×C4—161no—
153C2×C4—162no—
154C2×C4—162no—
155C2×C4—162no—
156C2×C4#34311yescharacteristic
157C2×C4#34411yescharacteristic
158C2×C4AΣL(1,𝔽16)11yescharacteristic
159C2×C4#35211yescharacteristic
160C8—82no—
161C8—161no—
162C8—162no—
163C22—21no—
164C22—21no—
165C22—41no—
166C22—41no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22—161no—
172C22#130511yescharacteristic
173C22#139011yescharacteristic
174C22#139111yescharacteristic
175C4—21no—
176C4—21no—
177C4—21no—
178C4—21no—
179C4—42no—
180C4—41no—
181C4—81no—
182C4—81no—
183C4—81no—
184C4—81no—
185C4—322no—
186C2—21no—
187C2—21no—
188C2—81no—
189C2—81no—
190C2#732011yescharacteristic
191C2#748111yescharacteristic
192C2#749011yescharacteristic
subgroupG/H
center, socleC22#1390
commutatorC22×Q8C42
Frattini, cosocleC4×#124C22
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×Q8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G
LECSG ▷ C4×#124 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #2983C4semidirect
10. #1461C4semidirect
13. #1272C4semidirect

Maximal subgroups

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

subgroupindex
#127982
#89642
#76492