Finite Group Databasebeta

← #93127

Subgroups — #93127

Order 512.

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

normal subgroups only

This group has 332 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#8556C211yesmaximal, characteristic
2#8545C211yesmaximal, characteristic
3#8309C211yesmaximal, characteristic
4C22×#126C2211yescharacteristic
5#1821—21no—
6#1819—21no—
7#1819—21no—
8#1819C412yes—
9#1817C412yescomplemented
10#1463C412yescomplemented
11#1462—21no—
12C2×#324—21no—
13C2×#323—21no—
14C2×#126—42no—
15C2×#126—42no—
16C2×#126—42no—
17C22×C4⋊C4C2×C411yescharacteristic
18C22×C4⋊C4D411yescharacteristic
19C22×C4⋊C4Q811yescharacteristic
20C22×C22⋊C4C2×C411yescharacteristic
21C22×C22⋊C4D411yescharacteristic
22C22×C22⋊C4D411yescharacteristic
23C22×C42C2×C411yescharacteristic
24#407—22no—
25C2×C22⋊C8—22no—
26#379—22no—
27#343—42no—
28#342—42no—
29#342—42no—
30#342—42no—
31#336—42no—
32#324—42no—
33#324—42no—
34#323—42no—
35#323—42no—
36C25C22⋊C411yescharacteristic
37C23×C4C4211yescharacteristic
38C23×C4C22⋊C411yescharacteristic
39C23×C4C22⋊C411yescharacteristic
40C23×C4C4⋊C411yescharacteristic
41C23×C4C4⋊C411yescharacteristic
42C23×C4C4⋊C411yescharacteristic
43C2×M16—42no—
44C22×C8—42no—
45#126—416no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—41no—
50C2×C4⋊C4—41no—
51C2×C4⋊C4—41no—
52C2×C4⋊C4—41no—
53C2×C4⋊C4—41no—
54C2×C4⋊C4—41no—
55C2×C22⋊C4—22no—
56C2×C22⋊C4—22no—
57C2×C22⋊C4—22no—
58C2×C22⋊C4—22no—
59C2×C22⋊C4—22no—
60C2×C22⋊C4—41no—
61C2×C22⋊C4—41no—
62C2×C22⋊C4—41no—
63C2×C22⋊C4—41no—
64C2×C22⋊C4—42no—
65C2×C22⋊C4—42no—
66C2×C22⋊C4—42no—
67C2×C42—22no—
68C2×C42—41no—
69C2×C42—41no—
70C22⋊C8—42no—
71C22⋊C8—42no—
72C22⋊C8—42no—
73C22⋊C8—42no—
74C24—21no—
75C24—21no—
76C24—42no—
77C24—81no—
78C24—81no—
79C24#9511yescharacteristic
80C24#9912yes—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4—81no—
100C22×C4—81no—
101C22×C4—81no—
102C22×C4—81no—
103C22×C4—82no—
104C22×C4#9912yes—
105C22×C4#9912yes—
106M16—84no—
107C2×C8—82no—
108C2×C8—82no—
109C2×C8—84no—
110C4⋊C4—44no—
111C4⋊C4—44no—
112C4⋊C4—44no—
113C22⋊C4—44no—
114C22⋊C4—44no—
115C22⋊C4—44no—
116C22⋊C4—84no—
117C22⋊C4—84no—
118C22⋊C4—84no—
119C42—44no—
120C23—22no—
121C23—22no—
122C23—41no—
123C23—41no—
124C23—42no—
125C23—42no—
126C23—42no—
127C23—42no—
128C23—42no—
129C23—81no—
130C23—81no—
131C23—81no—
132C23—81no—
133C23—164no—
134C23#34211yescharacteristic
135C23#34211yescharacteristic
136C23#34211yescharacteristic
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—81no—
150C2×C4—81no—
151C2×C4—81no—
152C2×C4—81no—
153C2×C4—81no—
154C2×C4—81no—
155C2×C4—81no—
156C2×C4—81no—
157C2×C4—81no—
158C2×C4—81no—
159C2×C4—81no—
160C2×C4—81no—
161C2×C4—162no—
162C2×C4—162no—
163C2×C4—162no—
164C8—162no—
165C8—162no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—21no—
174C22—44no—
175C22—42no—
176C22—42no—
177C22—42no—
178C22—162no—
179C22—162no—
180C22—162no—
181C22#130411yescharacteristic
182C22#139112yes—
183C4—42no—
184C4—42no—
185C4—81no—
186C4—81no—
187C4—81no—
188C4—81no—
189C4—322no—
190C2—22no—
191C2—41no—
192C2—41no—
193C2—82no—
194C2#730511yescharacteristic
195C2#731311yescharacteristic
196C2#731911yescharacteristic
subgroupG/H
center, socleC22#1304
commutatorC23×C4C42
Frattini, cosocleC22×#126C22
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#126 ◁ G
LECSG ▷ C22×#126 ▷ C24 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #1817C4semidirect
10. #1463C4semidirect

Maximal subgroups

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

subgroupindex
#85562
#85452
#83092