Finite Group Databasebeta

← #93489

Subgroups — #93489

Order 512.

556 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 70 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#10727C211yesmaximal, characteristic
2#8299C212yesmaximal
3C23×C4⋊C4C2211yescharacteristic
4C2×#407C412yes—
5#1779C414yes—
6C2×#343—22no—
7C2×#324—22no—
8C2×#324—22no—
9C24×C4C2×C411yescharacteristic
10C24×C4C2×C412yes—
11C22×M16—22no—
12C22×C4⋊C4—42no—
13C22×C4⋊C4—44no—
14C22×C4⋊C4D411yescharacteristic
15C22×C4⋊C4D412yes—
16C22×C4⋊C4Q811yescharacteristic
17#407—22no—
18#407—42no—
19#407C814yescomplemented
20C2×C22⋊C8—24no—
21C2×C22⋊C8—42no—
22C2×C22⋊C8—44no—
23#343—44no—
24#324—44no—
25#324—44no—
26C25C4211yescharacteristic
27C23×C4—22no—
28C23×C4—21no—
29C23×C4—21no—
30C23×C4—24no—
31C23×C4—42no—
32C23×C4—42no—
33C23×C4—42no—
34C23×C4—42no—
35C23×C4—84no—
36C23×C4C22⋊C411yescharacteristic
37C23×C4C22⋊C412yes—
38C23×C4C4⋊C411yescharacteristic
39C23×C4C4⋊C412yes—
40C23×C4C2×C812yes—
41C23×C4M1612yes—
42C2×M16—24no—
43C2×M16—42no—
44C2×M16—44no—
45C2×M16—44no—
46C22×C8—42no—
47C22×C8—44no—
48C2×C4⋊C4—24no—
49C2×C4⋊C4—24no—
50C2×C4⋊C4—24no—
51C2×C4⋊C4—42no—
52C2×C4⋊C4—41no—
53C2×C4⋊C4—42no—
54C2×C4⋊C4—41no—
55C2×C4⋊C4—416no—
56C22⋊C8—44no—
57C22⋊C8—44no—
58C22⋊C8—44no—
59C22⋊C8—44no—
60C22⋊C8—88no—
61C24—22no—
62C24—21no—
63C24—21no—
64C24—22no—
65C24—42no—
66C24—42no—
67C24#9511yescharacteristic
68C24C4×C811yescharacteristic
69C24#9711yescharacteristic
70C22×C4—22no—
71C22×C4—22no—
72C22×C4—22no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—22no—
85C22×C4—44no—
86C22×C4—44no—
87C22×C4—44no—
88C22×C4—44no—
89C22×C4—44no—
90C22×C4—44no—
91C22×C4—48no—
92C22×C4—48no—
93C22×C4—84no—
94C22×C4—82no—
95C22×C4—82no—
96C22×C4—84no—
97C22×C4—84no—
98C22×C4—84no—
99C22×C4—84no—
100C22×C4C22⋊C812yes—
101C22×C4#9912yes—
102C22×C4#9914yes—
103C22×C4#10011yescharacteristic
104C22×C4#10111yescharacteristic
105C22×C4C4⋊C812yes—
106M16—48no—
107M16—84no—
108C2×C8—44no—
109C2×C8—82no—
110C2×C8—82no—
111C2×C8—84no—
112C2×C8—88no—
113C4⋊C4—48no—
114C4⋊C4—44no—
115C4⋊C4—44no—
116C23—24no—
117C23—22no—
118C23—22no—
119C23—24no—
120C23—21no—
121C23—21no—
122C23—22no—
123C23—22no—
124C23—41no—
125C23—41no—
126C23—42no—
127C23—42no—
128C23—42no—
129C23—41no—
130C23—41no—
131C23—42no—
132C23—42no—
133C23—42no—
134C23—44no—
135C23—44no—
136C23—41no—
137C23—42no—
138C23—41no—
139C23#33611yescharacteristic
140C23#34211yescharacteristic
141C23#34212yes—
142C23#34311yescharacteristic
143C23#34412yes—
144C2×C4—24no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—24no—
151C2×C4—24no—
152C2×C4—44no—
153C2×C4—44no—
154C2×C4—44no—
155C2×C4—44no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—44no—
160C2×C4—44no—
161C2×C4—42no—
162C2×C4—82no—
163C2×C4—82no—
164C2×C4—816no—
165C2×C4—82no—
166C2×C4—82no—
167C2×C4#32312yes—
168C2×C4#32412yes—
169C8—84no—
170C8—164no—
171C22—22no—
172C22—21no—
173C22—21no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—24no—
178C22—24no—
179C22—41no—
180C22—41no—
181C22—42no—
182C22—41no—
183C22—41no—
184C22—42no—
185C22—42no—
186C22—44no—
187C22—44no—
188C22—42no—
189C22—42no—
190C22—41no—
191C22—41no—
192C22—42no—
193C22—42no—
194C22#128011yescharacteristic
195C22#128212yes—
196C22#130411yescharacteristic
197C22#130512yes—
198C22#130611yescharacteristic
199C4—22no—
200C4—22no—
201C4—42no—
202C4—44no—
203C4—82no—
204C2—21no—
205C2—22no—
206C2—22no—
207C2—21no—
208C2—42no—
209C2—42no—
210C2#703211yescharacteristic
211C2#731611yescharacteristic
212C2#731711yescharacteristic
subgroupG/H
center, socleC22#1304
commutatorC24C4×C8
Frattini, cosocleC23×C4⋊C4C22
prime radicalC25C42
LCSG ▷ C24 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C23×C4⋊C4 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #407C8semidirect

Maximal subgroups

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

subgroupindex
#107272
#82992