Finite Group Databasebeta

← #93104

Subgroups — #93104

Order 512.

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

normal subgroups only

This group has 542 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#8542C212yesmaximal
2#8307C211yesmaximal, characteristic
3C22×#121C2211yescharacteristic
4#1819—22no—
5#1781—22no—
6#1781C414yescomplemented
7#1779C412yescomplemented
8C2×#323—22no—
9C23×D4C2×C412yes—
10C23×D4D411yescharacteristic
11C24×C4C2×C411yescharacteristic
12C2×#121—42no—
13C2×#121—44no—
14C22×C4⋊C4Q811yescharacteristic
15C22×C22⋊C4D412yes—
16C2×#99—44no—
17C2×#99—44no—
18C2×C22⋊C8—22no—
19C2×C22⋊C8—42no—
20#379—24no—
21#342—44no—
22#323—44no—
23C25C4211yescharacteristic
24C25C22⋊C412yes—
25C22×D4—24no—
26C22×D4—24no—
27C22×D4—42no—
28C22×D4—42no—
29C22×D4—41no—
30C22×D4—82no—
31C22×D4—84no—
32C23×C4—22no—
33C23×C4—42no—
34C23×C4—82no—
35C23×C4C22⋊C411yescharacteristic
36C23×C4C4⋊C411yescharacteristic
37C23×C4C4⋊C412yes—
38C22×C8—42no—
39#121—416no—
40C2×C4⋊C4—41no—
41C2×C4⋊C4—42no—
42C2×C22⋊C4—24no—
43C2×C22⋊C4—24no—
44C2×C22⋊C4—42no—
45C2×C22⋊C4—44no—
46C2×C22⋊C4—44no—
47C2×C22⋊C4—44no—
48#99—88no—
49#99—88no—
50C22⋊C8—44no—
51C22⋊C8—84no—
52C24—21no—
53C24—22no—
54C24—22no—
55C24—22no—
56C24—42no—
57C24—44no—
58C24—42no—
59C24—82no—
60C24—81no—
61C24—82no—
62C24#9511yescharacteristic
63C24#9914yes—
64C2×D4—48no—
65C2×D4—44no—
66C2×D4—48no—
67C2×D4—48no—
68C2×D4—82no—
69C2×D4—84no—
70C2×D4—84no—
71C2×D4—82no—
72C2×D4—84no—
73C2×D4—84no—
74C2×D4—84no—
75C2×D4—84no—
76C22×C4—21no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—42no—
83C22×C4—44no—
84C22×C4—44no—
85C22×C4—84no—
86C22×C4—82no—
87C22×C4—81no—
88C22×C4—82no—
89C22×C4—84no—
90C22×C4—84no—
91C22×C4#9912yes—
92C2×C8—82no—
93C2×C8—84no—
94C4⋊C4—44no—
95C22⋊C4—48no—
96C22⋊C4—88no—
97C22⋊C4—88no—
98C22⋊C4—88no—
99C23—22no—
100C23—22no—
101C23—24no—
102C23—24no—
103C23—41no—
104C23—41no—
105C23—42no—
106C23—41no—
107C23—42no—
108C23—44no—
109C23—44no—
110C23—44no—
111C23—44no—
112C23—41no—
113C23—42no—
114C23—44no—
115C23—82no—
116C23—82no—
117C23—82no—
118C23—82no—
119C23—82no—
120C23—82no—
121C23—88no—
122C23—168no—
123C23#34212yes—
124C23#34211yescharacteristic
125D4—88no—
126D4—816no—
127C2×C4—22no—
128C2×C4—24no—
129C2×C4—22no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—41no—
133C2×C4—42no—
134C2×C4—44no—
135C2×C4—42no—
136C2×C4—82no—
137C2×C4—88no—
138C2×C4—82no—
139C2×C4—82no—
140C2×C4—164no—
141C2×C4—164no—
142C2×C4—164no—
143C8—162no—
144C22—21no—
145C22—21no—
146C22—22no—
147C22—22no—
148C22—22no—
149C22—44no—
150C22—44no—
151C22—44no—
152C22—44no—
153C22—44no—
154C22—42no—
155C22—82no—
156C22—84no—
157C22—84no—
158C22—81no—
159C22—81no—
160C22—164no—
161C22—164no—
162C22—164no—
163C22#130411yescharacteristic
164C22#139112yes—
165C4—42no—
166C4—42no—
167C4—82no—
168C4—324no—
169C2—22no—
170C2—42no—
171C2—84no—
172C2—82no—
173C2#730511yescharacteristic
174C2#731011yescharacteristic
175C2#731611yescharacteristic
subgroupG/H
center, socleC22#1304
commutatorC25C42
Frattini, prime radical, cosocleC22×#121C22
LCSG ▷ C25 ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ G
LECSG ▷ C22×#121 ▷ C24 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #1781C4semidirect
7. #1779C4semidirect

Maximal subgroups

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

subgroupindex
#85422
#83072