Finite Group Databasebeta

← #94501

Subgroups — #94501

Order 512.

294 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 46 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#11512C212yesmaximal
2#11373C211yesmaximal, characteristic
3C2×#384C2211yescharacteristic
4C2×#340—22no—
5C2×#336C411yescharacteristic
6#1463—22no—
7C4⋊#95C411yescharacteristic
8#1325—22no—
9#1325C414yescomplemented
10C23×C8—21no—
11C22×C4⋊C4—21no—
12C22×C4⋊C4—41no—
13C22×C4⋊C4D412yes—
14C22×C42C2×C411yescharacteristic
15#407—42no—
16#384—41no—
17#384C2×C412yes—
18#384D411yescharacteristic
19#384Q811yescharacteristic
20C2×#95—41no—
21#340—44no—
22#336—41no—
23#324—24no—
24#324—42no—
25C23×C4—21no—
26C23×C4—22no—
27C23×C4—41no—
28C23×C4C22⋊C411yescharacteristic
29C2×M16—82no—
30C22×C8—22no—
31C22×C8—41no—
32C22×C8—42no—
33C22×C8—42no—
34C22×C8—41no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4—22no—
37C2×C4⋊C4—42no—
38C2×C4⋊C4—42no—
39C2×C4⋊C4—42no—
40C2×C4⋊C4—42no—
41C2×C4⋊C4—42no—
42C2×C4⋊C4—41no—
43C2×C4⋊C4—41no—
44C2×C4⋊C4—41no—
45C2×C4⋊C4—82no—
46C2×C4⋊C4—82no—
47C2×C4⋊C4C22⋊C412yes—
48C2×C4⋊C4C4⋊C412yes—
49C2×C42—21no—
50C2×C42—42no—
51C2×C42C4211yescharacteristic
52C2×C42C4⋊C411yescharacteristic
53C22⋊C8—44no—
54C22⋊C8—82no—
55#95—82no—
56C24#9911yescharacteristic
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—22no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—22no—
63C22×C4—22no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—22no—
67C22×C4—22no—
68C22×C4—42no—
69C22×C4—42no—
70C22×C4—42no—
71C22×C4—42no—
72C22×C4—42no—
73C22×C4—82no—
74C22×C4—82no—
75C22×C4—82no—
76C22×C4—81no—
77C22×C4—82no—
78C22×C4—81no—
79C22×C4#9511yescharacteristic
80C22×C4#9911yescharacteristic
81M16—162no—
82M16—162no—
83C2×C8—44no—
84C2×C8—42no—
85C2×C8—82no—
86C2×C8—81no—
87C2×C8—81no—
88C2×C8—84no—
89C4⋊C4—42no—
90C4⋊C4—42no—
91C4⋊C4—44no—
92C4⋊C4—82no—
93C4⋊C4—82no—
94C4⋊C4—84no—
95C4⋊C4—82no—
96C4⋊C4—82no—
97C42—22no—
98C42—22no—
99C42—42no—
100C23—21no—
101C23—21no—
102C23—21no—
103C23—21no—
104C23—22no—
105C23#34211yescharacteristic
106C23#35311yescharacteristic
107C23#35411yescharacteristic
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—44no—
119C2×C4—44no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—82no—
127C2×C4—82no—
128C2×C4—82no—
129C2×C4—82no—
130C2×C4—82no—
131C2×C4—84no—
132C2×C4—162no—
133C2×C4—162no—
134C2×C4—161no—
135C2×C4—161no—
136C2×C4#34412yes—
137C2×C4#35511yescharacteristic
138C2×C4#35611yescharacteristic
139C8—82no—
140C8—164no—
141C22—22no—
142C22—22no—
143C22—22no—
144C22—22no—
145C22—21no—
146C22—21no—
147C22—21no—
148C22—21no—
149C22—21no—
150C22—21no—
151C22#130611yescharacteristic
152C22#139311yescharacteristic
153C22#139411yescharacteristic
154C22#140811yescharacteristic
155C22#140911yescharacteristic
156C22#141312yes—
157C4—21no—
158C4—22no—
159C4—21no—
160C4—42no—
161C4—42no—
162C4—82no—
163C4—82no—
164C4—162no—
165C2—21no—
166C2—21no—
167C2—21no—
168C2—21no—
169C2—22no—
170C2#732611yescharacteristic
171C2#749811yescharacteristic
172C2#750511yescharacteristic
subgroupG/H
center, socleC22#1393
commutatorC2×C42C42
Frattini, cosocleC2×#384C22
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G
LECSG ▷ C2×#384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #1325C4semidirect

Maximal subgroups

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

subgroupindex
#115122
#113732