Finite Group Databasebeta

← #149807

Subgroups — #149807

Order 512.

336 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 62 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#20764C211yesmaximal, characteristic
2#11512C212yesmaximal
3#7367C214yesmaximal, complemented
4C2×C4×M16C411yescharacteristic
5#2285C411yescharacteristic
6C2×#384C2211yescharacteristic
7#1836C412yes—
8#1836C2212yes—
9#1834—22no—
10#1736—22no—
11#1463—22no—
12#1325—22no—
13#1325C2214yes—
14C22×M16—21no—
15C2×#117—21no—
16C22×C4⋊C4—41no—
17C22×C4⋊C4D412yes—
18C22×C42C2×C411yescharacteristic
19#407—42no—
20C4×M16—24no—
21C2×#97C2×C412yes—
22C2×C4×C8—21no—
23#384—21no—
24#384—41no—
25#384C2×C411yescharacteristic
26#384D412yes—
27#384C2311yescharacteristic
28#382C2×C412yes—
29#380—41no—
30C4×C4⋊C4—41no—
31#340—24no—
32#340—24no—
33#324—24no—
34#324—42no—
35C23×C4—21no—
36C23×C4—22no—
37C23×C4C22⋊C411yescharacteristic
38C2×M16—24no—
39C2×M16—42no—
40C2×M16—42no—
41C2×M16—82no—
42C22×C8—22no—
43#117—44no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—42no—
47C2×C4⋊C4—42no—
48C2×C4⋊C4—41no—
49C2×C4⋊C4—42no—
50C2×C4⋊C4—41no—
51C2×C4⋊C4—41no—
52C2×C4⋊C4—41no—
53C2×C4⋊C4—84no—
54C2×C4⋊C4C22⋊C412yes—
55C2×C4⋊C4C2×D412yes—
56C2×C22⋊C4—41no—
57C2×C42—21no—
58C2×C42—22no—
59C2×C42—22no—
60C2×C42—41no—
61C2×C42C22⋊C411yescharacteristic
62C2×C42C22×C411yescharacteristic
63C22⋊C8—44no—
64C22⋊C8—82no—
65#97—22no—
66#97M1614yescomplemented
67C4×C8—24no—
68#95—42no—
69C24#9911yescharacteristic
70C22×C4—22no—
71C22×C4—21no—
72C22×C4—22no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—24no—
76C22×C4—21no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—42no—
80C22×C4—42no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—84no—
85C22×C4#9911yescharacteristic
86C22×C4C2×C22⋊C411yescharacteristic
87M16—48no—
88M16—164no—
89C2×C8—24no—
90C2×C8—42no—
91C2×C8—42no—
92C2×C8—84no—
93C4⋊C4—42no—
94C4⋊C4—42no—
95C4⋊C4—44no—
96C4⋊C4—82no—
97C4⋊C4—82no—
98C4⋊C4—82no—
99C4⋊C4—84no—
100C4⋊C4—82no—
101C4⋊C4—82no—
102C22⋊C4—82no—
103C42—24no—
104C42—21no—
105C42—21no—
106C42—82no—
107C42#10112yes—
108C42C2×M1612yes—
109C23—21no—
110C23—21no—
111C23—41no—
112C23—41no—
113C23#35412yes—
114C23C2×#9911yescharacteristic
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—24no—
118C2×C4—24no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—21no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—48no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—84no—
131C2×C4—82no—
132C2×C4—82no—
133C2×C4—82no—
134C2×C4—82no—
135C2×C4—82no—
136C2×C4—162no—
137C2×C4#35512yes—
138C2×C4#40711yescharacteristic
139C2×C4C2×#10111yescharacteristic
140C8—44no—
141C8—164no—
142C22—21no—
143C22—21no—
144C22—22no—
145C22—22no—
146C22—22no—
147C22—42no—
148C22—42no—
149C22#146311yescharacteristic
150C22C2×#35411yescharacteristic
151C22C2×#35511yescharacteristic
152C4—24no—
153C4—21no—
154C4—22no—
155C4—21no—
156C4—42no—
157C4—82no—
158C4—82no—
159C4—162no—
160C2—21no—
161C2—21no—
162C2—42no—
163C2#1151111yescharacteristic
164C2#1359511yescharacteristic
165C2#1359911yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#384C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7367C2semidirect
66. #97M16semidirect

Maximal subgroups

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

subgroupindex
#207642
#115122
#73672