Finite Group Databasebeta

← #149808

Subgroups — #149808

Order 512.

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

normal subgroups only

This group has 306 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#20765C211yesmaximal, characteristic
2#11513C212yesmaximal
3#7367C214yesmaximal, complemented
4C2×C4×M16C411yescharacteristic
5#2285C2211yescharacteristic
6C2×#382C411yescharacteristic
7#1836C412yes—
8#1836C2212yes—
9#1835—22no—
10#1736—22no—
11#1464—22no—
12#1325C2214yes—
13#1324—22no—
14C22×M16—21no—
15C2×#117D412yes—
16C22×C4⋊C4—21no—
17C22×C42C2×C411yescharacteristic
18#432—42no—
19C4×M16—24no—
20C2×#97C2×C412yes—
21C2×C4×C8—21no—
22#384C2×C411yescharacteristic
23#384C2311yescharacteristic
24#382—21no—
25#382—41no—
26#382C2×C412yes—
27#382D412yes—
28#380—41no—
29C4×C4⋊C4—41no—
30C2×#95—41no—
31#340—24no—
32#340—24no—
33#324—24no—
34#323—42no—
35C23×C4—21no—
36C23×C4—41no—
37C23×C4C22⋊C411yescharacteristic
38C2×M16—24no—
39C2×M16—42no—
40C2×M16—42no—
41C22×C8—22no—
42#117—42no—
43#117—42no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—44no—
46C2×C4⋊C4—41no—
47C2×C4⋊C4—41no—
48C2×C4⋊C4—41no—
49C2×C4⋊C4C22⋊C412yes—
50C2×C4⋊C4C2×D412yes—
51C2×C22⋊C4—22no—
52C2×C42—21no—
53C2×C42—22no—
54C2×C42—22no—
55C2×C42—22no—
56C2×C42C22⋊C411yescharacteristic
57C2×C42C22×C411yescharacteristic
58C4⋊C8—82no—
59C22⋊C8—44no—
60#97—22no—
61#97—82no—
62#97M1614yes—
63C4×C8—24no—
64#95—42no—
65#95—42no—
66#95—42no—
67C24#9911yescharacteristic
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—24no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—42no—
78C22×C4—81no—
79C22×C4—81no—
80C22×C4—82no—
81C22×C4—82no—
82C22×C4—82no—
83C22×C4#9911yescharacteristic
84C22×C4C2×C22⋊C411yescharacteristic
85M16—48no—
86C2×C8—24no—
87C2×C8—42no—
88C2×C8—42no—
89C2×C8—84no—
90C4⋊C4—42no—
91C4⋊C4—42no—
92C4⋊C4—44no—
93C4⋊C4—82no—
94C4⋊C4—82no—
95C4⋊C4—82no—
96C22⋊C4—82no—
97C42—24no—
98C42—21no—
99C42—21no—
100C42—42no—
101C42—42no—
102C42#10112yes—
103C42C2×M1612yes—
104C23—21no—
105C23—21no—
106C23—41no—
107C23—41no—
108C23#35512yes—
109C23C2×#9911yescharacteristic
110C2×C4—24no—
111C2×C4—24no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—84no—
125C2×C4—82no—
126C2×C4—82no—
127C2×C4—82no—
128C2×C4—82no—
129C2×C4—82no—
130C2×C4—164no—
131C2×C4#35412yes—
132C2×C4#40711yescharacteristic
133C2×C4C2×#10111yescharacteristic
134C8—44no—
135C8—164no—
136C22—21no—
137C22—21no—
138C22—22no—
139C22—42no—
140C22—42no—
141C22—41no—
142C22—41no—
143C22#146311yescharacteristic
144C22C2×#35411yescharacteristic
145C22C2×#35511yescharacteristic
146C4—21no—
147C4—22no—
148C4—21no—
149C4—42no—
150C4—42no—
151C4—82no—
152C4—82no—
153C4—162no—
154C2—21no—
155C2—21no—
156C2—42no—
157C2#1151111yescharacteristic
158C2#1359611yescharacteristic
159C2#1360011yescharacteristic
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#384C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7367C2semidirect

Maximal subgroups

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

subgroupindex
#207652
#115132
#73672