Finite Group Databasebeta

← #149802

Subgroups — #149802

Order 512.

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

normal subgroups only

This group has 312 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#20760C211yesmaximal, characteristic
2#11513C212yesmaximal
3#7366C214yesmaximal, complemented
4C22×C4×C8C411yescharacteristic
5#2285C411yescharacteristic
6#2285C2211yescharacteristic
7#1835C412yes—
8#1835C2212yes—
9#1834—22no—
10#1735—22no—
11#1464—22no—
12#1325—22no—
13#1324C2214yes—
14C23×C8—21no—
15C2×#117—21no—
16C2×#117D412yes—
17C22×C42C2×C411yescharacteristic
18#432—42no—
19C2×C4×C8—21no—
20C2×C4×C8—24no—
21C2×C4×C8C2×C412yes—
22#384—21no—
23#384D412yes—
24#382C2×C412yes—
25#382C2×C411yescharacteristic
26#382C2311yescharacteristic
27#380—41no—
28#380—41no—
29C4×C4⋊C4—41no—
30C4×C4⋊C4—41no—
31#340—24no—
32#340—24no—
33#324—24no—
34#323—42no—
35C23×C4—21no—
36C23×C4C22⋊C411yescharacteristic
37C22×C8—24no—
38C22×C8—22no—
39C22×C8—42no—
40C22×C8—42no—
41#117—42no—
42#117—42no—
43#117—44no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—42no—
46C2×C4⋊C4—42no—
47C2×C4⋊C4C22⋊C412yes—
48C2×C4⋊C4C2×D412yes—
49C2×C22⋊C4—22no—
50C2×C22⋊C4—41no—
51C2×C42—21no—
52C2×C42—22no—
53C2×C42—22no—
54C2×C42—22no—
55C2×C42—41no—
56C2×C42C22⋊C411yescharacteristic
57C2×C42C22×C411yescharacteristic
58C4⋊C8—82no—
59C22⋊C8—44no—
60#97—82no—
61C4×C8—24no—
62C4×C8—22no—
63C4×C8M1614yes—
64#95—41no—
65#95—41no—
66#95—42no—
67C24#9911yescharacteristic
68C22×C4—22no—
69C22×C4—21no—
70C22×C4—22no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—24no—
74C22×C4—21no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—44no—
78C22×C4—42no—
79C22×C4#9911yescharacteristic
80C22×C4C2×C22⋊C411yescharacteristic
81C2×C8—24no—
82C2×C8—48no—
83C2×C8—42no—
84C2×C8—42no—
85C2×C8—84no—
86C4⋊C4—42no—
87C4⋊C4—42no—
88C4⋊C4—44no—
89C4⋊C4—84no—
90C4⋊C4—84no—
91C22⋊C4—82no—
92C22⋊C4—82no—
93C42—24no—
94C42—21no—
95C42—21no—
96C42—42no—
97C42—42no—
98C42—82no—
99C42#10112yes—
100C42C2×M1612yes—
101C23—21no—
102C23—21no—
103C23—42no—
104C23#35512yes—
105C23C2×#9911yescharacteristic
106C2×C4—24no—
107C2×C4—24no—
108C2×C4—24no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—48no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—82no—
120C2×C4—82no—
121C2×C4—84no—
122C2×C4—82no—
123C2×C4#35412yes—
124C2×C4#40711yescharacteristic
125C2×C4C2×#10111yescharacteristic
126C8—44no—
127C8—164no—
128C22—22no—
129C22—22no—
130C22—42no—
131C22—42no—
132C22—42no—
133C22#146311yescharacteristic
134C22C2×#35411yescharacteristic
135C22C2×#35511yescharacteristic
136C4—24no—
137C4—21no—
138C4—22no—
139C4—21no—
140C4—42no—
141C4—82no—
142C4—82no—
143C4—162no—
144C2—21no—
145C2—21no—
146C2—42no—
147C2#1151111yescharacteristic
148C2#1360212yes—
subgroupG/H
center, socleC22C2×#354
commutatorC2×C42C22×C4
Frattini, cosocle#382C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G
LECSG ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7366C2semidirect

Maximal subgroups

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

subgroupindex
#207602
#115132
#73662