Finite Group Databasebeta

← #149814

Subgroups — #149814

Order 512.

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

normal subgroups only

This group has 372 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#20800C211yesmaximal, characteristic
2#11513C212yesmaximal
3#7371C214yesmaximal, complemented
4C2×#428—22no—
5C2×C4×M16C411yescharacteristic
6#2285C2211yescharacteristic
7C2×#384C411yescharacteristic
8#1847—22no—
9#1845C412yes—
10#1844C2212yes—
11#1464—22no—
12#1325C2214yes—
13#1324—22no—
14C22×M16—21no—
15C2×#117D412yes—
16C22×C4⋊C4—21no—
17C22×C4⋊C4—41no—
18C22×C42C2×C411yescharacteristic
19#432—42no—
20#428—44no—
21#428—44no—
22C2×#107—24no—
23C2×#106—24no—
24C4×M16—24no—
25C2×#97—21no—
26C2×C4×C8C2×C412yes—
27#384—21no—
28#384—41no—
29#384C2×C412yes—
30#384C2×C411yescharacteristic
31#384C2311yescharacteristic
32#382D412yes—
33#380—41no—
34C4×C4⋊C4—41no—
35#324—24no—
36#323—42no—
37C23×C4—21no—
38C23×C4—41no—
39C23×C4C22⋊C411yescharacteristic
40C2×M16—24no—
41C2×M16—24no—
42C2×M16—24no—
43C22×C8—22no—
44#117—42no—
45#117—42no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—42no—
48C2×C4⋊C4—42no—
49C2×C4⋊C4—41no—
50C2×C4⋊C4—41no—
51C2×C4⋊C4—42no—
52C2×C4⋊C4—42no—
53C2×C4⋊C4—41no—
54C2×C4⋊C4—84no—
55C2×C4⋊C4C22⋊C412yes—
56C2×C4⋊C4C2×D412yes—
57C2×C22⋊C4—22no—
58C2×C42—21no—
59C2×C42—22no—
60C2×C42—22no—
61C2×C42—22no—
62C2×C42C22⋊C411yescharacteristic
63C2×C42C22×C411yescharacteristic
64#107—48no—
65#106—48no—
66C4⋊C8—82no—
67C22⋊C8—44no—
68#97—24no—
69#97—82no—
70C4×C8—22no—
71C4×C8M1614yes—
72#95—42no—
73C24#9911yescharacteristic
74C22×C4—22no—
75C22×C4—21no—
76C22×C4—22no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—24no—
80C22×C4—21no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4—42no—
86C22×C4—81no—
87C22×C4—81no—
88C22×C4—84no—
89C22×C4#9911yescharacteristic
90C22×C4C2×C22⋊C411yescharacteristic
91M16—216no—
92C2×C8—24no—
93C2×C8—24no—
94C2×C8—24no—
95C2×C8—84no—
96C4⋊C4—42no—
97C4⋊C4—42no—
98C4⋊C4—44no—
99C4⋊C4—82no—
100C4⋊C4—82no—
101C4⋊C4—84no—
102C4⋊C4—84no—
103C4⋊C4—82no—
104C22⋊C4—82no—
105C42—24no—
106C42—21no—
107C42—21no—
108C42—42no—
109C42—42no—
110C42#10112yes—
111C42C2×M1612yes—
112C23—21no—
113C23—21no—
114C23—41no—
115C23—41no—
116C23#35612yes—
117C23C2×#9911yescharacteristic
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—24no—
121C2×C4—24no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—48no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—82no—
134C2×C4—82no—
135C2×C4—84no—
136C2×C4—82no—
137C2×C4—162no—
138C2×C4—162no—
139C2×C4#35312yes—
140C2×C4#40711yescharacteristic
141C2×C4C2×#10111yescharacteristic
142C8—28no—
143C8—164no—
144C22—21no—
145C22—21no—
146C22—22no—
147C22—42no—
148C22—42no—
149C22—41no—
150C22—41no—
151C22#146311yescharacteristic
152C22C2×#35311yescharacteristic
153C22C2×#35611yescharacteristic
154C4—24no—
155C4—21no—
156C4—22no—
157C4—21no—
158C4—42no—
159C4—82no—
160C4—82no—
161C4—162no—
162C2—21no—
163C2—21no—
164C2—42no—
165C2#1151211yescharacteristic
166C2#1358611yescharacteristic
167C2#1359111yescharacteristic
subgroupG/H
center, socleC22C2×#353
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. #7371C2semidirect

Maximal subgroups

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

subgroupindex
#208002
#115132
#73712