Finite Group Databasebeta

← #149786

Subgroups — #149786

Order 512.

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

normal subgroups only

This group has 366 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#21325C211yesmaximal, characteristic
2#11513C212yesmaximal
3#7358C214yesmaximal, complemented
4C22×#125C411yescharacteristic
5C2×C4×M16C411yescharacteristic
6#2285C2211yescharacteristic
7#1992C412yes—
8#1992C2212yes—
9#1991—22no—
10#1736—22no—
11#1464—22no—
12#1325C2214yes—
13#1324—22no—
14C22×M16—21no—
15C2×#125—24no—
16C2×#125—21no—
17C2×#125—42no—
18C2×#125—41no—
19C2×#125C2×C412yes—
20C2×#117D412yes—
21C22×C4⋊C4—21no—
22C22×C4⋊C4—41no—
23C22×C42C2×C411yescharacteristic
24#432—42no—
25C4×M16—42no—
26C2×#97C2×C412yes—
27C2×C4×C8—21no—
28#384C2×C411yescharacteristic
29#384C2311yescharacteristic
30#382D412yes—
31#380—41no—
32C4×C4⋊C4—41no—
33#340—24no—
34#340—24no—
35#324—24no—
36#323—42no—
37C23×C4—21no—
38C23×C4—41no—
39C23×C4C22⋊C411yescharacteristic
40C2×M16—24no—
41C2×M16—42no—
42C2×M16—42no—
43C22×C8—22no—
44#125—24no—
45#125—22no—
46#125—42no—
47#125—48no—
48#125—42no—
49#125M1614yes—
50#117—42no—
51#117—42no—
52C2×C4⋊C4—24no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—42no—
55C2×C4⋊C4—42no—
56C2×C4⋊C4—48no—
57C2×C4⋊C4—42no—
58C2×C4⋊C4—41no—
59C2×C4⋊C4—41no—
60C2×C4⋊C4—41no—
61C2×C4⋊C4C22⋊C412yes—
62C2×C4⋊C4C2×D412yes—
63C2×C22⋊C4—22no—
64C2×C42—21no—
65C2×C42—22no—
66C2×C42—22no—
67C2×C42—22no—
68C2×C42C22⋊C411yescharacteristic
69C2×C42C22×C411yescharacteristic
70C4⋊C8—82no—
71C22⋊C8—44no—
72#97—42no—
73#97—82no—
74C4×C8—42no—
75#95—42no—
76C24#9911yescharacteristic
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—24no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—44no—
87C22×C4—42no—
88C22×C4—81no—
89C22×C4—81no—
90C22×C4—82no—
91C22×C4—82no—
92C22×C4#9911yescharacteristic
93C22×C4C2×C22⋊C411yescharacteristic
94M16—48no—
95C2×C8—24no—
96C2×C8—42no—
97C2×C8—42no—
98C2×C8—84no—
99C4⋊C4—24no—
100C4⋊C4—44no—
101C4⋊C4—42no—
102C4⋊C4—42no—
103C4⋊C4—44no—
104C4⋊C4—48no—
105C4⋊C4—44no—
106C4⋊C4—42no—
107C4⋊C4—82no—
108C4⋊C4—82no—
109C22⋊C4—82no—
110C42—24no—
111C42—21no—
112C42—21no—
113C42—42no—
114C42—42no—
115C42#10012yes—
116C42C2×M1612yes—
117C23—21no—
118C23—21no—
119C23—41no—
120C23—41no—
121C23#35612yes—
122C23C2×#9911yescharacteristic
123C2×C4—24no—
124C2×C4—24no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—44no—
138C2×C4—84no—
139C2×C4—82no—
140C2×C4—82no—
141C2×C4—88no—
142C2×C4—82no—
143C2×C4—82no—
144C2×C4#35412yes—
145C2×C4#40711yescharacteristic
146C2×C4C2×#10011yescharacteristic
147C8—44no—
148C8—164no—
149C22—21no—
150C22—21no—
151C22—22no—
152C22—42no—
153C22—42no—
154C22—41no—
155C22—41no—
156C22#146211yescharacteristic
157C22C2×#35411yescharacteristic
158C22C2×#35611yescharacteristic
159C4—21no—
160C4—22no—
161C4—21no—
162C4—42no—
163C4—42no—
164C4—84no—
165C4—82no—
166C4—82no—
167C2—21no—
168C2—21no—
169C2—42no—
170C2#1150611yescharacteristic
171C2#1359611yescharacteristic
172C2#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. #7358C2semidirect

Maximal subgroups

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

subgroupindex
#213252
#115132
#73582