Finite Group Databasebeta

← #94531

Subgroups — #94531

Order 512.

205 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 205 conjugacy classes of subgroups — too many to draw. Only its 24 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#8540C211yesmaximal, characteristic
2#7351C212yesmaximal, complemented
3#3491C411yescharacteristic
4#1822C2211yescharacteristic
5#1818C411yescharacteristic
6#1468—22no—
7#1320—22no—
8#552—42no—
9#551—42no—
10#543C811yescharacteristic
11#539—41no—
12#534C811yescharacteristic
13#529—41no—
14C2×#123—21no—
15#518—21no—
16C2×#117C2×C411yescharacteristic
17#407D412yes—
18C2×C22⋊C8—21no—
19#349—42no—
20#343—41no—
21#336—41no—
22#324—42no—
23#323—42no—
24C22×D4—21no—
25C23×C4C22⋊C411yescharacteristic
26C2×M16—22no—
27C22×C8—41no—
28#126—42no—
29#126—42no—
30#125—42no—
31#125—41no—
32#124—41no—
33#123—42no—
34#123—41no—
35#123—41no—
36#123—41no—
37#123—42no—
38C22⋊Q8—42no—
39C22⋊Q8—42no—
40C22⋊Q8—41no—
41#121—41no—
42#120—41no—
43C4×Q8—41no—
44C4×D4—42no—
45C4×D4—42no—
46C4×D4—41no—
47#117—41no—
48#117—42no—
49#117—42no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—21no—
52C2×C22⋊C4—22no—
53C2×C22⋊C4—21no—
54C2×C22⋊C4—41no—
55C2×C22⋊C4C2×C811yescharacteristic
56C2×C22⋊C4M1611yescharacteristic
57C2×C42—21no—
58M32—82no—
59C22⋊C8—42no—
60C22⋊C8—42no—
61C24—21no—
62C24C22⋊C811yescharacteristic
63C2×Q8—41no—
64C2×D4—41no—
65C2×D4—41no—
66C2×D4—41no—
67C2×D4—82no—
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—42no—
74C22×C4—81no—
75C22×C4#9911yescharacteristic
76C22×C4#10011yescharacteristic
77M16—82no—
78C2×C8—42no—
79C2×C8—81no—
80C2×C8—82no—
81C4⋊C4—42no—
82C4⋊C4—42no—
83C4⋊C4—41no—
84C4⋊C4—42no—
85C4⋊C4—41no—
86C4⋊C4—42no—
87C4⋊C4—42no—
88C22⋊C4—42no—
89C22⋊C4—41no—
90C22⋊C4—42no—
91C22⋊C4—42no—
92C22⋊C4—41no—
93C22⋊C4—42no—
94C22⋊C4—41no—
95C22⋊C4—41no—
96C42—41no—
97C42—41no—
98C42—42no—
99C16—162no—
100C23—21no—
101C23—41no—
102C23—41no—
103C23—81no—
104C23—81no—
105C23—82no—
106C23#32311yescharacteristic
107Q8—161no—
108D4—82no—
109D4—82no—
110D4—161no—
111C2×C4—21no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—42no—
118C2×C4—82no—
119C2×C4—81no—
120C2×C4—82no—
121C2×C4—81no—
122C2×C4—81no—
123C2×C4—162no—
124C2×C4AΣL(1,𝔽16)12yes—
125C8—82no—
126C8—161no—
127C22—21no—
128C22—41no—
129C22—42no—
130C22—41no—
131C22—82no—
132C22—81no—
133C22—161no—
134C22—321no—
135C22#131611yescharacteristic
136C4—22no—
137C4—41no—
138C4—81no—
139C4—82no—
140C4—81no—
141C4—81no—
142C2—21no—
143C2—41no—
144C2—81no—
145C2—161no—
146C2#733911yescharacteristic
subgroupG/H
center, socleC2#7339
commutatorC2×C22⋊C4C2×C8
Frattini, cosocle#1822C22
prime radical#534C8
LCSG ▷ C2×C22⋊C4 ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #1822 ◁ G
LECSG ▷ #1822 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #7351C2semidirect

Maximal subgroups

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

subgroupindex
#85402
#73512