Finite Group Databasebeta

← #93209

Subgroups — #93209

Order 512.

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

normal subgroups only

This group has 176 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#9977C211yesmaximal, characteristic
2#7325C211yesmaximal, characteristic, complemented
3#7059C211yesmaximal, characteristic, complemented
4#3186C411yescharacteristic
5#3094—21no—
6#1818C2211yescharacteristic
7#1471—21no—
8#1464—21no—
9#1314—21no—
10#1293C411yescharacteristic
11C2×#121—21no—
12C2×#117D411yescharacteristic
13#482—42no—
14#481—41no—
15#480—41no—
16#480—41no—
17#466—41no—
18#447—41no—
19#432—41no—
20#428—41no—
21#418—41no—
22#407—21no—
23C2×C22⋊C8C2×C411yescharacteristic
24C2×C22⋊C8D411yescharacteristic
25C22⋊C16—41no—
26#336—41no—
27#324—41no—
28#323—22no—
29#323—42no—
30C22×D4—21no—
31C22×D4—41no—
32C23×C4C22⋊C411yescharacteristic
33C2×D8—41no—
34C2×M16—41no—
35C22×C8—21no—
36C22×C8—21no—
37#121—41no—
38#121—41no—
39#121—41no—
40#121—41no—
41#117—41no—
42#117—41no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4D811yescharacteristic
45C2×C4⋊C4SD1611yescharacteristic
46C2×C22⋊C4—21no—
47C2×C22⋊C4—41no—
48C2×C42—21no—
49C2×C16—81no—
50#107—42no—
51#107—41no—
52#106—42no—
53C4⋊C8—81no—
54#102—42no—
55#102—42no—
56#102—42no—
57C22⋊C8—22no—
58C22⋊C8—41no—
59C22⋊C8—41no—
60C22⋊C8—41no—
61C22⋊C8—42no—
62#97—81no—
63C24—41no—
64C24#10411yescharacteristic
65C2×D4—41no—
66C2×D4—41no—
67C2×D4—82no—
68C2×D4—81no—
69C2×D4—81no—
70C2×D4—162no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—41no—
76C22×C4—81no—
77C22×C4#9911yescharacteristic
78C22×C4#10211yescharacteristic
79D8—161no—
80M16—82no—
81C2×C8—42no—
82C2×C8—41no—
83C2×C8—41no—
84C2×C8—81no—
85C2×C8—81no—
86C2×C8—82no—
87C4⋊C4—41no—
88C4⋊C4—41no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C22⋊C4—81no—
92C22⋊C4—81no—
93C22⋊C4—81no—
94C42—41no—
95C42—41no—
96C16—161no—
97C23—21no—
98C23—41no—
99C23—41no—
100C23—41no—
101C23—82no—
102C23—81no—
103C23—161no—
104C23—161no—
105C23#32711yescharacteristic
106D4—82no—
107D4—82no—
108D4—162no—
109D4—162no—
110C2×C4—21no—
111C2×C4—41no—
112C2×C4—42no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—81no—
117C2×C4—81no—
118C2×C4—81no—
119C2×C4—81no—
120C2×C4—161no—
121C2×C4AΣL(1,𝔽16)11yescharacteristic
122C2×C4#35311yescharacteristic
123C8—81no—
124C8—81no—
125C8—82no—
126C8—162no—
127C22—21no—
128C22—21no—
129C22—21no—
130C22—41no—
131C22—41no—
132C22—41no—
133C22—161no—
134C22—161no—
135C22—161no—
136C22—161no—
137C22—322no—
138C22#134311yescharacteristic
139C4—21no—
140C4—21no—
141C4—41no—
142C4—81no—
143C4—81no—
144C4—161no—
145C2—41no—
146C2—81no—
147C2—321no—
148C2#739511yescharacteristic
149C2#739811yescharacteristic
150C2#740711yescharacteristic
subgroupG/H
center, socleC22#1343
commutatorC2×C22⋊C8C2×C4
Frattini, cosocle#1818C22
prime radical#3186C4
LCSG ▷ C2×C22⋊C8 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #1818 ◁ G
LECSG ▷ #1818 ▷ C23×C4 ▷ C23 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #7325C2semidirect
3. #7059C2semidirect

Maximal subgroups

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

subgroupindex
#99772
#73252
#70592