Finite Group Databasebeta

← #93230

Subgroups — #93230

Order 512.

200 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 26 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#10036C211yesmaximal, characteristic
2#7283C211yesmaximal, characteristic
3#7060C211yesmaximal, characteristic
4#3162C411yescharacteristic, complemented
5#3101—21no—
6#1955C2211yescharacteristic
7C4×#99—21no—
8#1481—21no—
9#1322—21no—
10#1275C411yescharacteristic, complemented
11#533—21no—
12C2×C4×D4D411yescharacteristic
13#501—41no—
14#501—41no—
15#483—41no—
16#476—41no—
17#461—41no—
18#450—41no—
19C4×SD16—42no—
20#431—21no—
21#427—41no—
22C2×C4⋊C8D411yescharacteristic
23C2×#99—41no—
24C2×C22⋊C8—41no—
25C2×#97C2×C411yescharacteristic
26C4×C22⋊C4—41no—
27C22⋊C16—41no—
28#342—22no—
29Q8⋊C8—42no—
30C22×D4D811yescharacteristic
31C23×C4—21no—
32C2×SD16—42no—
33C22×C8—21no—
34C22×C8—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37#125—41no—
38C22⋊Q8—41no—
39C22⋊Q8—41no—
40C4×Q8—41no—
41C4×Q8—41no—
42C4×D4—41no—
43C4×D4—41no—
44#117—41no—
45C2×C4⋊C4SD1611yescharacteristic
46C2×C22⋊C4—21no—
47C2×C22⋊C4—42no—
48C2×C42—81no—
49C2×C42C22⋊C411yescharacteristic
50C2×C16—81no—
51#107—41no—
52#107—41no—
53#106—42no—
54#106—42no—
55C4⋊C8—41no—
56C4⋊C8—41no—
57#103—42no—
58#102—42no—
59#99—44no—
60C22⋊C8—41no—
61C22⋊C8—82no—
62#97—41no—
63C4×C8—22no—
64#95—81no—
65C24—21no—
66C2×Q8—41no—
67C2×Q8—41no—
68C2×D4—41no—
69C2×D4—42no—
70C22×C4—21no—
71C22×C4—41no—
72C22×C4—41no—
73C22×C4—81no—
74C22×C4—81no—
75C22×C4—82no—
76C22×C4#9911yescharacteristic
77C22×C4#10211yescharacteristic
78C22×C4#10411yescharacteristic
79SD16—84no—
80C2×C8—22no—
81C2×C8—41no—
82C2×C8—41no—
83C2×C8—42no—
84C2×C8—41no—
85C2×C8—81no—
86C4⋊C4—41no—
87C4⋊C4—41no—
88C4⋊C4—41no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C4⋊C4—81no—
92C4⋊C4—81no—
93C22⋊C4—44no—
94C22⋊C4—81no—
95C22⋊C4—82no—
96C22⋊C4—81no—
97C42—21no—
98C42—21no—
99C42—81no—
100C42—161no—
101C42—161no—
102C16—161no—
103C23—22no—
104C23—41no—
105C23—41no—
106C23—81no—
107C23—81no—
108C23#32711yescharacteristic
109Q8—82no—
110Q8—82no—
111D4—42no—
112D4—81no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—81no—
121C2×C4—81no—
122C2×C4—81no—
123C2×C4—84no—
124C2×C4—81no—
125C2×C4—81no—
126C2×C4—161no—
127C2×C4—162no—
128C2×C4—162no—
129C2×C4AΣL(1,𝔽16)11yescharacteristic
130C2×C4#35311yescharacteristic
131C8—42no—
132C8—82no—
133C8—81no—
134C22—21no—
135C22—21no—
136C22—21no—
137C22—42no—
138C22—82no—
139C22—81no—
140C22—81no—
141C22—161no—
142C22#134311yescharacteristic
143C4—21no—
144C4—41no—
145C4—41no—
146C4—81no—
147C4—164no—
148C4—161no—
149C4—161no—
150C4#140711yescharacteristic
151C4#140911yescharacteristic
152C2—41no—
153C2—82no—
154C2#739611yescharacteristic
155C2#740311yescharacteristic
156C2#740611yescharacteristic
subgroupG/H
centerC2×C4#353
commutatorC2×#97C2×C4
Frattini, cosocle#1955C22
socleC22#1343
prime radicalC22×D4D8
LCSG ▷ C2×#97 ▷ C2×C42 ▷ C23 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1955 ◁ G
LECSG ▷ #1955 ▷ C2×C42 ▷ C23 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #3162C4semidirect
10. #1275C4semidirect

Maximal subgroups

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

subgroupindex
#100362
#72832
#70602