Finite Group Databasebeta

← #133965

Subgroups — #133965

Order 512.

874 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C2×#2861C211yesmaximal, characteristic
2#12860C214yesmaximal, complemented
3#8552C212yesmaximal
4C22×#125C411yescharacteristic
5C22×#124C2211yescharacteristic
6#2861—28no—
7C2×#391C2212yes—
8C2×#390C412yes—
9C2×#388—24no—
10C2×C43C411yescharacteristic
11#1784C2214yes—
12#1462—24no—
13#1394—28no—
14C23×Q8C2311yescharacteristic
15C23×D4C2×C411yescharacteristic
16C2×#125—24no—
17C2×#125—44no—
18C2×#124—24no—
19C2×#124—44no—
20C22×C4⋊C4—22no—
21C22×C4⋊C4C2×C412yes—
22C22×C22⋊C4—24no—
23C22×C22⋊C4D414yes—
24C22×C42—22no—
25C22×C42C2×C411yescharacteristic
26C22×C42C2×C412yes—
27C2×#101—28no—
28#407—24no—
29#391—24no—
30#391D418yes—
31#390—24no—
32#390—24no—
33#388—232no—
34C2×#95—24no—
35C43—24no—
36#323—48no—
37C25C22⋊C412yes—
38C22×Q8—42no—
39C22×Q8—44no—
40C22×Q8C2×D414yes—
41C22×D4—22no—
42C22×D4—42no—
43C22×D4—44no—
44C23×C4—22no—
45C23×C4—22no—
46C23×C4C22⋊C412yes—
47C23×C4C22×C411yescharacteristic
48C23×C4C2×D412yes—
49C2×M16—44no—
50#125—416no—
51#124—416no—
52C2×C4⋊C4—28no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—42no—
55C2×C4⋊C4—42no—
56C2×C4⋊C4—44no—
57C2×C4⋊C4—44no—
58C2×C4⋊C4C22⋊C414yes—
59C2×C22⋊C4—216no—
60C2×C22⋊C4—28no—
61C2×C22⋊C4—28no—
62C2×C22⋊C4—44no—
63C2×C22⋊C4—416no—
64C2×C42—24no—
65C2×C42—22no—
66C2×C42—28no—
67C2×C42—22no—
68C2×C42—48no—
69C2×C42—44no—
70C2×C42C22⋊C414yes—
71#101—416no—
72C22⋊C8—48no—
73#95—216no—
74C24—22no—
75C24—24no—
76C24—48no—
77C24—44no—
78C24C2×C22⋊C411yescharacteristic
79C2×Q8—28no—
80C2×Q8—44no—
81C2×Q8—44no—
82C2×Q8—88no—
83C2×D4—44no—
84C2×D4—44no—
85C2×D4—44no—
86C2×D4—88no—
87C22×C4—22no—
88C22×C4—24no—
89C22×C4—28no—
90C22×C4—24no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—24no—
94C22×C4—24no—
95C22×C4—44no—
96C22×C4—44no—
97C22×C4—44no—
98C22×C4—44no—
99C22×C4—48no—
100C22×C4—42no—
101C22×C4—42no—
102C22×C4#9914yes—
103C22×C4C2×C22⋊C412yes—
104C22×C4#12014yes—
105M16—88no—
106C2×C8—84no—
107C4⋊C4—44no—
108C4⋊C4—44no—
109C4⋊C4—416no—
110C22⋊C4—416no—
111C22⋊C4—432no—
112C42—28no—
113C42—44no—
114C42—416no—
115C42—44no—
116C23—24no—
117C23—22no—
118C23—48no—
119C23—42no—
120C23—44no—
121C23—48no—
122C23—816no—
123C23—84no—
124C23—84no—
125C23#37911yescharacteristic
126C23C2×#9912yes—
127Q8—88no—
128D4—88no—
129C2×C4—28no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—28no—
133C2×C4—24no—
134C2×C4—24no—
135C2×C4—28no—
136C2×C4—44no—
137C2×C4—44no—
138C2×C4—416no—
139C2×C4—44no—
140C2×C4—44no—
141C2×C4—44no—
142C2×C4—48no—
143C2×C4—44no—
144C2×C4—84no—
145C2×C4—84no—
146C2×C4—84no—
147C2×C4#41014yes—
148C8—164no—
149C22—28no—
150C22—22no—
151C22—22no—
152C22—22no—
153C22—88no—
154C22—82no—
155C22—84no—
156C22—168no—
157C22#178111yescharacteristic
158C22#178212yes—
159C22#213214yes—
160C4—24no—
161C4—48no—
162C4—44no—
163C4—44no—
164C4—84no—
165C2—22no—
166C2—24no—
167C2—162no—
168C2#854411yescharacteristic
169C2#1270112yes—
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC23×Q8C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×#124 ◁ G
LECSG ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12860C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28612
#128602
#85522