Finite Group Databasebeta

← #97482

Subgroups — #97482

Order 512.

762 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 90 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#35679C211yesmaximal, characteristic
2#8305C212yesmaximal
3#7646C214yesmaximal, complemented
4C22×#124C2211yescharacteristic
5C2×#519C411yescharacteristic
6#2482—24no—
7#2448—28no—
8#2310C411yescharacteristic
9#2309C412yes—
10#2296C2212yes—
11#1817C2214yes—
12C2×#323—24no—
13#1271—28no—
14C23×Q8C2×C411yescharacteristic
15C23×D4C2×C411yescharacteristic
16C2×#124—24no—
17C2×#124—44no—
18#519—44no—
19C2×C4×Q8—24no—
20C2×#117—22no—
21C2×#117C2×C412yes—
22C2×#117C2×C412yes—
23C22×C22⋊C4D414yes—
24C22×C42C2311yescharacteristic
25C2×C22⋊C8—24no—
26#399—28no—
27#397—28no—
28#391—24no—
29#390—24no—
30#390—24no—
31#389D418yes—
32#388—216no—
33#385—24no—
34#381—22no—
35#379—24no—
36C4×C4⋊C4—24no—
37#336—28no—
38#323—48no—
39C25C22⋊C412yes—
40C22×Q8—24no—
41C22×Q8—22no—
42C22×Q8—44no—
43C22×D4—22no—
44C22×D4—42no—
45C22×D4—44no—
46C23×C4C22⋊C412yes—
47C23×C4C22×C411yescharacteristic
48C23×C4C2×D412yes—
49C22×C8—44no—
50#124—416no—
51C4×Q8—416no—
52#117—44no—
53#117—44no—
54#117—44no—
55#117—44no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—28no—
58C2×C4⋊C4C22⋊C414yes—
59C2×C4⋊C4C22⋊C414yes—
60C2×C22⋊C4—22no—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—28no—
63C2×C22⋊C4—28no—
64C2×C22⋊C4—216no—
65C2×C22⋊C4—44no—
66C2×C42—24no—
67C2×C42—22no—
68C2×C42—22no—
69C2×C42—42no—
70C2×C42C2×D414yes—
71C22⋊C8—48no—
72#95—28no—
73#95—28no—
74C24—22no—
75C24—24no—
76C24—48no—
77C24—44no—
78C24C2×C22⋊C411yescharacteristic
79C2×Q8—28no—
80C2×Q8—416no—
81C2×Q8—44no—
82C2×Q8—44no—
83C2×D4—44no—
84C2×D4—44no—
85C2×D4—44no—
86C2×D4—88no—
87C22×C4—22no—
88C22×C4—24no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—24no—
92C22×C4—24no—
93C22×C4—24no—
94C22×C4—44no—
95C22×C4—44no—
96C22×C4#9914yes—
97C22×C4C2×C22⋊C412yes—
98C22×C4#12014yes—
99C2×C8—84no—
100C2×C8—88no—
101C4⋊C4—28no—
102C4⋊C4—44no—
103C4⋊C4—48no—
104C4⋊C4—88no—
105C22⋊C4—416no—
106C22⋊C4—44no—
107C22⋊C4—44no—
108C22⋊C4—816no—
109C42—44no—
110C42—44no—
111C42—44no—
112C42—48no—
113C23—24no—
114C23—22no—
115C23—48no—
116C23—48no—
117C23—42no—
118C23—44no—
119C23—84no—
120C23—84no—
121C23—168no—
122C23#37911yescharacteristic
123C23C2×#9912yes—
124Q8—416no—
125D4—88no—
126C2×C4—28no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—28no—
130C2×C4—24no—
131C2×C4—24no—
132C2×C4—44no—
133C2×C4—44no—
134C2×C4—44no—
135C2×C4—48no—
136C2×C4—48no—
137C2×C4—44no—
138C2×C4—44no—
139C2×C4—84no—
140C2×C4#41314yes—
141C8—164no—
142C22—24no—
143C22—22no—
144C22—44no—
145C22—84no—
146C22—88no—
147C22—82no—
148C22—168no—
149C22#178111yescharacteristic
150C22#178512yes—
151C22#179414yes—
152C4—24no—
153C4—44no—
154C4—44no—
155C4—84no—
156C4—84no—
157C2—42no—
158C2—162no—
159C2#846412yes—
160C2#847714yes—
161C2#854811yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7646C2semidirect

Maximal subgroups

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

subgroupindex
#356792
#83052
#76462