Finite Group Databasebeta

← #133939

Subgroups — #133939

Order 512.

718 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 82 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#35674C211yesmaximal, characteristic
2#12862C214yesmaximal
3#8551C212yesmaximal
4C22×C4⋊Q8C2211yescharacteristic
5C2×#519C411yescharacteristic
6#2484—24no—
7#2446—28no—
8#2310C411yescharacteristic
9#2310C2212yes—
10#2296C412yes—
11#1784C2214yes—
12#1463—24no—
13#1394—28no—
14C23×Q8C2×C411yescharacteristic
15C23×Q8C2311yescharacteristic
16C2×C4⋊Q8—24no—
17C2×C4⋊Q8—44no—
18#519—44no—
19C2×C4×Q8—24no—
20C2×#117—22no—
21C2×#117C2×C412yes—
22C2×#117C2×C412yes—
23C22×C4⋊C4D414yes—
24C22×C42C2×C411yescharacteristic
25C2×#101—28no—
26#407—24no—
27#400—28no—
28#398—28no—
29#391—24no—
30#391—24no—
31#391D418yes—
32#389—24no—
33#387—216no—
34#385—24no—
35#381—22no—
36#381—24no—
37C4×C4⋊C4—24no—
38#324—48no—
39C22×Q8—24no—
40C22×Q8—22no—
41C22×Q8—42no—
42C22×Q8—44no—
43C22×Q8—44no—
44C22×Q8C2×D414yes—
45C23×C4C22⋊C412yes—
46C23×C4C22⋊C412yes—
47C23×C4C22×C411yescharacteristic
48C23×C4C2×D412yes—
49C2×M16—44no—
50C4⋊Q8—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⋊C4—28no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—28no—
61C2×C4⋊C4—44no—
62C2×C4⋊C4C22⋊C414yes—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C42—24no—
66C2×C42—22no—
67C2×C42—22no—
68C2×C42—42no—
69C2×C42C22⋊C414yes—
70#101—416no—
71C22⋊C8—48no—
72#95—28no—
73#95—28no—
74#95—216no—
75C24C2×C22⋊C411yescharacteristic
76C2×Q8—28no—
77C2×Q8—28no—
78C2×Q8—44no—
79C2×Q8—44no—
80C2×Q8—416no—
81C2×Q8—44no—
82C2×Q8—44no—
83C2×Q8—88no—
84C22×C4—22no—
85C22×C4—24no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—24no—
89C22×C4—22no—
90C22×C4—24no—
91C22×C4—24no—
92C22×C4—24no—
93C22×C4—44no—
94C22×C4—44no—
95C22×C4—44no—
96C22×C4#9914yes—
97C22×C4C2×C22⋊C412yes—
98C22×C4#12014yes—
99M16—88no—
100C2×C8—84no—
101C4⋊C4—416no—
102C4⋊C4—44no—
103C4⋊C4—416no—
104C4⋊C4—48no—
105C4⋊C4—44no—
106C22⋊C4—44no—
107C22⋊C4—44no—
108C42—44no—
109C42—44no—
110C42—44no—
111C42—48no—
112C23—22no—
113C23—42no—
114C23#37911yescharacteristic
115C23C2×#9912yes—
116Q8—416no—
117Q8—88no—
118C2×C4—28no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—24no—
122C2×C4—28no—
123C2×C4—24no—
124C2×C4—24no—
125C2×C4—28no—
126C2×C4—42no—
127C2×C4—44no—
128C2×C4—44no—
129C2×C4—42no—
130C2×C4—44no—
131C2×C4—44no—
132C2×C4—48no—
133C2×C4—48no—
134C2×C4—44no—
135C2×C4—44no—
136C2×C4—84no—
137C2×C4#41014yes—
138C8—164no—
139C22—22no—
140C22—22no—
141C22—24no—
142C22—44no—
143C22#178111yescharacteristic
144C22#178212yes—
145C4—24no—
146C4—44no—
147C4—44no—
148C4—44no—
149C4—84no—
150C4—84no—
151C2—22no—
152C2—42no—
153C2#854311yescharacteristic
154C2#1271012yes—
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC23×Q8C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊Q8 ◁ G
LECSG ▷ C23×Q8 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#356742
#128622
#85512