Finite Group Databasebeta

← #134570

Subgroups — #134570

Order 512.

309 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 80 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#21657C211yesmaximal, characteristic
2#12659C212yesmaximal
3C2×#1356C212yesmaximal
4#7266C212yesmaximal
5C2×#491C2211yescharacteristic
6C4×C4⋊Q8C411yescharacteristic
7C4×#125C411yescharacteristic
8C2×#364—22no—
9C2×#347C2214yes—
10#1992C411yescharacteristic
11#1960C411yescharacteristic
12#1933—22no—
13#1836C2211yescharacteristic
14#1763—22no—
15C4×#97C2211yescharacteristic
16#1377—22no—
17#1356—24no—
18C2×C4⋊Q8C2×C411yescharacteristic
19C2×#125C2×C411yescharacteristic
20C2×C4×Q8—21no—
21C2×M32—24no—
22#491—22no—
23#491D412yes—
24#491Q812yes—
25C2×#103—22no—
26C2×#103—22no—
27C2×#97C2×C411yescharacteristic
28C2×#97D412yes—
29C2×#97C2311yescharacteristic
30C2×C4×C8D412yes—
31#391—21no—
32#389—21no—
33#389—41no—
34#387—41no—
35#384C2×C411yescharacteristic
36#382C2×C411yescharacteristic
37C4×C4⋊C4—21no—
38C4×C4⋊C4—41no—
39C4×C4⋊C4—41no—
40C43C2×C411yescharacteristic
41#364—28no—
42#347—24no—
43#347—24no—
44#340—22no—
45#340—22no—
46C22×Q8—21no—
47C22×C8C22⋊C412yes—
48C22×C8C2×D412yes—
49C4⋊Q8—21no—
50C4⋊Q8—41no—
51C4⋊Q8C4⋊C412yes—
52#125—21no—
53#125—41no—
54#125C4⋊C412yes—
55C4×Q8—44no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—41no—
60C2×C4⋊C4—41no—
61C2×C4⋊C4—41no—
62C2×C4⋊C4—41no—
63C2×C4⋊C4—41no—
64C2×C42—21no—
65C2×C42—21no—
66C2×C42—41no—
67C2×C42—41no—
68C2×C42C22⋊C411yescharacteristic
69C2×C42C22⋊C411yescharacteristic
70C2×C42C22×C411yescharacteristic
71M32—28no—
72M32—44no—
73C2×C16—44no—
74#103—24no—
75#103—24no—
76#103—44no—
77#97—41no—
78#97—42no—
79#97C2×D411yescharacteristic
80#97C2×Q811yescharacteristic
81#97#4112yes—
82C4×C8—24no—
83#95—41no—
84#95—41no—
85#95—42no—
86#95—41no—
87C2×Q8—22no—
88C2×Q8—41no—
89C2×Q8—42no—
90C2×Q8—42no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—41no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4#10011yescharacteristic
98C22×C4#10011yescharacteristic
99C22×C4C2×C22⋊C411yescharacteristic
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8#10414yes—
105C2×C8#12112yes—
106C2×C8C22⋊Q812yes—
107C4⋊C4—22no—
108C4⋊C4—42no—
109C4⋊C4—42no—
110C4⋊C4—42no—
111C4⋊C4—42no—
112C4⋊C4—41no—
113C4⋊C4—42no—
114C4⋊C4—42no—
115C4⋊C4—82no—
116C4⋊C4—82no—
117C4⋊C4—82no—
118C42—21no—
119C42—22no—
120C42—22no—
121C42—42no—
122C42—82no—
123C42—81no—
124C42—82no—
125C42C2×C4⋊C411yescharacteristic
126C42#11711yescharacteristic
127C16—48no—
128C23C2×#10011yescharacteristic
129Q8—44no—
130Q8—82no—
131C2×C4—22no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—81no—
145C2×C4—81no—
146C2×C4—81no—
147C2×C4—81no—
148C2×C4—81no—
149C2×C4—81no—
150C2×C4—82no—
151C2×C4—81no—
152C2×C4—81no—
153C2×C4#38011yescharacteristic
154C2×C4#41311yescharacteristic
155C2×C4C2×#10412yes—
156C2×C4#42112yes—
157C8—22no—
158C8—22no—
159C8—24no—
160C22—21no—
161C22—21no—
162C22#152511yescharacteristic
163C22#153311yescharacteristic
164C22#180011yescharacteristic
165C4—41no—
166C4—41no—
167C4—42no—
168C4—41no—
169C4—41no—
170C4—82no—
171C4—82no—
172C4—161no—
173C4—161no—
174C4#153212yes—
175C4#180612yes—
176C2—21no—
177C2—21no—
178C2#858311yescharacteristic
179C2#1191611yescharacteristic
180C2#1191711yescharacteristic
subgroupG/H
center, socleC22#1533
commutatorC2×C42C22×C4
Frattini, cosocleC2×#97C23
prime radicalC23C2×#100
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C2×#97 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#216572
#126592
C2×#13562
#72662