Finite Group Databasebeta

← #139689

Subgroups — #139689

Order 512.

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

normal subgroups only

This group has 317 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#33497C211yesmaximal, characteristic
2#28190C211yesmaximal, characteristic
3#11662C212yesmaximal
4#8621C211yesmaximal, characteristic
5#7218C212yesmaximal
6C2×C8×Q8C2211yescharacteristic
7C2×C4×Q16C2211yescharacteristic
8#2581—21no—
9#2183—24no—
10#2163—22no—
11#2162C2212yes—
12#2086C412yes—
13#2056C412yes—
14#1933—22no—
15#1835C2211yescharacteristic
16#1805—21no—
17#1568C2212yes—
18#1467—21no—
19#1332—24no—
20#1290—21no—
21C22×Q16C2×C411yescharacteristic
22C2×C4×Q8—21no—
23C2×C4×Q8D411yescharacteristic
24C8×Q8—22no—
25C8×Q8—22no—
26C4×Q16—22no—
27C4×Q16—22no—
28#432—22no—
29C2×#107C2×C411yescharacteristic
30C2×C4⋊C8C2×C412yes—
31C2×C4⋊C8D411yescharacteristic
32C2×#103—22no—
33C2×#103—21no—
34C2×C4×C8D412yes—
35C2×C4×C8C2311yescharacteristic
36#400—41no—
37#395—41no—
38#391—22no—
39#389—22no—
40#387—42no—
41#382—21no—
42C4⋊C16—24no—
43C4⋊C16—22no—
44C22⋊C16—22no—
45#346—24no—
46C4×C16D414yes—
47#340—22no—
48#340—21no—
49#340C2×C412yes—
50#335—22no—
51#334—22no—
52#324—42no—
53C22×Q8—21no—
54C22×Q8SD1611yescharacteristic
55C2×Q16—22no—
56C2×Q16—42no—
57C22×C8C22⋊C412yes—
58C22×C8C22×C411yescharacteristic
59C22×C8C2×D411yescharacteristic
60C4×Q8—22no—
61C4×Q8—22no—
62C4×Q8—42no—
63C4×Q8—42no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4—41no—
68C2×C4⋊C4D812yes—
69C2×C4⋊C4SD1611yescharacteristic
70C2×C42—41no—
71C2×C42C22⋊C412yes—
72C2×C42C2×D411yescharacteristic
73C2×C16—24no—
74#107—22no—
75C4⋊C8—24no—
76C4⋊C8—22no—
77C4⋊C8—42no—
78#103—44no—
79#103—41no—
80#103—41no—
81C22⋊C8—42no—
82#97—42no—
83C4×C8—22no—
84C4×C8—22no—
85C4×C8C2×D412yes—
86C4×C8#4112yes—
87#95—42no—
88#95—42no—
89#95—42no—
90#95—41no—
91#95—41no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—42no—
95C2×Q8—41no—
96C2×Q8—41no—
97C2×Q8—41no—
98C2×Q8—41no—
99C2×Q8—41no—
100C22×C4—41no—
101C22×C4—42no—
102C22×C4—41no—
103C22×C4#10212yes—
104C22×C4#10212yes—
105C22×C4C2×C22⋊C411yescharacteristic
106C22×C4C2×D811yescharacteristic
107C22×C4C2×SD1611yescharacteristic
108Q16—24no—
109Q16—42no—
110C2×C8—22no—
111C2×C8—41no—
112C2×C8—41no—
113C2×C8—42no—
114C2×C8—82no—
115C2×C8C4×D412yes—
116C2×C8#12011yescharacteristic
117C2×C8#12311yescharacteristic
118C4⋊C4—22no—
119C4⋊C4—24no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—41no—
123C4⋊C4—41no—
124C4⋊C4—84no—
125C4⋊C4—82no—
126C42—22no—
127C42—22no—
128C42—82no—
129C42#12112yes—
130C42#13512yes—
131C16—44no—
132C23C2×#10211yescharacteristic
133Q8—44no—
134Q8—42no—
135Q8—42no—
136Q8—81no—
137Q8—81no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—41no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—81no—
149C2×C4—81no—
150C2×C4—82no—
151C2×C4—82no—
152C2×C4—82no—
153C2×C4—82no—
154C2×C4—81no—
155C2×C4#38611yescharacteristic
156C2×C4#41611yescharacteristic
157C2×C4#46512yes—
158C2×C4#46612yes—
159C8—22no—
160C8—21no—
161C8—21no—
162C8—42no—
163C8—162no—
164C22—21no—
165C22—21no—
166C22#189311yescharacteristic
167C22#214312yes—
168C4—22no—
169C4—21no—
170C4—22no—
171C4—42no—
172C4—42no—
173C4—82no—
174C4—162no—
175C4—161no—
176C4#189111yescharacteristic
177C4#191211yescharacteristic
178C2—22no—
179C2#948611yescharacteristic
180C2#1306411yescharacteristic
181C2#1307511yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23C2×#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#334972
#281902
#116622
#86212
#72182