Finite Group Databasebeta

← #139385

Subgroups — #139385

Order 512.

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

normal subgroups only

This group has 317 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
1#33496C211yesmaximal, characteristic
2#28678C211yesmaximal, characteristic
3#11967C212yesmaximal
4#8621C211yesmaximal, characteristic
5#7299C212yesmaximal
6C2×#499C2211yescharacteristic
7C2×C8×Q8C2211yescharacteristic
8#2584—21no—
9Q8×C16—24no—
10#2162—22no—
11#2162C2212yes—
12#2087C412yes—
13#2086C412yes—
14#1989—21no—
15#1941—22no—
16C4×#106C2211yescharacteristic
17#1568C2212yes—
18#1467—21no—
19#1363—24no—
20#1290—21no—
21C2×#125—21no—
22C2×C4×Q8D411yescharacteristic
23#499—22no—
24#499—22no—
25C8×Q8—22no—
26C8×Q8—22no—
27#432—22no—
28C2×#107C2×C411yescharacteristic
29C2×#107C2×C411yescharacteristic
30C2×#106—21no—
31C2×#106—21no—
32C2×C4⋊C8C2×C412yes—
33C2×C4⋊C8D411yescharacteristic
34C2×C4×C8D412yes—
35C2×C4×C8C2311yescharacteristic
36#398—41no—
37#395—41no—
38#389—22no—
39#389—22no—
40#389—42no—
41C4×C4⋊C4—21no—
42C4⋊C16—24no—
43C4⋊C16D414yes—
44C22⋊C16—22no—
45C4×C16—24no—
46C4×C16—22no—
47#340—22no—
48#340—22no—
49#340C2×C412yes—
50#335—22no—
51#334—22no—
52#324—42no—
53C22×Q8SD1611yescharacteristic
54C22×C8C22⋊C412yes—
55C22×C8C22×C411yescharacteristic
56C22×C8C2×D411yescharacteristic
57#125—42no—
58#125—42no—
59C4×Q8—22no—
60C4×Q8—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4—41no—
66C2×C4⋊C4—41no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4D812yes—
69C2×C4⋊C4SD1611yescharacteristic
70C2×C42—41no—
71C2×C42C22⋊C412yes—
72C2×C42C2×D411yescharacteristic
73C2×C16—24no—
74#107—22no—
75#107—22no—
76#106—24no—
77#106—41no—
78#106—41no—
79C4⋊C8—24no—
80C4⋊C8—22no—
81C4⋊C8—42no—
82C22⋊C8—42no—
83#97—42no—
84C4×C8—22no—
85C4×C8—22no—
86C4×C8C2×D412yes—
87C4×C8#4112yes—
88#95—42no—
89#95—42no—
90#95—41no—
91C2×Q8—24no—
92C2×Q8—22no—
93C22×C4—42no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4#10212yes—
97C22×C4#10212yes—
98C22×C4C2×C22⋊C411yescharacteristic
99C22×C4C2×D811yescharacteristic
100C22×C4C2×SD1611yescharacteristic
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—41no—
104C2×C8—42no—
105C2×C8—82no—
106C2×C8C4×D412yes—
107C2×C8#12011yescharacteristic
108C2×C8#12311yescharacteristic
109C4⋊C4—22no—
110C4⋊C4—24no—
111C4⋊C4—24no—
112C4⋊C4—42no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—84no—
118C4⋊C4—82no—
119C4⋊C4—82no—
120C4⋊C4—162no—
121C42—22no—
122C42—22no—
123C42—161no—
124C42#12112yes—
125C42#13512yes—
126C16—44no—
127C23C2×#10211yescharacteristic
128Q8—28no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—41no—
136C2×C4—42no—
137C2×C4—44no—
138C2×C4—82no—
139C2×C4—82no—
140C2×C4—81no—
141C2×C4—81no—
142C2×C4—82no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4#35914yes—
146C2×C4#38611yescharacteristic
147C2×C4#41611yescharacteristic
148C2×C4#46512yes—
149C2×C4#46612yes—
150C8—21no—
151C8—22no—
152C8—42no—
153C8—162no—
154C8#45612yes—
155C22—21no—
156C22—21no—
157C22#189311yescharacteristic
158C22C2×#35912yes—
159C4—24no—
160C4—22no—
161C4—22no—
162C4—21no—
163C4—42no—
164C4—84no—
165C4—161no—
166C4—161no—
167C4#188611yescharacteristic
168C4#191511yescharacteristic
169C2—22no—
170C2#939411yescharacteristic
171C2#1306311yescharacteristic
172C2#1307711yescharacteristic
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
#334962
#286782
#119672
#86212
#72992