Finite Group Databasebeta

← #139373

Subgroups — #139373

Order 512.

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

normal subgroups only

This group has 339 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#33489C211yesmaximal, characteristic
2#28707C211yesmaximal, characteristic
3#11967C212yesmaximal, complemented
4#8620C211yesmaximal, characteristic
5#7299C212yesmaximal, complemented
6C2×#499C2211yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2562—21no—
9#2169—24no—
10#2162C2212yes—
11#2112—22no—
12#2076C412yes—
13#2075C412yes—
14#1989—21no—
15#1807—22no—
16C4×#106C2211yescharacteristic
17#1568C2212yes—
18#1466—21no—
19#1363—24no—
20#1290—21no—
21C23×C8D412yes—
22C2×#125—21no—
23C2×C4×D4D411yescharacteristic
24C2×M32—24no—
25#499—22no—
26#499—22no—
27C8×D4—22no—
28C8×D4—22no—
29#432—22no—
30C2×#107C2×C411yescharacteristic
31C2×#107C2×C411yescharacteristic
32C2×#106—21no—
33C2×#106—21no—
34C2×C4⋊C8D411yescharacteristic
35C2×C22⋊C8C2×C412yes—
36C2×C4×C8C2311yescharacteristic
37#400—41no—
38#394—41no—
39#388—42no—
40#380—22no—
41#380—22no—
42C4×C4⋊C4—21no—
43C4⋊C16D414yes—
44C22⋊C16—24no—
45C22⋊C16—22no—
46C4×C16—22no—
47#340—22no—
48#340—22no—
49#340C2×C412yes—
50#335—22no—
51#334—22no—
52#323—42no—
53C22×D4SD1611yescharacteristic
54C23×C4C22⋊C412yes—
55C22×C8—22no—
56C22×C8—22no—
57C22×C8—44no—
58C22×C8C22⋊C412yes—
59C22×C8C22×C411yescharacteristic
60C22×C8C2×D411yescharacteristic
61#125—42no—
62#125—42no—
63C4×D4—22no—
64C4×D4—22no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—41no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4SD1611yescharacteristic
71C2×C22⋊C4—42no—
72C2×C22⋊C4—42no—
73C2×C22⋊C4D812yes—
74C2×C42—41no—
75C2×C42C2×D411yescharacteristic
76M32—84no—
77C2×C16—24no—
78#107—22no—
79#107—22no—
80#106—42no—
81#106—41no—
82#106—41no—
83C4⋊C8—22no—
84C4⋊C8—42no—
85C22⋊C8—24no—
86C22⋊C8—42no—
87#97—42no—
88C4×C8C2×D412yes—
89C4×C8#4112yes—
90#95—42no—
91#95—42no—
92#95—41no—
93C24#10212yes—
94C2×D4—22no—
95C2×D4—42no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—42no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4#10212yes—
104C22×C4C2×C22⋊C411yescharacteristic
105C22×C4C2×D811yescharacteristic
106C22×C4C2×SD1611yescharacteristic
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—41no—
110C2×C8—44no—
111C2×C8—42no—
112C2×C8—82no—
113C2×C8—82no—
114C2×C8C4×D412yes—
115C2×C8#12011yescharacteristic
116C2×C8#12311yescharacteristic
117C4⋊C4—22no—
118C4⋊C4—42no—
119C4⋊C4—42no—
120C4⋊C4—41no—
121C4⋊C4—41no—
122C4⋊C4—41no—
123C4⋊C4—41no—
124C4⋊C4—82no—
125C4⋊C4—82no—
126C22⋊C4—24no—
127C22⋊C4—84no—
128C22⋊C4—162no—
129C42—161no—
130C42#12112yes—
131C42#13512yes—
132C16—44no—
133C23—22no—
134C23—22no—
135C23—22no—
136C23—42no—
137C23—42no—
138C23C2×#10211yescharacteristic
139D4—44no—
140C2×C4—22no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—82no—
149C2×C4—82no—
150C2×C4—81no—
151C2×C4—82no—
152C2×C4—81no—
153C2×C4—82no—
154C2×C4—82no—
155C2×C4—81no—
156C2×C4—81no—
157C2×C4#38611yescharacteristic
158C2×C4#41611yescharacteristic
159C2×C4#46512yes—
160C2×C4#46612yes—
161C8—21no—
162C8—22no—
163C8—42no—
164C8—162no—
165C8#45512yes—
166C22—21no—
167C22—21no—
168C22—44no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—82no—
173C22#189311yescharacteristic
174C22#214312yes—
175C4—22no—
176C4—22no—
177C4—21no—
178C4—42no—
179C4—162no—
180C4—161no—
181C4—161no—
182C4#188511yescharacteristic
183C4#191511yescharacteristic
184C2—22no—
185C2—82no—
186C2#939311yescharacteristic
187C2#1306411yescharacteristic
188C2#1307711yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11967C2semidirect
5. #7299C2semidirect

Maximal subgroups

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

subgroupindex
#334892
#287072
#119672
#86202
#72992