Finite Group Databasebeta

← #139669

Subgroups — #139669

Order 512.

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

normal subgroups only

This group has 367 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#33489C211yesmaximal, characteristic
2#28145C211yesmaximal, characteristic
3#11662C212yesmaximal, complemented
4#8620C211yesmaximal, characteristic
5#7218C212yesmaximal, complemented
6C2×C4×Q16C2211yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2536—21no—
9#2169—24no—
10#2162C2212yes—
11#2112—22no—
12#2075C412yes—
13#2001C412yes—
14#1892—22no—
15#1835C2211yescharacteristic
16#1805—21no—
17#1568C2212yes—
18#1466—21no—
19#1332—24no—
20#1290—21no—
21C22×Q16C2×C411yescharacteristic
22C23×C8D412yes—
23C2×C22⋊Q8—22no—
24C2×C4×Q8—21no—
25C2×C4×D4D411yescharacteristic
26C2×M32—24no—
27C4×Q16—22no—
28C4×Q16—22no—
29C8×D4—22no—
30C8×D4—22no—
31#432—22no—
32C2×#107C2×C411yescharacteristic
33C2×C4⋊C8D411yescharacteristic
34C2×#103—21no—
35C2×#103—22no—
36C2×C22⋊C8C2×C412yes—
37C2×C4×C8C2311yescharacteristic
38#398—41no—
39#394—41no—
40#388—42no—
41#382—21no—
42#380—22no—
43C4⋊C16—22no—
44C22⋊C16—24no—
45C22⋊C16—22no—
46C4×C16D414yes—
47#340—22no—
48#340—21no—
49#340C2×C412yes—
50#335—22no—
51#334—22no—
52#323—42no—
53C22×Q8—21no—
54C22×D4SD1611yescharacteristic
55C23×C4C22⋊C412yes—
56C2×Q16—22no—
57C2×Q16—42no—
58C22×C8—22no—
59C22×C8—22no—
60C22×C8—82no—
61C22×C8C22⋊C412yes—
62C22×C8C22×C411yescharacteristic
63C22×C8C2×D411yescharacteristic
64C22⋊Q8—44no—
65C22⋊Q8—82no—
66C4×Q8—42no—
67C4×Q8—42no—
68C4×D4—22no—
69C4×D4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—42no—
73C2×C4⋊C4—41no—
74C2×C4⋊C4SD1611yescharacteristic
75C2×C22⋊C4—42no—
76C2×C22⋊C4—42no—
77C2×C22⋊C4D812yes—
78C2×C42—41no—
79C2×C42C2×D411yescharacteristic
80M32—84no—
81C2×C16—24no—
82#107—22no—
83C4⋊C8—22no—
84C4⋊C8—42no—
85#103—41no—
86#103—41no—
87#103—82no—
88C22⋊C8—24no—
89C22⋊C8—42no—
90#97—42no—
91C4×C8C2×D412yes—
92C4×C8#4112yes—
93#95—42no—
94#95—41no—
95#95—41no—
96C24#10212yes—
97C2×Q8—22no—
98C2×Q8—41no—
99C2×Q8—41no—
100C2×Q8—41no—
101C2×Q8—41no—
102C2×Q8—41no—
103C2×D4—22no—
104C2×D4—42no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4—41no—
110C22×C4—42no—
111C22×C4—41no—
112C22×C4#10212yes—
113C22×C4C2×C22⋊C411yescharacteristic
114C22×C4C2×D811yescharacteristic
115C22×C4C2×SD1611yescharacteristic
116Q16—24no—
117Q16—42no—
118C2×C8—22no—
119C2×C8—41no—
120C2×C8—41no—
121C2×C8—44no—
122C2×C8—42no—
123C2×C8—82no—
124C2×C8—82no—
125C2×C8C4×D412yes—
126C2×C8#12011yescharacteristic
127C2×C8#12311yescharacteristic
128C4⋊C4—22no—
129C4⋊C4—42no—
130C4⋊C4—42no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C4⋊C4—84no—
134C4⋊C4—82no—
135C22⋊C4—24no—
136C22⋊C4—84no—
137C22⋊C4—84no—
138C42—82no—
139C42#12112yes—
140C42#13512yes—
141C16—44no—
142C23—22no—
143C23—22no—
144C23—22no—
145C23—42no—
146C23—42no—
147C23C2×#10211yescharacteristic
148Q8—42no—
149Q8—42no—
150Q8—81no—
151Q8—81no—
152D4—44no—
153C2×C4—22no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—82no—
163C2×C4—81no—
164C2×C4—81no—
165C2×C4—82no—
166C2×C4—82no—
167C2×C4—82no—
168C2×C4—82no—
169C2×C4—81no—
170C2×C4#38611yescharacteristic
171C2×C4#41611yescharacteristic
172C2×C4#46512yes—
173C2×C4#46612yes—
174C8—22no—
175C8—21no—
176C8—21no—
177C8—42no—
178C8—162no—
179C22—21no—
180C22—21no—
181C22—44no—
182C22—42no—
183C22—42no—
184C22—42no—
185C22—82no—
186C22#189311yescharacteristic
187C22#214312yes—
188C4—22no—
189C4—21no—
190C4—22no—
191C4—42no—
192C4—82no—
193C4—162no—
194C4—161no—
195C4#189111yescharacteristic
196C4#191111yescharacteristic
197C2—22no—
198C2—82no—
199C2#948511yescharacteristic
200C2#1306411yescharacteristic
201C2#1307511yescharacteristic
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. #11662C2semidirect
5. #7218C2semidirect

Maximal subgroups

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

subgroupindex
#334892
#281452
#116622
#86202
#72182