Finite Group Databasebeta

← #139668

Subgroups — #139668

Order 512.

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

normal subgroups only

This group has 399 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, complemented
2#28143C211yesmaximal, characteristic
3#11661C212yesmaximal, complemented
4#8620C211yesmaximal, characteristic, complemented
5#7218C212yesmaximal, complemented
6C2×C4×D8C2211yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2535—21no—
9#2169—24no—
10#2162C2212yes—
11#2112—22no—
12#2076C412yes—
13#1999C412yes—
14#1893—22no—
15#1835C2211yescharacteristic
16#1804—21no—
17#1568C2212yes—
18#1466—21no—
19#1331—24no—
20#1290—21no—
21C22×D8C2×C411yescharacteristic
22C23×C8D412yes—
23C2×#121—22no—
24C2×C4×D4—21no—
25C2×C4×D4D411yescharacteristic
26C2×M32—24no—
27C4×D8—22no—
28C4×D8—22no—
29C8×D4—22no—
30C8×D4—22no—
31#432—22no—
32C2×#107C2×C411yescharacteristic
33C2×C4⋊C8D411yescharacteristic
34C2×#102—21no—
35C2×#102—22no—
36C2×C22⋊C8C2×C412yes—
37C2×C4×C8C2311yescharacteristic
38#397—41no—
39#392—41no—
40#386—42no—
41#382—21no—
42#380—22no—
43C4⋊C16—22no—
44C22⋊C16—24no—
45C22⋊C16—22no—
46C4×C16D414yescomplemented
47#340—22no—
48#340—21no—
49#340C2×C412yes—
50#335—22no—
51#334—22no—
52#323—42no—
53C22×D4—21no—
54C22×D4—42no—
55C22×D4SD1611yescharacteristic
56C23×C4—41no—
57C23×C4C22⋊C412yes—
58C2×D8—22no—
59C2×D8—42no—
60C22×C8—22no—
61C22×C8—22no—
62C22×C8—82no—
63C22×C8C22⋊C412yes—
64C22×C8C22×C411yescharacteristic
65C22×C8C2×D411yescharacteristic
66#121—44no—
67#121—82no—
68C4×D4—22no—
69C4×D4—22no—
70C4×D4—42no—
71C4×D4—42no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4SD1611yescharacteristic
75C2×C22⋊C4—42no—
76C2×C22⋊C4—41no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4D812yes—
79C2×C42C2×D411yescharacteristic
80M32—84no—
81C2×C16—24no—
82#107—22no—
83C4⋊C8—22no—
84C4⋊C8—42no—
85#102—41no—
86#102—41no—
87#102—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—41no—
97C24#10212yes—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—42no—
101C2×D4—41no—
102C2×D4—41no—
103C2×D4—41no—
104C2×D4—41no—
105C2×D4—41no—
106C2×D4—84no—
107C2×D4—84no—
108C2×D4—162no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—41no—
115C22×C4—82no—
116C22×C4—162no—
117C22×C4#10212yes—
118C22×C4C2×C22⋊C411yescharacteristic
119C22×C4C2×D811yescharacteristic
120C22×C4C2×SD1611yescharacteristic
121D8—24no—
122D8—42no—
123C2×C8—22no—
124C2×C8—41no—
125C2×C8—41no—
126C2×C8—44no—
127C2×C8—42no—
128C2×C8—82no—
129C2×C8—82no—
130C2×C8C4×D412yes—
131C2×C8#12011yescharacteristic
132C2×C8#12311yescharacteristic
133C4⋊C4—22no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—41no—
137C4⋊C4—41no—
138C22⋊C4—24no—
139C22⋊C4—84no—
140C22⋊C4—82no—
141C22⋊C4—162no—
142C42#12112yes—
143C42#13512yes—
144C16—44no—
145C23—22no—
146C23—22no—
147C23—22no—
148C23—42no—
149C23—42no—
150C23—42no—
151C23—82no—
152C23—81no—
153C23—82no—
154C23—81no—
155C23C2×#10211yescharacteristic
156D4—44no—
157D4—42no—
158D4—42no—
159D4—81no—
160D4—81no—
161D4—168no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—82no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—81no—
176C2×C4—164no—
177C2×C4#38611yescharacteristic
178C2×C4#41611yescharacteristic
179C2×C4#46512yes—
180C2×C4#46612yes—
181C8—22no—
182C8—21no—
183C8—21no—
184C8—42no—
185C8—162no—
186C22—21no—
187C22—21no—
188C22—44no—
189C22—42no—
190C22—42no—
191C22—42no—
192C22—82no—
193C22—84no—
194C22—82no—
195C22—162no—
196C22—161no—
197C22—161no—
198C22#189311yescharacteristic
199C22#214312yes—
200C4—22no—
201C4—21no—
202C4—22no—
203C4—42no—
204C4—162no—
205C4—161no—
206C4#189111yescharacteristic
207C4#191111yescharacteristic
208C2—22no—
209C2—82no—
210C2—162no—
211C2#948511yescharacteristic
212C2#1306411yescharacteristic
213C2#1307411yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radical#28143C2
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
1. #33489C2semidirect
3. #11661C2semidirect
4. #8620C2semidirect
5. #7218C2semidirect
46. C4×C16D4semidirect

Maximal subgroups

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

subgroupindex
#334892
#281432
#116612
#86202
#72182