Finite Group Databasebeta

← #139422

Subgroups — #139422

Order 512.

331 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 84 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#28737C211yesmaximal, characteristic
3#11965C212yesmaximal
4#8623C211yesmaximal, characteristic
5#7301C212yesmaximal
6C2×#494C2211yescharacteristic
7C2×C8×Q8C2211yescharacteristic
8#2568—21no—
9#2183—24no—
10#2163—21no—
11#2163C2212yes—
12#2162—21no—
13#2056C412yes—
14#2056C412yes—
15#1981—21no—
16#1969—21no—
17#1969—21no—
18C4×#107C2211yescharacteristic
19#1568C2212yes—
20#1467—21no—
21#1362—24no—
22#1290—21no—
23C22×Q16—21no—
24C22×Q16C2×C412yes—
25C2×C4⋊Q8—21no—
26C2×C4×Q8D411yescharacteristic
27#494—22no—
28#494—22no—
29C8×Q8—22no—
30C8×Q8—22no—
31#432—22no—
32C2×#107—21no—
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C4⋊C8C2×C411yescharacteristic
35C2×C4⋊C8D411yescharacteristic
36C2×#103—22no—
37C2×#103—22no—
38C2×C4×C8D411yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8C2311yescharacteristic
41#400—41no—
42#400—41no—
43#391—22no—
44#391—22no—
45#391—41no—
46#382—41no—
47C4×C4⋊C4—21no—
48C4⋊C16—22no—
49C4⋊C16—22no—
50C4⋊C16D414yes—
51C22⋊C16—22no—
52#346—22no—
53#346—22no—
54C4×C16—22no—
55#340C2×C412yes—
56#335—22no—
57#334—22no—
58#324—41no—
59#324—41no—
60C22×Q8—22no—
61C22×Q8SD1611yescharacteristic
62C2×Q16—24no—
63C2×Q16—42no—
64C2×Q16—42no—
65C2×Q16—41no—
66C2×Q16—41no—
67C2×Q16—82no—
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22×C411yescharacteristic
71C22×C8C2×D411yescharacteristic
72C4⋊Q8—42no—
73C4⋊Q8—42no—
74C4×Q8—22no—
75C4×Q8—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4D811yescharacteristic
80C2×C4⋊C4D811yescharacteristic
81C2×C4⋊C4SD1611yescharacteristic
82C2×C42—41no—
83C2×C42C22⋊C411yescharacteristic
84C2×C42C22⋊C411yescharacteristic
85C2×C42C2×D411yescharacteristic
86C2×C16—24no—
87#107—42no—
88C4⋊C8—22no—
89C4⋊C8—22no—
90C4⋊C8—22no—
91C4⋊C8—42no—
92#103—82no—
93#103—82no—
94C22⋊C8—42no—
95#97—42no—
96C4×C8—21no—
97C4×C8—21no—
98C4×C8—21no—
99C4×C8—21no—
100C4×C8C2×D412yes—
101C4×C8#4112yes—
102#95—42no—
103#95—42no—
104#95—42no—
105#95—42no—
106#95—41no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—41no—
110C2×Q8—42no—
111C2×Q8—42no—
112C2×Q8—42no—
113C2×Q8—42no—
114C2×Q8—42no—
115C2×Q8—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4#10211yescharacteristic
119C22×C4#10211yescharacteristic
120C22×C4#10211yescharacteristic
121C22×C4#10211yescharacteristic
122C22×C4C2×C22⋊C411yescharacteristic
123C22×C4C2×D811yescharacteristic
124C22×C4C2×SD1611yescharacteristic
125Q16—84no—
126Q16—82no—
127Q16—82no—
128C2×C8—22no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—41no—
132C2×C8—41no—
133C2×C8—41no—
134C2×C8—82no—
135C2×C8C4×D412yes—
136C2×C8#12011yescharacteristic
137C2×C8#12311yescharacteristic
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—82no—
144C4⋊C4—82no—
145C42—21no—
146C42—21no—
147C42—21no—
148C42—21no—
149C42—82no—
150C42#12112yes—
151C42#13512yes—
152C16—44no—
153C23C2×#10211yescharacteristic
154Q8—44no—
155Q8—82no—
156Q8—82no—
157Q8—82no—
158Q8—82no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4—82no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4#36212yes—
180C2×C4#38611yescharacteristic
181C2×C4#41611yescharacteristic
182C2×C4#46512yes—
183C2×C4#46612yes—
184C8—21no—
185C8—22no—
186C8—41no—
187C8—41no—
188C8—162no—
189C8#45412yes—
190C22—21no—
191C22—21no—
192C22#189311yescharacteristic
193C22C2×#36211yescharacteristic
194C22#214811yescharacteristic
195C4—22no—
196C4—22no—
197C4—22no—
198C4—21no—
199C4—41no—
200C4—41no—
201C4—41no—
202C4—162no—
203C4—162no—
204C4#188811yescharacteristic
205C4#191411yescharacteristic
206C2—21no—
207C2—21no—
208C2#939811yescharacteristic
209C2#1306611yescharacteristic
210C2#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
#287372
#119652
#86232
#73012