Finite Group Databasebeta

← #120213

Subgroups — #120213

Order 512.

422 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 126 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#21973C211yesmaximal, characteristic
2#17819C211yesmaximal, characteristic
3#8110C212yesmaximal
4#8002C212yesmaximal
5#7987C211yesmaximal, characteristic
6#2920C2212yes—
7#2920C2212yes—
8C2×#431C2211yescharacteristic
9C22×C4⋊C8C2211yescharacteristic
10#2283C2211yescharacteristic
11#1939—22no—
12#1850C414yes—
13#1843—21no—
14#1832—21no—
15#1574—24no—
16#1568—22no—
17#1305—24no—
18#1292—24no—
19#1290—21no—
20#1290—21no—
21C23×C8C2×C412yes—
22C22×C42C2311yescharacteristic
23#432—22no—
24#432—22no—
25#432—22no—
26#432Q814yes—
27#431—22no—
28#431—22no—
29#431—22no—
30#431—22no—
31#431—22no—
32#431—22no—
33#423—22no—
34#423D414yes—
35C2×C4⋊C8—22no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—24no—
40C2×C4⋊C8D411yescharacteristic
41C2×C4⋊C8D411yescharacteristic
42#407—22no—
43#407—22no—
44C2×C22⋊C8C2×C412yes—
45C4×M16—22no—
46C4×M16—22no—
47C2×C4×C8D412yes—
48#383—21no—
49#382—21no—
50#381Q812yes—
51#380D412yes—
52C4×C4⋊C4—21no—
53C4×C22⋊C4C2×C412yes—
54#376—21no—
55#336—24no—
56#335—24no—
57#334—24no—
58C23×C4C22×C411yescharacteristic
59C23×C4C2×D411yescharacteristic
60C23×C4C2×Q811yescharacteristic
61C2×M16—24no—
62C2×M16—42no—
63C22×C8—22no—
64C22×C8—24no—
65C22×C8—82no—
66C22×C8C22⋊C412yes—
67C22×C8C22⋊C412yes—
68C2×C4⋊C4—22no—
69C2×C22⋊C4C4⋊C414yes—
70C2×C42—22no—
71C2×C42C2×D412yes—
72C2×C42C2×D411yescharacteristic
73C2×C42C2×D411yescharacteristic
74C2×C42C2×Q812yes—
75C2×C42#4112yes—
76C2×C42#4111yescharacteristic
77C2×C42#4112yes—
78C2×C42#4111yescharacteristic
79C4⋊C8—24no—
80C4⋊C8—22no—
81C4⋊C8—22no—
82C4⋊C8—24no—
83C4⋊C8—24no—
84C4⋊C8—41no—
85C4⋊C8—42no—
86C4⋊C8—42no—
87C4⋊C8—41no—
88C4⋊C8—84no—
89C22⋊C8—24no—
90C22⋊C8—24no—
91C22⋊C8—42no—
92C22⋊C8—42no—
93C22⋊C8—42no—
94#97—24no—
95#97—42no—
96C4×C8—22no—
97C4×C8—22no—
98C4×C8—42no—
99C4×C8—42no—
100#95—22no—
101#95—22no—
102#95—22no—
103#95—22no—
104C24C2×C4⋊C411yescharacteristic
105C22×C4—24no—
106C22×C4—41no—
107C22×C4—41no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4C2×C22⋊C411yescharacteristic
111C22×C4#11711yescharacteristic
112C22×C4C4×D412yes—
113C22×C4C4×Q812yes—
114C22×C4#12112yes—
115C22×C4#12111yescharacteristic
116C22×C4#12111yescharacteristic
117C22×C4C22⋊Q811yescharacteristic
118C22×C4C22⋊Q811yescharacteristic
119C22×C4C22⋊Q812yes—
120C22×C4#12512yes—
121C22×C4C4⋊Q812yes—
122M16—84no—
123M16—84no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—42no—
127C2×C8—42no—
128C2×C8—44no—
129C2×C8—42no—
130C2×C8—44no—
131C2×C8—44no—
132C2×C8—84no—
133C4⋊C4—44no—
134C22⋊C4—44no—
135C42—42no—
136C42—42no—
137C42#12011yescharacteristic
138C42#12112yes—
139C42C22⋊Q812yes—
140C42C22⋊Q812yes—
141C42#12311yescharacteristic
142C42#12312yes—
143C42#12411yescharacteristic
144C42#12512yes—
145C42#12612yes—
146C42C4⋊Q811yescharacteristic
147C23—22no—
148C23—41no—
149C23—41no—
150C23#38011yescharacteristic
151C23#38912yes—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—24no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—44no—
167C2×C4—44no—
168C2×C4—44no—
169C2×C4#38611yescharacteristic
170C2×C4#38712yes—
171C2×C4#39111yescharacteristic
172C2×C4#39411yescharacteristic
173C2×C4#39511yescharacteristic
174C2×C4#39612yes—
175C2×C4#39812yes—
176C2×C4#39811yescharacteristic
177C2×C4#40011yescharacteristic
178C8—42no—
179C8—42no—
180C8—84no—
181C8—84no—
182C8—82no—
183C22—21no—
184C22—21no—
185C22—22no—
186C22—22no—
187C22—22no—
188C22—44no—
189C22#144111yescharacteristic
190C22#181312yes—
191C4—22no—
192C4—21no—
193C4—21no—
194C4—22no—
195C4—21no—
196C4—22no—
197C4—21no—
198C4—84no—
199C4#191411yescharacteristic
200C4#191511yescharacteristic
201C4#205312yes—
202C2—22no—
203C2—42no—
204C2#1120611yescharacteristic
205C2#1123011yescharacteristic
206C2#1123611yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C4⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#219732
#178192
#81102
#80022
#79872