Finite Group Databasebeta

← #120147

Subgroups — #120147

Order 512.

598 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 138 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#20765C211yesmaximal, characteristic
2#14168C211yesmaximal, characteristic
3C2×#1300C212yesmaximal
4#8030C211yesmaximal, characteristic
5#8022C212yesmaximal
6C2×#432C2212yes—
7C2×C4×M16C2211yescharacteristic
8#2285C2211yescharacteristic
9C2×#382C2211yescharacteristic
10C2×#376C2212yes—
11#1836C414yescomplemented
12#1835—22no—
13#1736—22no—
14#1441—22no—
15#1436—24no—
16#1300—24no—
17#1295—22no—
18#1294—24no—
19C22×M16—21no—
20C2×#117—21no—
21C22×C4⋊C4—22no—
22C22×C4⋊C4D412yes—
23C22×C42—22no—
24C22×C42C2311yescharacteristic
25#432—22no—
26#432—22no—
27#432—22no—
28#432Q814yes—
29C2×C4⋊C8—22no—
30C2×C22⋊C8—22no—
31C4×M16—21no—
32C4×M16—21no—
33C4×M16D412yes—
34C4×M16Q812yes—
35C2×#97—22no—
36C2×#97C2×C412yes—
37C2×C4×C8—21no—
38#384C2×C412yes—
39#382—22no—
40#382—21no—
41#382C2×C412yes—
42#382D412yes—
43#380—22no—
44C4×C4⋊C4—22no—
45#376—22no—
46#376—22no—
47#376—22no—
48#376—22no—
49C2×#95—22no—
50C2×#95—24no—
51C2×#95—22no—
52C2×#95—22no—
53C2×#95D414yes—
54#340—28no—
55C23×C4—22no—
56C23×C4—24no—
57C23×C4C2×D411yescharacteristic
58C23×C4C2×D412yes—
59C2×M16—22no—
60C2×M16—22no—
61C2×M16—42no—
62C2×M16—42no—
63C22×C8—22no—
64C22×C8—42no—
65#117—44no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—42no—
71C2×C4⋊C4—48no—
72C2×C4⋊C4—82no—
73C2×C4⋊C4C22⋊C414yes—
74C2×C22⋊C4—41no—
75C2×C42—48no—
76C2×C42—42no—
77C2×C42—42no—
78C2×C42—41no—
79C2×C42C22×C411yescharacteristic
80C2×C42C2×D411yescharacteristic
81C2×C42C2×D411yescharacteristic
82C2×C42C2×Q811yescharacteristic
83C2×C42C2×Q812yes—
84C2×C42#4111yescharacteristic
85C2×C42#4111yescharacteristic
86C2×C42#4112yes—
87C2×C42#4112yes—
88C4⋊C8—24no—
89C4⋊C8—42no—
90C22⋊C8—24no—
91C22⋊C8—42no—
92#97—22no—
93#97—24no—
94#97—42no—
95#97C4⋊C414yescomplemented
96C4×C8—22no—
97C4×C8—22no—
98#95—24no—
99#95—24no—
100#95—24no—
101#95—24no—
102#95—48no—
103#95—42no—
104#95—42no—
105#95—42no—
106#95—82no—
107C24#12011yescharacteristic
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—28no—
111C22×C4—24no—
112C22×C4—44no—
113C22×C4—44no—
114C22×C4—44no—
115C22×C4—44no—
116C22×C4—44no—
117C22×C4—44no—
118C22×C4—44no—
119C22×C4—44no—
120C22×C4—42no—
121C22×C4—44no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4C2×C22⋊C411yescharacteristic
125C22×C4C4×D412yes—
126C22×C4#12111yescharacteristic
127C22×C4#12112yes—
128C22×C4#12112yes—
129C22×C4C22⋊Q811yescharacteristic
130C22×C4C22⋊Q812yes—
131C22×C4#12311yescharacteristic
132C22×C4#12312yes—
133C22×C4#12411yescharacteristic
134C22×C4C4⋊Q811yescharacteristic
135C22×C4C4⋊Q812yes—
136M16—44no—
137M16—44no—
138C2×C8—24no—
139C2×C8—42no—
140C2×C8—42no—
141C2×C8—44no—
142C2×C8—84no—
143C4⋊C4—44no—
144C4⋊C4—44no—
145C4⋊C4—82no—
146C4⋊C4—84no—
147C4⋊C4—88no—
148C22⋊C4—82no—
149C42—48no—
150C42—48no—
151C42—82no—
152C42C2×C4⋊C411yescharacteristic
153C42#11711yescharacteristic
154C42C4×Q812yes—
155C42#12111yescharacteristic
156C42C22⋊Q811yescharacteristic
157C42C22⋊Q812yes—
158C42C22⋊Q812yes—
159C42#12512yes—
160C42#12512yes—
161C42#12612yes—
162C23—24no—
163C23#38611yescharacteristic
164C23#39411yescharacteristic
165C23#39511yescharacteristic
166C23#45412yes—
167C23#45512yes—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—48no—
183C2×C4—48no—
184C2×C4—48no—
185C2×C4—44no—
186C2×C4—48no—
187C2×C4—82no—
188C2×C4—84no—
189C2×C4—84no—
190C2×C4—84no—
191C2×C4—82no—
192C2×C4—84no—
193C2×C4—84no—
194C2×C4—82no—
195C2×C4—84no—
196C2×C4#38011yescharacteristic
197C2×C4#38712yes—
198C2×C4#38912yes—
199C2×C4#39111yescharacteristic
200C2×C4#39612yes—
201C2×C4#39811yescharacteristic
202C2×C4#39812yes—
203C2×C4#40011yescharacteristic
204C8—44no—
205C8—84no—
206C22—22no—
207C22—22no—
208C22—22no—
209C22—22no—
210C22—24no—
211C22#144111yescharacteristic
212C22#188511yescharacteristic
213C22#188811yescharacteristic
214C22#190614yes—
215C22#204012yes—
216C22#206112yes—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—21no—
221C4—22no—
222C4—22no—
223C4—22no—
224C4—22no—
225C4—88no—
226C4—84no—
227C4—162no—
228C2—22no—
229C2—22no—
230C2#943912yes—
231C2#944212yes—
232C2#1121111yescharacteristic
233C2#1121512yes—
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #1836C4semidirect
95. #97C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#207652
#141682
C2×#13002
#80302
#80222