Finite Group Databasebeta

← #120549

Subgroups — #120549

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#17815C211yesmaximal, characteristic
3#8082C212yesmaximal
4#8006C211yesmaximal, characteristic
5#7976C212yesmaximal
6#2920C2212yes—
7#2920C2212yes—
8C2×#431C2211yescharacteristic
9C22×C4⋊C8C2211yescharacteristic
10#2283C2211yescharacteristic
11#1939—21no—
12#1939—21no—
13#1850—21no—
14#1850—21no—
15#1843C412yes—
16#1832C412yes—
17#1572—24no—
18#1567—21no—
19#1566—21no—
20#1301—24no—
21#1293—21no—
22#1293—21no—
23#1288—24no—
24C23×C8—21no—
25C22×C42C2311yescharacteristic
26#432—22no—
27#432—22no—
28#432—22no—
29#432Q814yes—
30#431—22no—
31#431—22no—
32#431—22no—
33#431—22no—
34#431—22no—
35#431D414yes—
36#423—22no—
37#423—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—22no—
43C2×C4⋊C8C2×C412yes—
44C2×C4⋊C8C2×C411yescharacteristic
45C2×C4⋊C8C2×C411yescharacteristic
46#407—22no—
47#407—22no—
48C2×C22⋊C8D411yescharacteristic
49C2×C22⋊C8D411yescharacteristic
50C4×M16—22no—
51C4×M16—22no—
52C2×C4×C8D411yescharacteristic
53C2×C4×C8D411yescharacteristic
54#383D412yes—
55#382—21no—
56#381—21no—
57#380Q812yes—
58C4×C4⋊C4—21no—
59C4×C22⋊C4—21no—
60#376C2×C412yes—
61#336—22no—
62#336—22no—
63#335—24no—
64#334—24no—
65C23×C4C2×D411yescharacteristic
66C23×C4C2×Q811yescharacteristic
67C23×C4#4111yescharacteristic
68C2×M16—42no—
69C2×M16—42no—
70C22×C8—22no—
71C22×C8—42no—
72C22×C8—82no—
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C2×C4⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C42—22no—
80C2×C42C22×C411yescharacteristic
81C2×C42C2×D412yes—
82C2×C42C2×D411yescharacteristic
83C2×C42C2×D411yescharacteristic
84C2×C42C2×Q812yes—
85C2×C42#4112yes—
86C2×C42#4112yes—
87C2×C42#4111yescharacteristic
88C4⋊C8—22no—
89C4⋊C8—22no—
90C4⋊C8—24no—
91C4⋊C8—24no—
92C4⋊C8—24no—
93C4⋊C8—41no—
94C4⋊C8—42no—
95C4⋊C8—41no—
96C4⋊C8—42no—
97C4⋊C8—82no—
98C4⋊C8—82no—
99C22⋊C8—22no—
100C22⋊C8—22no—
101C22⋊C8—24no—
102C22⋊C8—24no—
103C22⋊C8—42no—
104C22⋊C8—42no—
105#97—24no—
106#97—42no—
107C4×C8—21no—
108C4×C8—21no—
109C4×C8—24no—
110C4×C8—21no—
111C4×C8—21no—
112C4×C8—42no—
113#95—22no—
114#95—22no—
115#95—22no—
116#95C4⋊C414yes—
117C24C22⋊Q811yescharacteristic
118C22×C4—24no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4C2×C22⋊C411yescharacteristic
124C22×C4C2×C4⋊C411yescharacteristic
125C22×C4#11711yescharacteristic
126C22×C4#12112yes—
127C22×C4#12111yescharacteristic
128C22×C4#12111yescharacteristic
129C22×C4C22⋊Q811yescharacteristic
130C22×C4C22⋊Q812yes—
131C22×C4#12312yes—
132C22×C4#12512yes—
133C22×C4#12512yes—
134C22×C4C4⋊Q812yes—
135M16—84no—
136M16—82no—
137M16—82no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—42no—
147C2×C8—42no—
148C2×C8—42no—
149C2×C8—44no—
150C2×C8—44no—
151C2×C8—84no—
152C4⋊C4—44no—
153C22⋊C4—44no—
154C42—42no—
155C42—42no—
156C42C4×D412yes—
157C42C4×Q812yes—
158C42#12011yescharacteristic
159C42#12112yes—
160C42C22⋊Q812yes—
161C42C22⋊Q812yes—
162C42#12311yescharacteristic
163C42#12411yescharacteristic
164C42#12612yes—
165C42C4⋊Q811yescharacteristic
166C23—22no—
167C23—41no—
168C23—41no—
169C23#38011yescharacteristic
170C23#39812yes—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—44no—
186C2×C4—44no—
187C2×C4—44no—
188C2×C4#38611yescharacteristic
189C2×C4#38712yes—
190C2×C4#38912yes—
191C2×C4#39111yescharacteristic
192C2×C4#39411yescharacteristic
193C2×C4#39511yescharacteristic
194C2×C4#39612yes—
195C2×C4#39811yescharacteristic
196C2×C4#40011yescharacteristic
197C8—41no—
198C8—41no—
199C8—41no—
200C8—41no—
201C8—84no—
202C8—84no—
203C8—82no—
204C22—21no—
205C22—21no—
206C22—22no—
207C22—22no—
208C22—22no—
209C22—44no—
210C22#144111yescharacteristic
211C22#181412yes—
212C4—22no—
213C4—21no—
214C4—22no—
215C4—21no—
216C4—21no—
217C4—22no—
218C4—21no—
219C4—84no—
220C4#190911yescharacteristic
221C4#191011yescharacteristic
222C4#210312yes—
223C2—22no—
224C2—42no—
225C2#1122511yescharacteristic
226C2#1123711yescharacteristic
227C2#1124111yescharacteristic
subgroupG/H
center, socleC22#1441
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ 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
#178152
#80822
#80062
#79762