Finite Group Databasebeta

← #120548

Subgroups — #120548

Order 512.

455 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 134 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#21969C211yesmaximal, characteristic
3#8065C212yesmaximal
4C2×#1292C211yesmaximal, characteristic
5#7960C212yesmaximal
6#2920C2212yes—
7C2×#431C2211yescharacteristic
8C2×#423C2212yes—
9C22×C4⋊C8C2211yescharacteristic
10#2283C2211yescharacteristic
11#1950—24no—
12#1939—22no—
13#1850—22no—
14#1843C412yes—
15#1832C412yes—
16#1810—22no—
17#1299—24no—
18#1292—22no—
19#1286—22no—
20#1286—22no—
21C22×M16—22no—
22C23×C8D412yes—
23C22×C42C2311yescharacteristic
24#432—22no—
25#432—22no—
26#431—22no—
27#431—22no—
28#431—22no—
29#431—22no—
30C2×#108—24no—
31#423—22no—
32#423—22no—
33#423—22no—
34#423—22no—
35#423Q814yes—
36C2×C4⋊C8—22no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—24no—
41C2×C4⋊C8C2×C411yescharacteristic
42C2×C4⋊C8C2×C411yescharacteristic
43C2×C4⋊C8C2×C412yes—
44C2×C4⋊C8D412yes—
45#407—22no—
46C2×C22⋊C8—21no—
47C4×M16D414yes—
48C2×C4×C8—21no—
49#383—21no—
50#382D411yescharacteristic
51#382D411yescharacteristic
52#381Q812yes—
53#380—21no—
54C4×C4⋊C4—21no—
55C4×C22⋊C4—21no—
56#376C2×C412yes—
57#341—24no—
58#336—24no—
59C23×C4C2×D411yescharacteristic
60C23×C4C2×Q811yescharacteristic
61C23×C4#4111yescharacteristic
62C2×M16—24no—
63C2×M16—24no—
64C2×M16—42no—
65C2×M16—82no—
66C2×M16—82no—
67C22×C8—22no—
68C22×C8—22no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—42no—
72C22×C8—82no—
73C22×C8C22⋊C414yes—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C22⋊C4—22no—
77C2×C42—22no—
78C2×C42C22×C411yescharacteristic
79C2×C42C2×D412yes—
80C2×C42C2×D411yescharacteristic
81C2×C42C2×D411yescharacteristic
82C2×C42C2×Q812yes—
83C2×C42#4112yes—
84C2×C42#4112yes—
85C2×C42#4111yescharacteristic
86#108—84no—
87C4⋊C8—24no—
88C4⋊C8—22no—
89C4⋊C8—22no—
90C4⋊C8—24no—
91C4⋊C8—24no—
92C4⋊C8—24no—
93C4⋊C8—42no—
94C4⋊C8—42no—
95C4⋊C8—42no—
96C4⋊C8—42no—
97C4⋊C8—84no—
98C22⋊C8—41no—
99C22⋊C8—42no—
100C22⋊C8—41no—
101C22⋊C8—42no—
102#97—24no—
103C4×C8—24no—
104C4×C8—42no—
105#95—22no—
106#95—21no—
107#95—21no—
108#95—22no—
109#95C4⋊C412yes—
110#95C4⋊C412yes—
111C24C22⋊Q811yescharacteristic
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—42no—
120C22×C4C2×C22⋊C411yescharacteristic
121C22×C4C2×C4⋊C411yescharacteristic
122C22×C4#11711yescharacteristic
123C22×C4#12011yescharacteristic
124C22×C4#12111yescharacteristic
125C22×C4#12111yescharacteristic
126C22×C4C22⋊Q812yes—
127C22×C4C22⋊Q811yescharacteristic
128C22×C4#12311yescharacteristic
129C22×C4#12411yescharacteristic
130C22×C4#12512yes—
131C22×C4#12612yes—
132C22×C4C4⋊Q812yes—
133C22×C4C4⋊Q811yescharacteristic
134M16—44no—
135M16—44no—
136M16—82no—
137M16—82no—
138M16—84no—
139C2×C8—24no—
140C2×C8—44no—
141C2×C8—44no—
142C2×C8—42no—
143C2×C8—42no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—44no—
147C2×C8—44no—
148C2×C8—82no—
149C2×C8—84no—
150C4⋊C4—42no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C22⋊C4—44no—
154C42—42no—
155C42—42no—
156C42C4×D412yes—
157C42C4×Q812yes—
158C42#12112yes—
159C42#12112yes—
160C42C22⋊Q812yes—
161C42C22⋊Q812yes—
162C42#12312yes—
163C42#12512yes—
164C23—21no—
165C23—21no—
166C23—22no—
167C23—41no—
168C23#38011yescharacteristic
169C23#39511yescharacteristic
170C23#40011yescharacteristic
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—24no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4#38611yescharacteristic
197C2×C4#38712yes—
198C2×C4#38912yes—
199C2×C4#39111yescharacteristic
200C2×C4#39411yescharacteristic
201C2×C4#39612yes—
202C2×C4#39812yes—
203C2×C4#39811yescharacteristic
204C2×C4#47112yes—
205C2×C4#47211yescharacteristic
206C2×C4#47311yescharacteristic
207C8—44no—
208C8—84no—
209C8—84no—
210C8—81no—
211C8—81no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—42no—
220C22—42no—
221C22#144111yescharacteristic
222C22#181511yescharacteristic
223C22#181611yescharacteristic
224C22#210312yes—
225C4—21no—
226C4—21no—
227C4—21no—
228C4—22no—
229C4—22no—
230C4—22no—
231C4—82no—
232C4—82no—
233C4#191111yescharacteristic
234C4#191211yescharacteristic
235C4#206312yes—
236C4#206412yes—
237C2—21no—
238C2—21no—
239C2—22no—
240C2—41no—
241C2#1122411yescharacteristic
242C2#1123711yescharacteristic
243C2#1124211yescharacteristic
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
#219692
#80652
C2×#12922
#79602