Finite Group Databasebeta

← #120539

Subgroups — #120539

Order 512.

453 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 136 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#21972C211yesmaximal, characteristic
2#17819C211yesmaximal, characteristic
3#8082C212yesmaximal
4#8006C211yesmaximal, characteristic
5#7856C212yesmaximal
6#2920C2212yes—
7#2920C2212yes—
8C2×#431C2211yescharacteristic
9C22×C4⋊C8C2211yescharacteristic
10C4×#117C2211yescharacteristic
11#1939—22no—
12#1850—22no—
13#1842C412yes—
14#1831C412yes—
15#1574—24no—
16#1568—22no—
17#1301—24no—
18#1293—22no—
19#1274—24no—
20C23×C8—21no—
21C2×#117Q811yescharacteristic
22C2×#117Q811yescharacteristic
23C22×C42C2311yescharacteristic
24#432—22no—
25#432—22no—
26#432—22no—
27#432—22no—
28#431—22no—
29#431—22no—
30#431—22no—
31#431—22no—
32#431D414yes—
33#431Q814yes—
34#423—22no—
35#423—22no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—24no—
40C2×C4⋊C8C2×C412yes—
41C2×C4⋊C8C2×C411yescharacteristic
42C2×C4⋊C8C2×C411yescharacteristic
43#407—22no—
44#407—22no—
45C2×C22⋊C8D412yes—
46C4×M16—22no—
47C4×M16—22no—
48C2×C4×C8D412yes—
49C4×C4⋊C4—21no—
50C4×C4⋊C4—21no—
51C4×C22⋊C4—21no—
52C4×C22⋊C4—21no—
53#376D412yes—
54C43C2×C412yes—
55#336—24no—
56#335—24no—
57#334—24no—
58C23×C4C2×D411yescharacteristic
59C23×C4C2×Q811yescharacteristic
60C23×C4#4111yescharacteristic
61C2×M16—42no—
62C2×M16—42no—
63C22×C8—22no—
64C22×C8—42no—
65C22×C8—82no—
66C22×C8C22⋊C412yes—
67C22×C8C22⋊C412yes—
68#117—24no—
69#117—42no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C42—22no—
75C2×C42—22no—
76C2×C42C4⋊C412yes—
77C2×C42C4⋊C412yes—
78C2×C42C22×C411yescharacteristic
79C2×C42C2×D412yes—
80C2×C42C2×D411yescharacteristic
81C2×C42C2×D411yescharacteristic
82C2×C42C2×Q812yes—
83C2×C42#4112yes—
84C2×C42#4112yes—
85C2×C42#4111yescharacteristic
86C4⋊C8—22no—
87C4⋊C8—22no—
88C4⋊C8—24no—
89C4⋊C8—24no—
90C4⋊C8—24no—
91C4⋊C8—41no—
92C4⋊C8—42no—
93C4⋊C8—42no—
94C4⋊C8—41no—
95C4⋊C8—84no—
96C22⋊C8—24no—
97C22⋊C8—24no—
98C22⋊C8—24no—
99C22⋊C8—42no—
100C22⋊C8—42no—
101#97—42no—
102#97—42no—
103C4×C8—22no—
104C4×C8—24no—
105C4×C8—24no—
106C4×C8—22no—
107#95—22no—
108#95—22no—
109C24C22⋊Q811yescharacteristic
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4—81no—
117C22×C4C2×C22⋊C411yescharacteristic
118C22×C4C2×C4⋊C411yescharacteristic
119C22×C4#11711yescharacteristic
120C22×C4#12112yes—
121C22×C4#12111yescharacteristic
122C22×C4#12111yescharacteristic
123C22×C4C22⋊Q811yescharacteristic
124C22×C4C22⋊Q812yes—
125C22×C4#12312yes—
126C22×C4#12512yes—
127C22×C4#12512yes—
128C22×C4C4⋊Q812yes—
129M16—82no—
130M16—82no—
131M16—84no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—42no—
135C2×C8—42no—
136C2×C8—44no—
137C2×C8—42no—
138C2×C8—44no—
139C2×C8—44no—
140C2×C8—84no—
141C4⋊C4—24no—
142C4⋊C4—42no—
143C22⋊C4—24no—
144C22⋊C4—42no—
145C42—22no—
146C42—24no—
147C42—42no—
148C42—42no—
149C42—82no—
150C42#10814yes—
151C42C4×D412yes—
152C42C4×Q812yes—
153C42#12011yescharacteristic
154C42#12112yes—
155C42C22⋊Q812yes—
156C42C22⋊Q812yes—
157C42#12311yescharacteristic
158C42#12411yescharacteristic
159C42#12612yes—
160C42C4⋊Q811yescharacteristic
161C23—21no—
162C23—21no—
163C23—41no—
164C23—41no—
165C23#38011yescharacteristic
166C23#39812yes—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4#38611yescharacteristic
188C2×C4#38712yes—
189C2×C4#38912yes—
190C2×C4#39111yescharacteristic
191C2×C4#39411yescharacteristic
192C2×C4#39511yescharacteristic
193C2×C4#39612yes—
194C2×C4#39811yescharacteristic
195C2×C4#40011yescharacteristic
196C2×C4C2×#10812yes—
197C2×C4#43012yes—
198C8—42no—
199C8—42no—
200C8—84no—
201C8—84no—
202C8—82no—
203C22—21no—
204C22—21no—
205C22—22no—
206C22—22no—
207C22—22no—
208C22—42no—
209C22—42no—
210C22#144111yescharacteristic
211C22#181011yescharacteristic
212C22#181111yescharacteristic
213C4—22no—
214C4—21no—
215C4—21no—
216C4—21no—
217C4—21no—
218C4—22no—
219C4—21no—
220C4—44no—
221C4—82no—
222C4#190911yescharacteristic
223C4#191011yescharacteristic
224C4#195014yes—
225C2—21no—
226C2—21no—
227C2—42no—
228C2#1122511yescharacteristic
229C2#1123311yescharacteristic
230C2#1124511yescharacteristic
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
#219722
#178192
#80822
#80062
#78562