Finite Group Databasebeta

← #120530

Subgroups — #120530

Order 512.

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

normal subgroups only

This group has 451 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#21976C211yesmaximal, characteristic
2#21969C211yesmaximal, characteristic
3#7995C212yesmaximal
4C2×#1290C211yesmaximal, characteristic
5#7949C212yesmaximal
6#2920C2212yes—
7C2×#432C2212yes—
8C2×#432C2211yescharacteristic
9C22×C4⋊C8C2211yescharacteristic
10C4×#117C2211yescharacteristic
11#1950—24no—
12#1940—22no—
13#1850—22no—
14#1842C412yes—
15#1831C412yes—
16#1810—22no—
17#1291—24no—
18#1290—22no—
19#1285—24no—
20C23×C8D412yes—
21C2×#117—21no—
22C22×C42C2311yescharacteristic
23#432—22no—
24#432—22no—
25#432—22no—
26#432—22no—
27#432—22no—
28#432—22no—
29#432—22no—
30#432Q814yes—
31#431—22no—
32#431—22no—
33C2×#108—24no—
34#423—22no—
35C2×C4⋊C8—22no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—24no—
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C4⋊C8C2×C411yescharacteristic
41C2×C4⋊C8C2×C412yes—
42C2×C4⋊C8D412yes—
43#407—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—21no—
46C4×M16D414yes—
47C2×#97—22no—
48C2×#97—21no—
49C4×C4⋊C4—21no—
50C4×C4⋊C4D412yes—
51C4×C22⋊C4—21no—
52C4×C22⋊C4Q811yescharacteristic
53C4×C22⋊C4Q811yescharacteristic
54#376—21no—
55C43C2×C412yes—
56#341—24no—
57#336—24no—
58C23×C4C2×D411yescharacteristic
59C23×C4C2×Q811yescharacteristic
60C23×C4#4111yescharacteristic
61C2×M16—24no—
62C22×C8—22no—
63C22×C8—22no—
64C22×C8—21no—
65C22×C8—21no—
66C22×C8—42no—
67C22×C8—82no—
68C22×C8C22⋊C414yes—
69#117—44no—
70C2×C4⋊C4—22no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C42—22no—
74C2×C42—22no—
75C2×C42C4⋊C412yes—
76C2×C42C4⋊C412yes—
77C2×C42C22×C411yescharacteristic
78C2×C42C2×D412yes—
79C2×C42C2×D411yescharacteristic
80C2×C42C2×D411yescharacteristic
81C2×C42C2×Q812yes—
82C2×C42#4112yes—
83C2×C42#4112yes—
84C2×C42#4111yescharacteristic
85#108—84no—
86C4⋊C8—24no—
87C4⋊C8—22no—
88C4⋊C8—22no—
89C4⋊C8—24no—
90C4⋊C8—24no—
91C4⋊C8—42no—
92C4⋊C8—42no—
93C4⋊C8—42no—
94C4⋊C8—84no—
95C22⋊C8—24no—
96C22⋊C8—41no—
97C22⋊C8—42no—
98C22⋊C8—42no—
99C22⋊C8—41no—
100C22⋊C8—42no—
101#97—24no—
102#97—24no—
103#97—42no—
104#97—42no—
105C4×C8—24no—
106#95—22no—
107#95—21no—
108#95—21no—
109C24C22⋊Q811yescharacteristic
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4C2×C22⋊C411yescharacteristic
117C22×C4C2×C4⋊C411yescharacteristic
118C22×C4#11711yescharacteristic
119C22×C4#12011yescharacteristic
120C22×C4#12111yescharacteristic
121C22×C4#12111yescharacteristic
122C22×C4C22⋊Q811yescharacteristic
123C22×C4C22⋊Q812yes—
124C22×C4#12311yescharacteristic
125C22×C4#12411yescharacteristic
126C22×C4#12512yes—
127C22×C4#12612yes—
128C22×C4C4⋊Q812yes—
129C22×C4C4⋊Q811yescharacteristic
130M16—44no—
131M16—44no—
132C2×C8—24no—
133C2×C8—44no—
134C2×C8—44no—
135C2×C8—42no—
136C2×C8—42no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—44no—
140C2×C8—44no—
141C2×C8—82no—
142C2×C8—84no—
143C4⋊C4—44no—
144C22⋊C4—24no—
145C22⋊C4—42no—
146C42—42no—
147C42—42no—
148C42—42no—
149C42—42no—
150C42—44no—
151C42C4×D412yes—
152C42C4×Q812yes—
153C42#12112yes—
154C42#12112yes—
155C42C22⋊Q812yes—
156C42C22⋊Q812yes—
157C42#12312yes—
158C42#12512yes—
159C23—22no—
160C23—22no—
161C23—41no—
162C23#38011yescharacteristic
163C23#39511yescharacteristic
164C23#40011yescharacteristic
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4#38611yescharacteristic
189C2×C4#38712yes—
190C2×C4#38912yes—
191C2×C4#39111yescharacteristic
192C2×C4#39411yescharacteristic
193C2×C4#39612yes—
194C2×C4#39812yes—
195C2×C4#39811yescharacteristic
196C8—44no—
197C8—84no—
198C8—84no—
199C8—81no—
200C8—81no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—44no—
209C22#144111yescharacteristic
210C22#181312yes—
211C22#210212yes—
212C4—21no—
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—22no—
218C4—22no—
219C4—44no—
220C4—82no—
221C4#191111yescharacteristic
222C4#191211yescharacteristic
223C2—22no—
224C2—22no—
225C2—41no—
226C2#1122411yescharacteristic
227C2#1123011yescharacteristic
228C2#1124411yescharacteristic
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
#219762
#219692
#79952
C2×#12902
#79492