Finite Group Databasebeta

← #120150

Subgroups — #120150

Order 512.

520 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 154 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#21952C211yesmaximal, characteristic
2#20764C211yesmaximal, characteristic
3C2×#1299C212yesmaximal
4#8043C212yesmaximal
5#8041C211yesmaximal, characteristic
6C2×#431C2212yes—
7C2×#423C2212yes—
8C2×C4×M16C2211yescharacteristic
9#2285C2211yescharacteristic
10C2×#384C2211yescharacteristic
11#1939—24no—
12#1836C414yes—
13#1834—22no—
14#1798—22no—
15#1736—22no—
16#1299—24no—
17#1296—24no—
18#1296—22no—
19C22×M16—21no—
20C22×M16—22no—
21C2×#117D412yes—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—22no—
24C22×C42C2311yescharacteristic
25#431—22no—
26#431—22no—
27#431—22no—
28#431—22no—
29#423—22no—
30#423—22no—
31#423—22no—
32#423Q814yes—
33C2×C4⋊C8—22no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8—24no—
36C2×C4⋊C8D414yes—
37C2×C22⋊C8—22no—
38C4×M16—21no—
39C4×M16—21no—
40C4×M16D412yes—
41C4×M16Q812yes—
42C2×#97C2×C412yes—
43C2×C4×C8—21no—
44C2×C4×C8—22no—
45#384—22no—
46#384—21no—
47#384C2×C412yes—
48#384D412yes—
49#382C2×C412yes—
50#380—22no—
51C4×C4⋊C4—22no—
52#340—28no—
53#336—24no—
54C23×C4—41no—
55C23×C4C2×D411yescharacteristic
56C23×C4#4112yes—
57C2×M16—22no—
58C2×M16—22no—
59C2×M16—24no—
60C2×M16—42no—
61C2×M16—42no—
62C2×M16—42no—
63C2×M16—82no—
64C2×M16—82no—
65C22×C8—22no—
66C22×C8—24no—
67C22×C8—42no—
68#117—44no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—44no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—82no—
77C2×C4⋊C4—82no—
78C2×C4⋊C4C22⋊C414yes—
79C2×C22⋊C4—22no—
80C2×C42—22no—
81C2×C42C22×C411yescharacteristic
82C2×C42C2×D411yescharacteristic
83C2×C42C2×D411yescharacteristic
84C2×C42C2×D412yes—
85C2×C42C2×Q811yescharacteristic
86C2×C42C2×Q812yes—
87C2×C42#4111yescharacteristic
88C2×C42#4111yescharacteristic
89C2×C42#4112yes—
90C4⋊C8—24no—
91C4⋊C8—24no—
92C4⋊C8—24no—
93C4⋊C8—24no—
94C4⋊C8—42no—
95C4⋊C8—42no—
96C4⋊C8—84no—
97C22⋊C8—42no—
98C22⋊C8—42no—
99#97—22no—
100#97C4⋊C414yes—
101C4×C8—22no—
102C4×C8—22no—
103C4×C8—42no—
104C4×C8—42no—
105#95—24no—
106C24#12311yescharacteristic
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—24no—
114C22×C4—42no—
115C22×C4—82no—
116C22×C4—84no—
117C22×C4C2×C22⋊C411yescharacteristic
118C22×C4C4×D412yes—
119C22×C4#12011yescharacteristic
120C22×C4#12111yescharacteristic
121C22×C4#12112yes—
122C22×C4C22⋊Q811yescharacteristic
123C22×C4C22⋊Q812yes—
124C22×C4#12411yescharacteristic
125C22×C4#12512yes—
126C22×C4#12512yes—
127C22×C4#12612yes—
128C22×C4C4⋊Q811yescharacteristic
129M16—44no—
130M16—44no—
131M16—84no—
132M16—84no—
133C2×C8—24no—
134C2×C8—42no—
135C2×C8—42no—
136C2×C8—44no—
137C2×C8—44no—
138C2×C8—44no—
139C2×C8—44no—
140C2×C8—82no—
141C2×C8—82no—
142C4⋊C4—44no—
143C4⋊C4—44no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—84no—
149C4⋊C4—82no—
150C4⋊C4—82no—
151C4⋊C4—82no—
152C22⋊C4—82no—
153C42—44no—
154C42C2×C4⋊C411yescharacteristic
155C42#11711yescharacteristic
156C42C4×Q812yes—
157C42#12111yescharacteristic
158C42#12112yes—
159C42C22⋊Q811yescharacteristic
160C42C22⋊Q812yes—
161C42C22⋊Q812yes—
162C42#12312yes—
163C42C4⋊Q812yes—
164C23—21no—
165C23—21no—
166C23—22no—
167C23—22no—
168C23#38611yescharacteristic
169C23#39811yescharacteristic
170C23#40011yescharacteristic
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—44no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4—82no—
191C2×C4—84no—
192C2×C4—164no—
193C2×C4#38011yescharacteristic
194C2×C4#38712yes—
195C2×C4#38912yes—
196C2×C4#39111yescharacteristic
197C2×C4#39411yescharacteristic
198C2×C4#39511yescharacteristic
199C2×C4#39612yes—
200C2×C4#39812yes—
201C2×C4#45412yes—
202C2×C4#45512yes—
203C2×C4#47112yes—
204C2×C4#47112yes—
205C2×C4#47212yes—
206C2×C4#47312yes—
207C8—44no—
208C8—84no—
209C8—84no—
210C22—21no—
211C22—21no—
212C22—22no—
213C22—22no—
214C22—42no—
215C22—41no—
216C22—41no—
217C22#144111yescharacteristic
218C22#188511yescharacteristic
219C22#188811yescharacteristic
220C22#190712yes—
221C22#190812yes—
222C4—21no—
223C4—21no—
224C4—22no—
225C4—22no—
226C4—84no—
227C4—162no—
228C4#204012yes—
229C4#206112yes—
230C4#206312yes—
231C4#206412yes—
232C4#208912yes—
233C4#209012yes—
234C2—41no—
235C2—41no—
236C2#943312yes—
237C2#943611yescharacteristic
238C2#943811yescharacteristic
239C2#1121111yescharacteristic
240C2#1121612yes—
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ 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
#219522
#207642
C2×#12992
#80432
#80412