Finite Group Databasebeta

← #120149

Subgroups — #120149

Order 512.

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

normal subgroups only

This group has 564 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#20800C211yesmaximal, characteristic
3C2×#1299C212yesmaximal
4#8043C212yesmaximal
5#8041C211yesmaximal, characteristic
6C2×#431C2212yes—
7C2×#428—22no—
8C2×#423C2212yes—
9C2×C4×M16C2211yescharacteristic
10#2285C2211yescharacteristic
11C2×#384C2211yescharacteristic
12#1939—24no—
13#1847C414yes—
14#1845—21no—
15#1844—21no—
16#1798—22no—
17#1299—24no—
18#1296—24no—
19#1296—21no—
20#1296—21no—
21C22×M16—21no—
22C22×M16—22no—
23C2×#117D412yes—
24C22×C4⋊C4—21no—
25C22×C4⋊C4—22no—
26C22×C42C2311yescharacteristic
27#431—22no—
28#431—22no—
29#431—22no—
30#431—22no—
31#428—48no—
32C2×#107—24no—
33C2×#106—24no—
34#423—22no—
35#423—22no—
36#423—22no—
37#423Q814yes—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—24no—
41C2×C4⋊C8D414yes—
42C2×C22⋊C8—22no—
43C4×M16—21no—
44C4×M16—21no—
45C4×M16D412yes—
46C4×M16Q812yes—
47C2×#97C2×C412yes—
48C2×C4×C8—21no—
49C2×C4×C8—22no—
50#384—22no—
51#384—21no—
52#384C2×C412yes—
53#384D411yescharacteristic
54#384D411yescharacteristic
55#382C2×C412yes—
56#380—22no—
57C4×C4⋊C4—22no—
58#336—24no—
59C23×C4—41no—
60C23×C4C2×D411yescharacteristic
61C23×C4#4112yes—
62C2×M16—24no—
63C2×M16—24no—
64C2×M16—22no—
65C2×M16—22no—
66C2×M16—24no—
67C2×M16—42no—
68C2×M16—82no—
69C2×M16—82no—
70C22×C8—22no—
71C22×C8—24no—
72C22×C8—42no—
73#117—44no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—44no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—82no—
82C2×C4⋊C4—82no—
83C2×C4⋊C4C22⋊C412yes—
84C2×C4⋊C4C22⋊C412yes—
85C2×C22⋊C4—22no—
86C2×C42—22no—
87C2×C42C22×C411yescharacteristic
88C2×C42C2×D411yescharacteristic
89C2×C42C2×D411yescharacteristic
90C2×C42C2×D412yes—
91C2×C42C2×Q811yescharacteristic
92C2×C42C2×Q812yes—
93C2×C42#4111yescharacteristic
94C2×C42#4111yescharacteristic
95C2×C42#4112yes—
96#107—48no—
97#106—48no—
98C4⋊C8—24no—
99C4⋊C8—24no—
100C4⋊C8—24no—
101C4⋊C8—24no—
102C4⋊C8—42no—
103C4⋊C8—42no—
104C4⋊C8—84no—
105C22⋊C8—42no—
106C22⋊C8—42no—
107#97—22no—
108#97C4⋊C414yes—
109C4×C8—22no—
110C4×C8—22no—
111C4×C8—42no—
112C4×C8—42no—
113#95—24no—
114C24#12311yescharacteristic
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—42no—
124C22×C4—82no—
125C22×C4—84no—
126C22×C4C2×C22⋊C411yescharacteristic
127C22×C4C4×D412yes—
128C22×C4#12011yescharacteristic
129C22×C4#12111yescharacteristic
130C22×C4#12112yes—
131C22×C4C22⋊Q811yescharacteristic
132C22×C4C22⋊Q812yes—
133C22×C4#12411yescharacteristic
134C22×C4#12512yes—
135C22×C4#12512yes—
136C22×C4#12612yes—
137C22×C4C4⋊Q811yescharacteristic
138M16—28no—
139M16—28no—
140M16—84no—
141M16—84no—
142C2×C8—24no—
143C2×C8—24no—
144C2×C8—24no—
145C2×C8—44no—
146C2×C8—44no—
147C2×C8—44no—
148C2×C8—44no—
149C2×C8—82no—
150C2×C8—82no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—44no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—84no—
159C4⋊C4—82no—
160C4⋊C4—82no—
161C4⋊C4—82no—
162C22⋊C4—82no—
163C42—44no—
164C42C2×C4⋊C411yescharacteristic
165C42#11711yescharacteristic
166C42C4×Q812yes—
167C42#12111yescharacteristic
168C42#12112yes—
169C42C22⋊Q811yescharacteristic
170C42C22⋊Q812yes—
171C42C22⋊Q812yes—
172C42#12312yes—
173C42C4⋊Q812yes—
174C23—21no—
175C23—21no—
176C23—22no—
177C23—22no—
178C23#38611yescharacteristic
179C23#39811yescharacteristic
180C23#40011yescharacteristic
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—82no—
202C2×C4—84no—
203C2×C4—164no—
204C2×C4#38011yescharacteristic
205C2×C4#38712yes—
206C2×C4#38912yes—
207C2×C4#39111yescharacteristic
208C2×C4#39411yescharacteristic
209C2×C4#39511yescharacteristic
210C2×C4#39612yes—
211C2×C4#39812yes—
212C2×C4#45312yes—
213C2×C4#45612yes—
214C2×C4#47112yes—
215C2×C4#47112yes—
216C2×C4#47212yes—
217C2×C4#47312yes—
218C8—28no—
219C8—84no—
220C8—84no—
221C22—21no—
222C22—21no—
223C22—22no—
224C22—22no—
225C22—42no—
226C22—41no—
227C22—41no—
228C22#144111yescharacteristic
229C22#188611yescharacteristic
230C22#188711yescharacteristic
231C22#190712yes—
232C22#190812yes—
233C4—21no—
234C4—21no—
235C4—22no—
236C4—22no—
237C4—82no—
238C4—82no—
239C4—162no—
240C4#203912yes—
241C4#206012yes—
242C4#206312yes—
243C4#206412yes—
244C4#208912yes—
245C4#209012yes—
246C2—41no—
247C2—41no—
248C2#943412yes—
249C2#943511yescharacteristic
250C2#943711yescharacteristic
251C2#1121011yescharacteristic
252C2#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
#208002
C2×#12992
#80432
#80412