Finite Group Databasebeta

← #140902

Subgroups — #140902

Order 512.

514 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 118 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#33172C211yesmaximal, characteristic
2#12599C211yesmaximal, characteristic
3C2×#1348C214yesmaximal
4#10842C211yesmaximal, characteristic
5C2×#494C411yescharacteristic, complemented
6C2×#467—21no—
7C2×#467C2212yes—
8C22×C4×C8C2211yescharacteristic
9#2857—21no—
10C2×#382—21no—
11C2×C22⋊C16C2212yes—
12C2×#346—21no—
13#1969—21no—
14#1969C412yescomplemented
15#1956—21no—
16#1845C411yescharacteristic, complemented
17C2×#340—21no—
18C2×#340C2212yes—
19#1374—21no—
20#1348—28no—
21C22×Q16—21no—
22C22×Q16C2×C412yes—
23C23×C8D411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×C4⋊Q8—21no—
26C2×C22⋊Q8—22no—
27C22×C4⋊C4—22no—
28C22×C42D411yescharacteristic
29C22×C16—22no—
30#494—42no—
31#467—24no—
32#467—24no—
33#467—42no—
34#467—42no—
35C2×#107—21no—
36C2×#107C2×C412yes—
37C2×#103—22no—
38C2×#103—22no—
39C2×C4×C8—42no—
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44#391—22no—
45#389—42no—
46#384—21no—
47#382—24no—
48#382—42no—
49#380—42no—
50C2×#95—42no—
51C22⋊C16—22no—
52C22⋊C16D414yes—
53#346—42no—
54#340—24no—
55#340—42no—
56C22×Q8—22no—
57C23×C4—42no—
58C23×C4D812yes—
59C23×C4C2×D411yescharacteristic
60C2×Q16—24no—
61C2×Q16—42no—
62C2×Q16—42no—
63C2×Q16—42no—
64C2×Q16—42no—
65C2×Q16—42no—
66C22×C8—41no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8—41no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8D812yes—
73C22×C8SD1612yes—
74C22×C8C22×C411yescharacteristic
75C22×C8C2×D411yescharacteristic
76C22×C8C2×D412yes—
77C22×C8#4112yes—
78C4⋊Q8—42no—
79C4⋊Q8—42no—
80C22⋊Q8—42no—
81C22⋊Q8—42no—
82C22⋊Q8—42no—
83C22⋊Q8—42no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C42—22no—
96C2×C42—22no—
97C2×C42C22⋊C411yescharacteristic
98C2×C42C22⋊C411yescharacteristic
99C2×C42SD1612yes—
100C2×C16—24no—
101C2×C16—42no—
102C2×C16—42no—
103#107—24no—
104#107—42no—
105#103—44no—
106#103—44no—
107C4×C8—22no—
108C4×C8—22no—
109C4×C8—41no—
110C4×C8—41no—
111#95—48no—
112#95—44no—
113C24C2×D811yescharacteristic
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—42no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—24no—
125C22×C4—24no—
126C22×C4—42no—
127C22×C4—44no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—82no—
131C22×C4—84no—
132C22×C4—82no—
133C22×C4—84no—
134C22×C4#10212yes—
135C22×C4#10212yes—
136C22×C4C2×C22⋊C411yescharacteristic
137C22×C4#12411yescharacteristic
138C22×C4#12711yescharacteristic
139C22×C4C2×D811yescharacteristic
140C22×C4C2×SD1611yescharacteristic
141C22×C4C2×SD1611yescharacteristic
142Q16—84no—
143Q16—84no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—42no—
153C2×C8—42no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8#10212yes—
157C2×C8#10212yes—
158C2×C8C4×D412yes—
159C2×C8#12112yes—
160C4⋊C4—44no—
161C4⋊C4—44no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—84no—
165C4⋊C4—82no—
166C4⋊C4—82no—
167C4⋊C4—84no—
168C4⋊C4—82no—
169C4⋊C4—82no—
170C22⋊C4—82no—
171C22⋊C4—82no—
172C42—24no—
173C42—21no—
174C42—21no—
175C42#9912yes—
176C42#10412yes—
177C16—44no—
178C23—22no—
179C23—21no—
180C23—21no—
181C23C2×#10211yescharacteristic
182C23#48611yescharacteristic
183C23#49311yescharacteristic
184C23#51014yes—
185Q8—82no—
186Q8—82no—
187Q8—82no—
188Q8—82no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—48no—
205C2×C4—84no—
206C2×C4—88no—
207C2×C4—84no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—164no—
213C2×C4—162no—
214C2×C4—162no—
215C2×C4#32714yes—
216C2×C4#39011yescharacteristic
217C2×C4C2×#9911yescharacteristic
218C2×C4#41011yescharacteristic
219C2×C4C2×#10211yescharacteristic
220C2×C4C2×#10411yescharacteristic
221C2×C4#42111yescharacteristic
222C2×C4#48611yescharacteristic
223C2×C4#49211yescharacteristic
224C8—22no—
225C8—22no—
226C8—22no—
227C8—22no—
228C22—24no—
229C22—21no—
230C22—21no—
231C22—24no—
232C22—21no—
233C22—21no—
234C22C2×#32711yescharacteristic
235C22#151711yescharacteristic
236C22#195711yescharacteristic
237C22#214112yes—
238C22#214512yes—
239C22#224512yes—
240C22#225112yes—
241C4—24no—
242C4—21no—
243C4—21no—
244C4—42no—
245C4—164no—
246C4—162no—
247C4—162no—
248C4#150211yescharacteristic
249C4#151911yescharacteristic
250C4#196411yescharacteristic
251C4#198011yescharacteristic
252C2—22no—
253C2—22no—
254C2#968011yescharacteristic
255C2#1170612yes—
256C2#1185912yes—
257C2#1320711yescharacteristic
258C2#1321111yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. C2×#494C4semidirect
14. #1969C4semidirect
16. #1845C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#331722
#125992
C2×#13482
#108422