Finite Group Databasebeta

← #140911

Subgroups — #140911

Order 512.

528 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 134 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
2C4×C22⋊C16C211yesmaximal, characteristic
3C2×#1348C214yesmaximal
4#10862C211yesmaximal, characteristic
5C2×#494C411yescharacteristic, complemented
6C2×#467—21no—
7C2×#467C2212yes—
8C22×#107—21no—
9C22×C4×C8C2211yescharacteristic
10#2857—21no—
11C2×C22⋊C16C2212yes—
12C2×C4×C16—21no—
13#1969—21no—
14#1969C412yescomplemented
15#1956—21no—
16#1845C411yescharacteristic, complemented
17C2×#340C2212yes—
18C4⋊#95—21no—
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—24no—
36C2×#107—21no—
37C2×#107—42no—
38C2×#107—42no—
39C2×#107C2×C412yes—
40C2×#103—22no—
41C2×#103—22no—
42C2×C4×C8—42no—
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47#391—22no—
48#389—42no—
49#384—21no—
50#380—42no—
51C2×#95—42no—
52C22⋊C16—22no—
53C22⋊C16D414yes—
54C4×C16—42no—
55#340—24no—
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—22no—
67C22×C8—22no—
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—41no—
96C2×C42—41no—
97C2×C42C22⋊C411yescharacteristic
98C2×C42C22⋊C411yescharacteristic
99C2×C42SD1612yes—
100C2×C16—24no—
101C2×C16—42no—
102C2×C16—42no—
103#107—24no—
104#107—48no—
105#107—42no—
106#103—44no—
107#103—44no—
108C4×C8—22no—
109C4×C8—22no—
110C4×C8—41no—
111C4×C8—41no—
112#95—44no—
113#95—84no—
114C24C2×D811yescharacteristic
115C2×Q8—42no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—42no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—42no—
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×C4SD3214yes—
137C22×C4Q3214yes—
138C22×C4C2×C22⋊C411yescharacteristic
139C22×C4#12411yescharacteristic
140C22×C4#12711yescharacteristic
141C22×C4C2×D811yescharacteristic
142C22×C4C2×SD1611yescharacteristic
143C22×C4C2×SD1611yescharacteristic
144Q16—84no—
145Q16—84no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—24no—
153C2×C8—24no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8#9912yes—
157C2×C8#10212yes—
158C2×C8#10212yes—
159C2×C8#10412yes—
160C2×C8C4×D412yes—
161C2×C8#12112yes—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—84no—
167C4⋊C4—82no—
168C4⋊C4—82no—
169C4⋊C4—84no—
170C4⋊C4—82no—
171C4⋊C4—82no—
172C22⋊C4—82no—
173C22⋊C4—82no—
174C42—21no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—42no—
179C16—44no—
180C23—22no—
181C23—21no—
182C23—21no—
183C23C2×#10211yescharacteristic
184C23#48611yescharacteristic
185C23#49311yescharacteristic
186C23C2×SD3212yes—
187C23C2×Q3212yes—
188Q8—82no—
189Q8—82no—
190Q8—82no—
191Q8—82no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—84no—
209C2×C4—88no—
210C2×C4—84no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—164no—
216C2×C4—162no—
217C2×C4—162no—
218C2×C4#35814yes—
219C2×C4#35814yes—
220C2×C4#39011yescharacteristic
221C2×C4C2×#9911yescharacteristic
222C2×C4#41011yescharacteristic
223C2×C4C2×#10211yescharacteristic
224C2×C4C2×#10411yescharacteristic
225C2×C4#42111yescharacteristic
226C2×C4#48611yescharacteristic
227C2×C4#49211yescharacteristic
228C8—22no—
229C8—22no—
230C8—22no—
231C8#32714yes—
232C22—22no—
233C22—22no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—22no—
239C22—22no—
240C22#150211yescharacteristic
241C22#151911yescharacteristic
242C22#195711yescharacteristic
243C22C2×#35812yes—
244C22#214512yes—
245C22#224112yes—
246C22#224711yescharacteristic
247C22#224811yescharacteristic
248C4—24no—
249C4—24no—
250C4—21no—
251C4—21no—
252C4—164no—
253C4—162no—
254C4—162no—
255C4C2×#32711yescharacteristic
256C4#151711yescharacteristic
257C4#196411yescharacteristic
258C4#198011yescharacteristic
259C2—21no—
260C2—22no—
261C2—21no—
262C2#968011yescharacteristic
263C2#1174312yes—
264C2#1188712yes—
265C2#1320311yescharacteristic
266C2#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
C4×C22⋊C162
C2×#13482
#108622