Finite Group Databasebeta

← #140912

Subgroups — #140912

Order 512.

592 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 122 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#33171C211yesmaximal, characteristic
2#12599C211yesmaximal, characteristic, complemented
3C2×#1347C214yesmaximal, complemented
4#10861C211yesmaximal, characteristic, complemented
5C2×#493C411yescharacteristic, complemented
6C2×#466—21no—
7C2×#466C2212yes—
8C22×#106—21no—
9C22×C4×C8C2211yescharacteristic
10#2856—21no—
11C2×C22⋊C16C2212yes—
12C2×#346—21no—
13#1970—21no—
14#1970C412yescomplemented
15#1957—21no—
16#1845C411yescharacteristic, complemented
17C2×#340C2212yes—
18C4⋊#95—21no—
19#1374—21no—
20#1347—28no—
21C22×D8—21no—
22C22×D8C2×C412yes—
23C23×C8D411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×#127—21no—
26C2×#121—22no—
27C2×#121—42no—
28C22×C4⋊C4—22no—
29C22×C42D411yescharacteristic
30C22×C16—22no—
31#493—42no—
32#466—24no—
33#466—24no—
34#466—42no—
35#466—42no—
36C2×#107—21no—
37C2×#107C2×C412yes—
38C2×#106—24no—
39C2×#106—42no—
40C2×#106—42no—
41C2×#102—22no—
42C2×#102—22no—
43C2×C4×C8—42no—
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48#390—22no—
49#390—42no—
50#384—21no—
51C2×#95—42no—
52C22⋊C16—22no—
53C22⋊C16D414yescomplemented
54#346—42no—
55#340—24no—
56C22×D4—22no—
57C22×D4—42no—
58C22×D4—42no—
59C23×C4—42no—
60C23×C4D812yes—
61C23×C4C2×D411yescharacteristic
62C2×D8—24no—
63C2×D8—42no—
64C2×D8—42no—
65C2×D8—42no—
66C2×D8—42no—
67C2×D8—42no—
68C22×C8—22no—
69C22×C8—22no—
70C22×C8—41no—
71C22×C8—41no—
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8D812yes—
75C22×C8SD1612yes—
76C22×C8C22×C411yescharacteristic
77C22×C8C2×D411yescharacteristic
78C22×C8C2×D412yes—
79C22×C8#4112yes—
80#127—42no—
81#127—42no—
82#121—42no—
83#121—42no—
84#121—42no—
85#121—42no—
86#121—88no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C22⋊C4—44no—
96C2×C22⋊C4—42no—
97C2×C42—41no—
98C2×C42—41no—
99C2×C42C22⋊C411yescharacteristic
100C2×C42C22⋊C411yescharacteristic
101C2×C42SD1612yes—
102C2×C16—24no—
103C2×C16—42no—
104C2×C16—42no—
105#107—24no—
106#107—42no—
107#106—48no—
108#102—44no—
109#102—44no—
110C4×C8—22no—
111C4×C8—22no—
112C4×C8—41no—
113C4×C8—41no—
114#95—84no—
115C24—42no—
116C24C2×D811yescharacteristic
117C2×D4—42no—
118C2×D4—42no—
119C2×D4—42no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4—42no—
123C2×D4—84no—
124C2×D4—84no—
125C2×D4—82no—
126C2×D4—82no—
127C2×D4—88no—
128C2×D4—84no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—42no—
136C22×C4—44no—
137C22×C4—42no—
138C22×C4—84no—
139C22×C4—82no—
140C22×C4—82no—
141C22×C4—84no—
142C22×C4#10212yes—
143C22×C4#10212yes—
144C22×C4C2×C22⋊C411yescharacteristic
145C22×C4#12411yescharacteristic
146C22×C4#12711yescharacteristic
147C22×C4C2×D811yescharacteristic
148C22×C4C2×SD1611yescharacteristic
149C22×C4C2×SD1611yescharacteristic
150D8—84no—
151D8—84no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—24no—
159C2×C8—24no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8#9912yes—
163C2×C8#10212yes—
164C2×C8#10212yes—
165C2×C8#10412yes—
166C2×C8C4×D412yes—
167C2×C8#12112yes—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—82no—
173C4⋊C4—82no—
174C4⋊C4—84no—
175C22⋊C4—88no—
176C22⋊C4—82no—
177C22⋊C4—82no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—42no—
183C16—44no—
184C23—22no—
185C23—21no—
186C23—21no—
187C23—84no—
188C23—84no—
189C23—82no—
190C23—82no—
191C23—82no—
192C23C2×#10211yescharacteristic
193C23#48611yescharacteristic
194C23#49311yescharacteristic
195C23#50914yes—
196D4—82no—
197D4—82no—
198D4—82no—
199D4—82no—
200D4—168no—
201D4—164no—
202D4—164no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—48no—
219C2×C4—84no—
220C2×C4—88no—
221C2×C4—84no—
222C2×C4—82no—
223C2×C4—164no—
224C2×C4—162no—
225C2×C4—162no—
226C2×C4#36014yes—
227C2×C4#39011yescharacteristic
228C2×C4C2×#9911yescharacteristic
229C2×C4#41011yescharacteristic
230C2×C4C2×#10211yescharacteristic
231C2×C4C2×#10411yescharacteristic
232C2×C4#42111yescharacteristic
233C2×C4#48611yescharacteristic
234C2×C4#49211yescharacteristic
235C8—22no—
236C8—22no—
237C8—22no—
238C8#32714yes—
239C22—24no—
240C22—21no—
241C22—21no—
242C22—21no—
243C22—21no—
244C22—24no—
245C22—164no—
246C22—162no—
247C22—162no—
248C22—162no—
249C22—164no—
250C22#150211yescharacteristic
251C22#151911yescharacteristic
252C22#195711yescharacteristic
253C22#214012yes—
254C22C2×#36012yes—
255C22#224412yes—
256C22#225012yes—
257C4—24no—
258C4—21no—
259C4—21no—
260C4—42no—
261C4—164no—
262C4—162no—
263C4C2×#32711yescharacteristic
264C4#151711yescharacteristic
265C4#196411yescharacteristic
266C4#198011yescharacteristic
267C2—22no—
268C2—22no—
269C2—322no—
270C2#968011yescharacteristic
271C2#1174412yes—
272C2#1188812yes—
273C2#1320611yescharacteristic
274C2#1321011yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radical#33171C2
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
2. #12599C2semidirect
3. C2×#1347C2semidirect
4. #10861C2semidirect
5. C2×#493C4semidirect
14. #1970C4semidirect
16. #1845C4semidirect
53. C22⋊C16D4semidirect

Maximal subgroups

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

subgroupindex
#331712
#125992
C2×#13472
#108612