Finite Group Databasebeta

← #140908

Subgroups — #140908

Order 512.

550 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 114 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×#1349C214yesmaximal, complemented
4#10844C211yesmaximal, characteristic, complemented
5C2×#493C411yescharacteristic
6C2×#466—21no—
7C2×#466C2212yes—
8C2×#423—21no—
9C22×C4×C8C2211yescharacteristic
10#2856—21no—
11C2×C22⋊C16C2212yes—
12C2×#346—21no—
13#1970—21no—
14#1970C412yes—
15#1957—21no—
16#1845C411yescharacteristic
17C2×#341—21no—
18C2×#341C2212yes—
19#1374—21no—
20#1349—28no—
21C22×D8—21no—
22C22×D8C2×C412yes—
23C22×M16—22no—
24C23×C8D411yescharacteristic
25C23×C8C2311yescharacteristic
26C2×#127—21no—
27C2×#121—22no—
28C2×#121—42no—
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—
38#423—24no—
39#423—42no—
40C2×C4⋊C8—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—
51C22⋊C16—22no—
52C22⋊C16D414yescomplemented
53#346—42no—
54#341—24no—
55#341—42no—
56C22×D4—22no—
57C22×D4—42no—
58C22×D4—42no—
59C23×C4SD1612yes—
60C23×C4C2×D411yescharacteristic
61C2×D8—24no—
62C2×D8—42no—
63C2×D8—42no—
64C2×D8—42no—
65C2×D8—42no—
66C2×D8—42no—
67C2×M16—24no—
68C2×M16—42no—
69C2×M16—42no—
70C2×M16—42no—
71C2×M16—42no—
72C2×M16—42no—
73C22×C8—41no—
74C22×C8—41no—
75C22×C8—41no—
76C22×C8—41no—
77C22×C8—42no—
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22⋊C411yescharacteristic
80C22×C8D812yes—
81C22×C8SD1612yes—
82C22×C8C22×C411yescharacteristic
83C22×C8C2×D411yescharacteristic
84C22×C8C2×D412yes—
85C22×C8#4112yes—
86#127—42no—
87#127—42no—
88#121—42no—
89#121—42no—
90#121—42no—
91#121—42no—
92#121—88no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—42no—
95C2×C22⋊C4—44no—
96C2×C22⋊C4—42no—
97C2×C42—22no—
98C2×C42—22no—
99C2×C42C22⋊C411yescharacteristic
100C2×C42C22⋊C411yescharacteristic
101C2×C42D812yes—
102C2×C16—24no—
103C2×C16—42no—
104C2×C16—42no—
105#107—24no—
106#107—42no—
107C4⋊C8—48no—
108#102—44no—
109#102—44no—
110C4×C8—22no—
111C4×C8—22no—
112C4×C8—41no—
113C4×C8—41no—
114C24—42no—
115C24C2×SD1611yescharacteristic
116C2×D4—42no—
117C2×D4—42no—
118C2×D4—42no—
119C2×D4—42no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4—84no—
123C2×D4—84no—
124C2×D4—88no—
125C2×D4—82no—
126C2×D4—82no—
127C2×D4—84no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—24no—
133C22×C4—24no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4#10212yes—
137C22×C4#10212yes—
138C22×C4C2×C22⋊C411yescharacteristic
139C22×C4#12411yescharacteristic
140C22×C4#12711yescharacteristic
141C22×C4C2×D811yescharacteristic
142C22×C4C2×D811yescharacteristic
143C22×C4C2×SD1611yescharacteristic
144D8—84no—
145D8—84no—
146M16—44no—
147M16—44no—
148M16—82no—
149M16—82no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—44no—
163C2×C8—82no—
164C2×C8—82no—
165C2×C8#10212yes—
166C2×C8#10212yes—
167C2×C8C4×D412yes—
168C2×C8#12112yes—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—84no—
172C22⋊C4—88no—
173C22⋊C4—82no—
174C22⋊C4—82no—
175C42—24no—
176C42—21no—
177C42—21no—
178C42#9912yes—
179C42#10412yes—
180C16—44no—
181C23—22no—
182C23—22no—
183C23—21no—
184C23—21no—
185C23—84no—
186C23—84no—
187C23—82no—
188C23—82no—
189C23—82no—
190C23C2×#10211yescharacteristic
191C23#48611yescharacteristic
192C23#49211yescharacteristic
193D4—82no—
194D4—82no—
195D4—82no—
196D4—82no—
197D4—168no—
198D4—164no—
199D4—164no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—21no—
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—48no—
215C2×C4—84no—
216C2×C4—82no—
217C2×C4#32714yes—
218C2×C4#36114yes—
219C2×C4#39011yescharacteristic
220C2×C4C2×#9911yescharacteristic
221C2×C4#41011yescharacteristic
222C2×C4C2×#10211yescharacteristic
223C2×C4C2×#10411yescharacteristic
224C2×C4#42111yescharacteristic
225C2×C4#48611yescharacteristic
226C2×C4#49311yescharacteristic
227C8—22no—
228C8—22no—
229C8—22no—
230C8—22no—
231C8—84no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—42no—
240C22—164no—
241C22—162no—
242C22—162no—
243C22—164no—
244C22—162no—
245C22C2×#32711yescharacteristic
246C22#151711yescharacteristic
247C22#195711yescharacteristic
248C22#214312yes—
249C22C2×#36112yes—
250C4—24no—
251C4—24no—
252C4—21no—
253C4—21no—
254C4—162no—
255C4#150211yescharacteristic
256C4#151911yescharacteristic
257C4#196411yescharacteristic
258C4#198011yescharacteristic
259C2—41no—
260C2—41no—
261C2—322no—
262C2#968011yescharacteristic
263C2#1171112yes—
264C2#1186412yes—
265C2#1320911yescharacteristic
266C2#1321311yescharacteristic
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×#1349C2semidirect
4. #10844C2semidirect
52. 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×#13492
#108442