Finite Group Databasebeta

← #139587

Subgroups — #139587

Order 512.

368 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 86 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#33486C211yesmaximal, characteristic
2#27659C211yesmaximal, characteristic
3#12869C212yesmaximal, complemented
4#8611C211yesmaximal, characteristic
5#7232C212yesmaximal, complemented
6C2×C8×D4C2211yescharacteristic
7C2×#427C2211yescharacteristic
8#2485—21no—
9D4×C16—24no—
10#2162—21no—
11C2×C22⋊C16—21no—
12C2×C22⋊C16C2212yes—
13#2087C411yescharacteristic
14#2086C411yescharacteristic
15#2076C411yescharacteristic
16#2075C411yescharacteristic
17#1941—21no—
18#1825—21no—
19#1810C2212yes—
20#1807—21no—
21#1807C2211yescharacteristic
22#1465—21no—
23#1382—24no—
24#1293—21no—
25C23×C8D411yescharacteristic
26C23×C8C2311yescharacteristic
27C2×C4×D4D411yescharacteristic
28C2×#117—21no—
29C22×C16—22no—
30C22×C16D414yes—
31C8×D4—22no—
32C8×D4—22no—
33C2×#108—22no—
34#427—22no—
35#427—22no—
36C2×#107C2×C411yescharacteristic
37C2×#107C2×C411yescharacteristic
38C2×#106—21no—
39C2×#106—21no—
40C2×C4⋊C8C2×C411yescharacteristic
41#407—22no—
42C2×C22⋊C8C2×C411yescharacteristic
43C2×C22⋊C8D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45#397—41no—
46#394—41no—
47#389—21no—
48#389—21no—
49#388—41no—
50#387—41no—
51#380—21no—
52#380—21no—
53#380—21no—
54C4⋊C16—22no—
55C22⋊C16—22no—
56C22⋊C16—22no—
57C22⋊C16—22no—
58C4×C16—22no—
59#341—22no—
60#340—21no—
61#340—21no—
62#340—21no—
63#340—21no—
64#340C2×C411yescharacteristic
65#340C2×C411yescharacteristic
66#324—41no—
67#323—41no—
68C22×D4SD1611yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C2×M16—42no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—42no—
75C22×C8—42no—
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78C22×C8C22×C411yescharacteristic
79C22×C8C2×D412yes—
80C22×C8C2×D411yescharacteristic
81C22×C8#4112yes—
82C4×D4—22no—
83C4×D4—22no—
84#117—42no—
85#117—42no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4D811yescharacteristic
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4D811yescharacteristic
98C2×C22⋊C4SD1611yescharacteristic
99C2×C42—41no—
100C2×C42C22⋊C411yescharacteristic
101C2×C16—24no—
102C2×C16—44no—
103C2×C16—82no—
104#108—44no—
105#107—22no—
106#107—22no—
107#106—42no—
108#106—41no—
109#106—41no—
110C4⋊C8—22no—
111C22⋊C8—22no—
112C22⋊C8—22no—
113C22⋊C8—42no—
114C22⋊C8—42no—
115C4×C8—21no—
116C4×C8—21no—
117#95—41no—
118#95—41no—
119#95—41no—
120#95—41no—
121#95—41no—
122C24#10211yescharacteristic
123C24C2×SD1611yescharacteristic
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4#10211yescharacteristic
138C22×C4#10211yescharacteristic
139C22×C4#10211yescharacteristic
140C22×C4C2×C22⋊C411yescharacteristic
141C22×C4#12112yes—
142C22×C4C2×D811yescharacteristic
143C22×C4#13512yes—
144M16—82no—
145M16—82no—
146C2×C8—22no—
147C2×C8—24no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—41no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—81no—
156C2×C8—82no—
157C2×C8C4×D412yes—
158C2×C8#12011yescharacteristic
159C2×C8#12311yescharacteristic
160C4⋊C4—22no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—82no—
170C4⋊C4—161no—
171C22⋊C4—22no—
172C22⋊C4—22no—
173C22⋊C4—82no—
174C22⋊C4—82no—
175C22⋊C4—161no—
176C22⋊C4—161no—
177C42—21no—
178C42—21no—
179C42—82no—
180C16—44no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—21no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23#35912yes—
190C23C2×#10211yescharacteristic
191C23#46512yes—
192D4—28no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4#35912yes—
221C2×C4#38611yescharacteristic
222C2×C4#41611yescharacteristic
223C2×C4#46612yes—
224C8—21no—
225C8—22no—
226C8—41no—
227C8—41no—
228C8—162no—
229C8#45812yes—
230C22—21no—
231C22—21no—
232C22—22no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22—42no—
237C22—41no—
238C22—41no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22#189311yescharacteristic
243C22C2×#35911yescharacteristic
244C22C2×#35911yescharacteristic
245C4—22no—
246C4—22no—
247C4—21no—
248C4—41no—
249C4—41no—
250C4—161no—
251C4—161no—
252C4—161no—
253C4—161no—
254C4#189011yescharacteristic
255C4#190911yescharacteristic
256C2—21no—
257C2—21no—
258C2—42no—
259C2—42no—
260C2#947011yescharacteristic
261C2#1306311yescharacteristic
262C2#1307611yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC22×D4SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #12869C2semidirect
5. #7232C2semidirect

Maximal subgroups

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

subgroupindex
#334862
#276592
#128692
#86112
#72322