Finite Group Databasebeta

← #138531

Subgroups — #138531

Order 512.

426 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 88 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#29770C211yesmaximal, characteristic
2#21960C211yesmaximal, characteristic, complemented
3#11886C212yesmaximal
4C2×#1322C211yesmaximal, characteristic, complemented
5#7230C212yesmaximal, complemented
6C2×#466C2211yescharacteristic
7C22×C4⋊C8C2211yescharacteristic
8#2854—21no—
9#2283—21no—
10C2×C22⋊C16C2212yes—
11#2086C412yescomplemented
12#2067—22no—
13#2054C412yescomplemented
14#1949—24no—
15#1893—21no—
16#1808C2211yescharacteristic
17#1808C2212yes—
18#1807—21no—
19#1347—24no—
20#1322—22no—
21C22×D8C2×C411yescharacteristic
22C23×C8D411yescharacteristic
23C23×C8C2311yescharacteristic
24C2×#123—41no—
25C2×#121—21no—
26C22×C42D411yescharacteristic
27C22×C16—22no—
28#466—22no—
29#466—22no—
30C2×#107—21no—
31C2×#107—22no—
32C2×#107C2×C411yescharacteristic
33C2×#106—22no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8—22no—
36C2×C4⋊C8C2×C412yes—
37C2×C4⋊C8D412yes—
38C2×#102—21no—
39C2×#102—22no—
40#390—22no—
41#389—22no—
42#388—42no—
43#386—41no—
44#383—41no—
45#382—22no—
46#381—41no—
47#380—22no—
48#380—21no—
49C4×C4⋊C4—22no—
50C4×C22⋊C4—41no—
51#376—41no—
52C22⋊C16—22no—
53C22⋊C16—22no—
54#340—22no—
55#340—22no—
56#340C2×C412yes—
57#340D414yes—
58C22×D4—21no—
59C22×D4—41no—
60C23×C4D811yescharacteristic
61C23×C4D811yescharacteristic
62C23×C4C2×D411yescharacteristic
63C2×D8—22no—
64C2×D8—81no—
65C22×C8—21no—
66C22×C8—21no—
67C22×C8—81no—
68C22×C8—81no—
69C22×C8C22⋊C412yes—
70C22×C8C22×C411yescharacteristic
71C22×C8C2×D412yes—
72C22×C8C2×D411yescharacteristic
73C22×C8#4112yes—
74#123—84no—
75#121—41no—
76#121—41no—
77#121—41no—
78#121—41no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—42no—
83C2×C22⋊C4—41no—
84C2×C22⋊C4—42no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—41no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—42no—
93C2×C42C22⋊C412yes—
94C2×C42SD1612yes—
95C2×C16—42no—
96C2×C16—82no—
97#107—22no—
98#107—81no—
99#107—82no—
100#106—82no—
101C4⋊C8—24no—
102C4⋊C8—24no—
103#102—42no—
104#102—82no—
105#95—42no—
106#95—42no—
107#95—42no—
108#95—42no—
109#95—41no—
110#95—42no—
111C24—41no—
112C24C2×D811yescharacteristic
113C2×D4—42no—
114C2×D4—42no—
115C2×D4—41no—
116C2×D4—41no—
117C2×D4—81no—
118C2×D4—81no—
119C2×D4—82no—
120C2×D4—82no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—42no—
130C22×C4—41no—
131C22×C4—44no—
132C22×C4—81no—
133C22×C4#10212yes—
134C22×C4#10212yes—
135C22×C4C2×C22⋊C411yescharacteristic
136C22×C4#12011yescharacteristic
137C22×C4#12311yescharacteristic
138C22×C4C2×SD1611yescharacteristic
139C22×C4AGL(1,ℤ/8)11yescharacteristic
140C22×C4#13711yescharacteristic
141D8—82no—
142C2×C8—22no—
143C2×C8—41no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—41no—
147C2×C8—42no—
148C2×C8—42no—
149C2×C8—81no—
150C2×C8C4×D412yes—
151C2×C8#12112yes—
152C4⋊C4—42no—
153C4⋊C4—44no—
154C4⋊C4—44no—
155C4⋊C4—42no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—84no—
159C22⋊C4—84no—
160C22⋊C4—84no—
161C22⋊C4—82no—
162C22⋊C4—82no—
163C22⋊C4—82no—
164C22⋊C4—81no—
165C22⋊C4—81no—
166C42—22no—
167C42—22no—
168C42—22no—
169C42—22no—
170C42—84no—
171C16—82no—
172C23—41no—
173C23—41no—
174C23—82no—
175C23—82no—
176C23—81no—
177C23—81no—
178C23—81no—
179C23C2×#10211yescharacteristic
180C23#44711yescharacteristic
181C23#48011yescharacteristic
182C23#50812yes—
183C23#50912yes—
184D4—82no—
185D4—81no—
186D4—81no—
187D4—162no—
188D4—162no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—81no—
207C2×C4—84no—
208C2×C4—82no—
209C2×C4—84no—
210C2×C4—82no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—82no—
214C2×C4—81no—
215C2×C4—84no—
216C2×C4#38611yescharacteristic
217C2×C4#41711yescharacteristic
218C2×C4#44811yescharacteristic
219C2×C4#48111yescharacteristic
220C8—22no—
221C8—22no—
222C8—42no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—42no—
230C22—42no—
231C22—162no—
232C22—162no—
233C22—161no—
234C22—161no—
235C22—161no—
236C22#187411yescharacteristic
237C22#213811yescharacteristic
238C22#214011yescharacteristic
239C22#218612yes—
240C22#223412yes—
241C4—21no—
242C4—21no—
243C4—21no—
244C4—42no—
245C4—42no—
246C4—162no—
247C4—164no—
248C4—161no—
249C4#191211yescharacteristic
250C4#191311yescharacteristic
251C2—21no—
252C2—22no—
253C2—21no—
254C2—22no—
255C2—321no—
256C2#920411yescharacteristic
257C2#1299211yescharacteristic
258C2#1300811yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ 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. #21960C2semidirect
4. C2×#1322C2semidirect
5. #7230C2semidirect
11. #2086C4semidirect
13. #2054C4semidirect

Maximal subgroups

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

subgroupindex
#297702
#219602
#118862
C2×#13222
#72302