Finite Group Databasebeta

← #138285

Subgroups — #138285

Order 512.

569 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 569 conjugacy classes of subgroups — too many to draw. Only its 110 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#33129C211yesmaximal, characteristic
2C2×C4×#106C211yesmaximal, characteristic, complemented
3#12844C211yesmaximal, characteristic, complemented
4C2×#1347C212yesmaximal
5#7258C212yesmaximal, complemented
6C2×#466C2211yescharacteristic
7C22×#106—21no—
8C22×C4×C8C2211yescharacteristic
9#2854—21no—
10C2×C4×C4⋊C4—21no—
11C2×C22⋊C16C2212yes—
12#1970C412yescomplemented
13#1942C412yescomplemented
14#1934—22no—
15#1893—21no—
16#1808C2211yescharacteristic
17C4×#106—24no—
18C2×#340C2212yes—
19#1374—22no—
20#1347—24no—
21C22×D8C2×C411yescharacteristic
22C23×C8D411yescharacteristic
23C23×C8C2311yescharacteristic
24C2×#123—41no—
25C2×#121—21no—
26C22×C4⋊C4—22no—
27C22×C42—41no—
28C22×C42D411yescharacteristic
29C22×C16—22no—
30#466—22no—
31#466—22no—
32C2×#107C2×C411yescharacteristic
33C2×#107C2×C412yes—
34C2×#106—28no—
35C2×#106—24no—
36C2×#102—22no—
37C2×#102—21no—
38C2×C4×C8—22no—
39C2×C4×C8—22no—
40C2×C4×C8C2×C412yes—
41C2×C4×C8D412yes—
42#390—22no—
43#389—22no—
44#388—42no—
45#386—41no—
46#380—21no—
47C4×C4⋊C4—24no—
48C4×C4⋊C4—42no—
49C2×#95—41no—
50C22⋊C16—22no—
51C22⋊C16—22no—
52#340—22no—
53#340—21no—
54#340—22no—
55#340D414yes—
56C22×D4—21no—
57C22×D4—41no—
58C23×C4—42no—
59C23×C4D811yescharacteristic
60C23×C4D811yescharacteristic
61C23×C4C2×D411yescharacteristic
62C2×D8—22no—
63C2×D8—42no—
64C22×C8—21no—
65C22×C8—21no—
66C22×C8—24no—
67C22×C8—42no—
68C22×C8C22⋊C412yes—
69C22×C8C22×C411yescharacteristic
70C22×C8C2×D411yescharacteristic
71C22×C8C2×D412yes—
72C22×C8#4112yes—
73#123—84no—
74#121—41no—
75#121—41no—
76#121—41no—
77#121—41no—
78C2×C4⋊C4—28no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—44no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C22⋊C4—41no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—42no—
88C2×C22⋊C4—41no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—44no—
94C2×C42—84no—
95C2×C42C22⋊C412yes—
96C2×C42SD1612yes—
97C2×C16—44no—
98C2×C16—42no—
99#107—22no—
100#107—42no—
101#106—216no—
102#102—44no—
103#102—42no—
104C4×C8—24no—
105C4×C8—24no—
106#95—44no—
107#95—42no—
108#95—41no—
109C24—41no—
110C24C2×D811yescharacteristic
111C2×D4—41no—
112C2×D4—42no—
113C2×D4—41no—
114C2×D4—42no—
115C2×D4—81no—
116C2×D4—81no—
117C2×D4—82no—
118C2×D4—82no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—24no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—48no—
129C22×C4—44no—
130C22×C4—42no—
131C22×C4—41no—
132C22×C4—84no—
133C22×C4—82no—
134C22×C4—82no—
135C22×C4#10212yes—
136C22×C4#10212yes—
137C22×C4C2×C22⋊C411yescharacteristic
138C22×C4#12011yescharacteristic
139C22×C4#12311yescharacteristic
140C22×C4C2×SD1611yescharacteristic
141C22×C4AGL(1,ℤ/8)11yescharacteristic
142C22×C4#13711yescharacteristic
143D8—82no—
144C2×C8—24no—
145C2×C8—24no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—42no—
151C2×C8—42no—
152C2×C8#10414yes—
153C2×C8C4×D412yes—
154C2×C8#12112yes—
155C4⋊C4—28no—
156C4⋊C4—44no—
157C4⋊C4—48no—
158C4⋊C4—41no—
159C4⋊C4—42no—
160C4⋊C4—41no—
161C4⋊C4—42no—
162C4⋊C4—84no—
163C22⋊C4—84no—
164C22⋊C4—82no—
165C22⋊C4—84no—
166C22⋊C4—81no—
167C22⋊C4—81no—
168C42—22no—
169C42—22no—
170C42—22no—
171C42—22no—
172C42—88no—
173C16—82no—
174C23—22no—
175C23—22no—
176C23—81no—
177C23—81no—
178C23—81no—
179C23—82no—
180C23—82no—
181C23C2×#10211yescharacteristic
182C23#44711yescharacteristic
183C23#48011yescharacteristic
184C23#50812yes—
185C23#50912yes—
186D4—81no—
187D4—81no—
188D4—82no—
189D4—162no—
190D4—162no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—48no—
208C2×C4—84no—
209C2×C4—88no—
210C2×C4—88no—
211C2×C4—84no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—82no—
215C2×C4—81no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4#36014yes—
219C2×C4#38611yescharacteristic
220C2×C4#41012yes—
221C2×C4#41711yescharacteristic
222C2×C4C2×#10412yes—
223C2×C4#44811yescharacteristic
224C2×C4#48111yescharacteristic
225C8—24no—
226C8—22no—
227C8—22no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—24no—
233C22—24no—
234C22—161no—
235C22—161no—
236C22—161no—
237C22—162no—
238C22—162no—
239C22#150212yes—
240C22#187411yescharacteristic
241C22#213811yescharacteristic
242C22#214011yescharacteristic
243C22C2×#36012yes—
244C22#218612yes—
245C22#223412yes—
246C4—21no—
247C4—21no—
248C4—24no—
249C4—42no—
250C4—88no—
251C4—161no—
252C4—162no—
253C4#150912yes—
254C4#189611yescharacteristic
255C4#189911yescharacteristic
256C2—22no—
257C2—22no—
258C2—321no—
259C2#914811yescharacteristic
260C2#1174112yes—
261C2#1174412yes—
262C2#1299211yescharacteristic
263C2#1300111yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC2×#466C22
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. C2×C4×#106C2semidirect
3. #12844C2semidirect
5. #7258C2semidirect
12. #1970C4semidirect
13. #1942C4semidirect

Maximal subgroups

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

subgroupindex
#331292
C2×C4×#1062
#128442
C2×#13472
#72582