Finite Group Databasebeta

← #138294

Subgroups — #138294

Order 512.

509 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 509 conjugacy classes of subgroups — too many to draw. Only its 92 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#33135C211yesmaximal, characteristic
2#20451C211yesmaximal, characteristic, complemented
3#12846C211yesmaximal, characteristic, complemented
4#11705C212yesmaximal
5#7262C212yesmaximal, complemented
6C2×#466C2211yescharacteristic
7C2×#427—21no—
8C22×#97C2211yescharacteristic
9#2854—21no—
10C4×#117—21no—
11#2112C2212yes—
12#1972C412yescomplemented
13#1944C412yescomplemented
14#1936—22no—
15#1893—21no—
16#1807C2211yescharacteristic
17#1776—24no—
18#1735C2212yes—
19#1377—22no—
20#1347—24no—
21C22×D8C2×C411yescharacteristic
22C23×C8D411yescharacteristic
23C23×C8C2311yescharacteristic
24C2×#123—41no—
25C2×#121—21no—
26C2×#117—22no—
27C22×C42D411yescharacteristic
28C2×M32—22no—
29#466—22no—
30#466—22no—
31#427—28no—
32C2×#107—22no—
33C2×#107C2×C411yescharacteristic
34C2×#106—22no—
35C2×#106C2×C412yes—
36C2×#102—22no—
37C2×#102—21no—
38C2×#97—22no—
39C2×#97—22no—
40C2×#97C2×C412yes—
41C2×#97D412yes—
42#390—22no—
43#389—22no—
44#388—42no—
45#386—41no—
46#380—21no—
47C4×C4⋊C4—22no—
48C4×C4⋊C4—22no—
49C4×C22⋊C4—41no—
50C4×C22⋊C4—41no—
51#376—41no—
52C43—41no—
53C22⋊C16—22no—
54C22⋊C16—22no—
55#340—22no—
56#340—21no—
57#340—22no—
58#340D414yes—
59C22×D4—21no—
60C22×D4—41no—
61C23×C4D811yescharacteristic
62C23×C4D811yescharacteristic
63C23×C4C2×D411yescharacteristic
64C2×D8—22no—
65C2×D8—42no—
66C22×C8—21no—
67C22×C8—21no—
68C22×C8—24no—
69C22×C8—42no—
70C22×C8C22⋊C412yes—
71C22×C8C22×C411yescharacteristic
72C22×C8C2×D411yescharacteristic
73C22×C8C2×D412yes—
74C22×C8#4112yes—
75#123—84no—
76#121—41no—
77#121—41no—
78#121—41no—
79#121—41no—
80#117—28no—
81#117—44no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—42no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—42no—
90C2×C22⋊C4—41no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—42no—
96C2×C42—42no—
97C2×C42—42no—
98C2×C42C22⋊C412yes—
99C2×C42SD1612yes—
100M32—44no—
101C2×C16—42no—
102#107—28no—
103#107—22no—
104#106—28no—
105#106—42no—
106#102—44no—
107#102—42no—
108#97—24no—
109#97—24no—
110#95—42no—
111#95—42no—
112#95—42no—
113#95—41no—
114C24—41no—
115C24C2×D811yescharacteristic
116C2×D4—41no—
117C2×D4—42no—
118C2×D4—41no—
119C2×D4—42no—
120C2×D4—82no—
121C2×D4—82no—
122C2×D4—81no—
123C2×D4—81no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—42no—
131C22×C4—44no—
132C22×C4—42no—
133C22×C4—41no—
134C22×C4—81no—
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—22no—
145C2×C8—22no—
146C2×C8—24no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—42no—
152C2×C8—42no—
153C2×C8#10414yes—
154C2×C8C4×D412yes—
155C2×C8#12112yes—
156C4⋊C4—28no—
157C4⋊C4—44no—
158C4⋊C4—41no—
159C4⋊C4—42no—
160C4⋊C4—41no—
161C4⋊C4—42no—
162C4⋊C4—84no—
163C22⋊C4—48no—
164C22⋊C4—84no—
165C22⋊C4—84no—
166C22⋊C4—82no—
167C22⋊C4—81no—
168C22⋊C4—81no—
169C42—22no—
170C42—22no—
171C42—22no—
172C42—22no—
173C42—48no—
174C42—84no—
175C42—84no—
176C16—82no—
177C23—22no—
178C23—22no—
179C23—41no—
180C23—81no—
181C23—82no—
182C23—81no—
183C23—81no—
184C23—82no—
185C23C2×#10211yescharacteristic
186C23#44711yescharacteristic
187C23#48011yescharacteristic
188D4—81no—
189D4—81no—
190D4—82no—
191D4—162no—
192D4—162no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—48no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—84no—
210C2×C4—84no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—82no—
217C2×C4#38611yescharacteristic
218C2×C4#41012yes—
219C2×C4#41711yescharacteristic
220C2×C4C2×#10412yes—
221C2×C4#44811yescharacteristic
222C2×C4#48111yescharacteristic
223C8—24no—
224C8—22no—
225C8—22no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—44no—
234C22—162no—
235C22—162no—
236C22—161no—
237C22—161no—
238C22—161no—
239C22#150212yes—
240C22#187411yescharacteristic
241C22#219012yes—
242C4—21no—
243C4—21no—
244C4—42no—
245C4—42no—
246C4—88no—
247C4—162no—
248C4—161no—
249C4#150912yes—
250C4#189611yescharacteristic
251C4#189911yescharacteristic
252C2—22no—
253C2—22no—
254C2—41no—
255C2—321no—
256C2#914811yescharacteristic
257C2#1299911yescharacteristic
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×#97 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20451C2semidirect
3. #12846C2semidirect
5. #7262C2semidirect
12. #1972C4semidirect
13. #1944C4semidirect

Maximal subgroups

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

subgroupindex
#331352
#204512
#128462
#117052
#72622