Finite Group Databasebeta

← #108244

Subgroups — #108244

Order 512.

956 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 956 conjugacy classes of subgroups — too many to draw. Only its 190 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#60124C211yesmaximal, characteristic
2#20688C211yesmaximal, characteristic
3#14343C211yesmaximal, characteristic
4#7983C214yesmaximal
5C22×#117C2211yescharacteristic
6#2924—28no—
7#2921—22no—
8#2920—22no—
9#2905C412yescomplemented
10C2×#407C2212yes—
11#2873C412yescomplemented
12#2280C2212yes—
13C2×#380C2212yes—
14#1820C414yescomplemented
15#1819—22no—
16#1740—22no—
17#1445C414yescomplemented
18#1440—22no—
19C4⋊#95—22no—
20#1289—28no—
21C24×C4C2311yescharacteristic
22C22×M16C2×C414yes—
23C2×#117—24no—
24C2×#117—24no—
25C2×#117—22no—
26C2×#117C2×C411yescharacteristic
27C2×#117C2×C411yescharacteristic
28C2×#117C2×C412yes—
29C2×#117D412yes—
30C2×#117Q812yes—
31C22×C4⋊C4—22no—
32C22×C4⋊C4C2×C412yes—
33C22×C22⋊C4—22no—
34C22×C22⋊C4D412yes—
35C22×C42D412yes—
36#433—216no—
37#432—28no—
38#431—28no—
39#424—28no—
40#423—24no—
41#407—22no—
42#407—22no—
43#407—24no—
44C2×C22⋊C8D418yes—
45#405—28no—
46C4×M16—24no—
47#381C2×C414yes—
48#380—24no—
49#380—24no—
50#380—24no—
51#379—22no—
52C4×C22⋊C4—24no—
53C2×#95—28no—
54C2×#95C2×C414yes—
55#342—28no—
56C25C2×D411yescharacteristic
57C23×C4—24no—
58C23×C4—44no—
59C23×C4C22⋊C414yes—
60C23×C4C22×C411yescharacteristic
61C23×C4C22×C411yescharacteristic
62C23×C4C22×C411yescharacteristic
63C23×C4C2×D411yescharacteristic
64C23×C4C2×D411yescharacteristic
65C23×C4C2×D414yes—
66C23×C4C2×Q811yescharacteristic
67C23×C4#4114yes—
68C2×M16—24no—
69C2×M16—24no—
70C2×M16—48no—
71C22×C8C22⋊C418yes—
72#117—22no—
73#117—22no—
74#117—22no—
75#117—22no—
76#117—28no—
77#117—24no—
78#117—28no—
79#117—24no—
80#117—48no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—48no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4C4212yes—
88C2×C4⋊C4C4212yes—
89C2×C4⋊C4C4⋊C414yes—
90C2×C22⋊C4—28no—
91C2×C22⋊C4—24no—
92C2×C22⋊C4—44no—
93C2×C22⋊C4—44no—
94C2×C22⋊C4—84no—
95C2×C22⋊C4C22⋊C414yes—
96C2×C22⋊C4C4⋊C412yes—
97C2×C22⋊C4C4⋊C412yes—
98C2×C42—24no—
99C2×C42—24no—
100C2×C42—42no—
101C2×C42C22⋊C412yes—
102C2×C42C22⋊C412yes—
103C2×C42C4⋊C414yes—
104C4⋊C8—216no—
105C22⋊C8—28no—
106C22⋊C8—28no—
107#97—28no—
108C4×C8—28no—
109#95—28no—
110#95—28no—
111#95—88no—
112C24—42no—
113C24—42no—
114C24—42no—
115C24C2×C22⋊C411yescharacteristic
116C24C2×C22⋊C411yescharacteristic
117C24C2×C4⋊C411yescharacteristic
118C24#12114yes—
119C22×C4—24no—
120C22×C4—24no—
121C22×C4—24no—
122C22×C4—24no—
123C22×C4—24no—
124C22×C4—48no—
125C22×C4—48no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—44no—
130C22×C4—44no—
131C22×C4—44no—
132C22×C4—48no—
133C22×C4—48no—
134C22×C4—44no—
135C22×C4—44no—
136C22×C4—44no—
137C22×C4#9514yes—
138C22×C4#9514yes—
139C22×C4C2×C4211yescharacteristic
140C22×C4C2×C22⋊C412yes—
141C22×C4C2×C22⋊C411yescharacteristic
142C22×C4C2×C4⋊C411yescharacteristic
143C22×C4C2×C4⋊C411yescharacteristic
144C22×C4#11712yes—
145C22×C4C4×D414yes—
146C22×C4C4×D414yes—
147C22×C4#12012yes—
148C22×C4#12012yes—
149C22×C4C22⋊Q814yes—
150C22×C4#12312yes—
151C22×C4#12312yes—
152M16—48no—
153M16—48no—
154C2×C8—28no—
155C2×C8—48no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—28no—
159C4⋊C4—84no—
160C4⋊C4—84no—
161C4⋊C4—84no—
162C22⋊C4—24no—
163C22⋊C4—24no—
164C22⋊C4—28no—
165C22⋊C4—48no—
166C22⋊C4—84no—
167C22⋊C4—84no—
168C22⋊C4—88no—
169C42—22no—
170C42—22no—
171C42—22no—
172C42—22no—
173C42—28no—
174C42—84no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—24no—
179C23—24no—
180C23—22no—
181C23—22no—
182C23—44no—
183C23—42no—
184C23—42no—
185C23—44no—
186C23—82no—
187C23—84no—
188C23—82no—
189C23C2×#9511yescharacteristic
190C23#38012yes—
191C23#38612yes—
192C23#38612yes—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—84no—
212C2×C4—84no—
213C2×C4—88no—
214C2×C4—84no—
215C2×C4—88no—
216C2×C4—88no—
217C2×C4—84no—
218C2×C4—84no—
219C2×C4C4×C22⋊C412yes—
220C2×C4#37911yescharacteristic
221C2×C4#38111yescharacteristic
222C2×C4#38512yes—
223C2×C4#38512yes—
224C2×C4#41014yes—
225C8—48no—
226C22—22no—
227C22—24no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—22no—
234C22—24no—
235C22—24no—
236C22—84no—
237C22—84no—
238C22—82no—
239C22—82no—
240C22C22⋊#9511yescharacteristic
241C22#174012yes—
242C22#190912yes—
243C22#191012yes—
244C4—22no—
245C4—22no—
246C4—22no—
247C4—44no—
248C4—44no—
249C4—164no—
250C4—164no—
251C4#178212yes—
252C4#182012yes—
253C2—22no—
254C2—22no—
255C2—22no—
256C2—42no—
257C2—42no—
258C2#1072611yescharacteristic
259C2#1074212yes—
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #2905C4semidirect
11. #2873C4semidirect
14. #1820C4semidirect
17. #1445C4semidirect

Maximal subgroups

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

subgroupindex
#601242
#206882
#143432
#79832