Finite Group Databasebeta

← #107136

Subgroups — #107136

Order 512.

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

normal subgroups only

This group has 996 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#13853C211yesmaximal, characteristic
4#8051C214yesmaximal
5C22×#117C2211yescharacteristic
6#2924—28no—
7#2921—22no—
8#2920—22no—
9#2905C412yescomplemented
10C2×#407C2212yes—
11#2873C412yescomplemented
12#2280C2212yes—
13C2×C4×C22⋊C4C2212yes—
14#1820C414yescomplemented
15#1819—22no—
16#1740—22no—
17#1445C414yescomplemented
18#1440—22no—
19C4×#95—22no—
20#1297—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⋊C4C2×C412yes—
32C22×C22⋊C4—22no—
33C22×C22⋊C4D412yes—
34C22×C42—22no—
35C22×C42D412yes—
36#433—216no—
37#432—28no—
38#431—28no—
39#424—28no—
40#423—24no—
41#407—22no—
42#407—22no—
43#407D418yes—
44C2×C22⋊C8—24no—
45#405—28no—
46C4×M16—24no—
47#381C2×C414yes—
48#380—24no—
49#379—22no—
50C4×C22⋊C4—24no—
51C4×C22⋊C4—24no—
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—48no—
70C2×M16C22⋊C418yes—
71C22×C8—24no—
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—42no—
84C2×C4⋊C4C4212yes—
85C2×C4⋊C4C4212yes—
86C2×C4⋊C4C4⋊C414yes—
87C2×C22⋊C4—28no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—48no—
90C2×C22⋊C4—44no—
91C2×C22⋊C4—44no—
92C2×C22⋊C4C22⋊C414yes—
93C2×C22⋊C4C4⋊C412yes—
94C2×C22⋊C4C4⋊C412yes—
95C2×C42—24no—
96C2×C42—24no—
97C2×C42—48no—
98C2×C42—42no—
99C2×C42—42no—
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—416no—
112C24—44no—
113C24—42no—
114C24C2×C22⋊C411yescharacteristic
115C24C2×C22⋊C411yescharacteristic
116C24C2×C4⋊C411yescharacteristic
117C24#12114yes—
118C22×C4—24no—
119C22×C4—24no—
120C22×C4—24no—
121C22×C4—24no—
122C22×C4—24no—
123C22×C4—28no—
124C22×C4—48no—
125C22×C4—48no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—48no—
130C22×C4—48no—
131C22×C4—48no—
132C22×C4—44no—
133C22×C4—44no—
134C22×C4—44no—
135C22×C4#9514yes—
136C22×C4#9514yes—
137C22×C4C2×C4211yescharacteristic
138C22×C4C2×C22⋊C412yes—
139C22×C4C2×C22⋊C411yescharacteristic
140C22×C4C2×C4⋊C411yescharacteristic
141C22×C4C2×C4⋊C411yescharacteristic
142C22×C4#11712yes—
143C22×C4C4×D414yes—
144C22×C4C4×D414yes—
145C22×C4#12012yes—
146C22×C4#12012yes—
147C22×C4C22⋊Q814yes—
148C22×C4#12312yes—
149C22×C4#12312yes—
150M16—48no—
151M16—48no—
152C2×C8—28no—
153C2×C8—48no—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—28no—
157C4⋊C4—84no—
158C22⋊C4—24no—
159C22⋊C4—24no—
160C22⋊C4—28no—
161C22⋊C4—416no—
162C22⋊C4—48no—
163C22⋊C4—88no—
164C42—22no—
165C42—22no—
166C42—22no—
167C42—22no—
168C42—28no—
169C42—84no—
170C42—84no—
171C42—84no—
172C23—24no—
173C23—22no—
174C23—28no—
175C23—22no—
176C23—22no—
177C23—44no—
178C23—44no—
179C23—42no—
180C23—42no—
181C23—88no—
182C23C2×#9511yescharacteristic
183C23#38012yes—
184C23#38612yes—
185C23#38612yes—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—48no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—48no—
204C2×C4—416no—
205C2×C4—84no—
206C2×C4—84no—
207C2×C4—88no—
208C2×C4—84no—
209C2×C4—84no—
210C2×C4—88no—
211C2×C4C4×C22⋊C412yes—
212C2×C4#37911yescharacteristic
213C2×C4#38111yescharacteristic
214C2×C4#38512yes—
215C2×C4#38512yes—
216C2×C4#41014yes—
217C8—48no—
218C22—28no—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22—22no—
224C22—24no—
225C22—24no—
226C22—88no—
227C22—82no—
228C22—82no—
229C22C22⋊#9511yescharacteristic
230C22#174012yes—
231C22#188414yes—
232C4—22no—
233C4—22no—
234C4—22no—
235C4—44no—
236C4—44no—
237C4—88no—
238C4—164no—
239C4#178212yes—
240C4#182012yes—
241C2—24no—
242C2—22no—
243C2—42no—
244C2—42no—
245C2#1072611yescharacteristic
246C2#1075112yes—
subgroupG/H
centerC2×C4#379
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
socleC22C22⋊#95
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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
#138532
#80512