Finite Group Databasebeta

← #97038

Subgroups — #97038

Order 512.

804 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C2×#2629C211yesmaximal, characteristic
2#8300C212yesmaximal
3#7593C214yesmaximal, complemented
4C22×#125C411yescharacteristic
5C22×#117C2211yescharacteristic
6#2629—28no—
7C2×#398—22no—
8C2×#396—22no—
9C2×#389C2212yes—
10C2×#384C411yescharacteristic
11C2×#380C412yes—
12#1817C2214yes—
13C2×#324—22no—
14C2×#323—22no—
15#1271—28no—
16C24×C4C2×C411yescharacteristic
17C2×#125—24no—
18C2×#125—44no—
19C2×#117—24no—
20C2×#117—44no—
21C22×C4⋊C4—22no—
22C22×C4⋊C4—22no—
23C22×C4⋊C4—22no—
24C22×C4⋊C4C2×C412yes—
25C22×C4⋊C4C2×C412yes—
26C22×C4⋊C4D412yes—
27C22×C22⋊C4—22no—
28C22×C22⋊C4D412yes—
29C22×C42C2×C411yescharacteristic
30C22×C42C2311yescharacteristic
31C2×C22⋊C8—24no—
32#398—216no—
33#396—216no—
34#389—24no—
35#389D418yes—
36#384—24no—
37#380—24no—
38#380—24no—
39C2×#95—22no—
40C2×#95—22no—
41#336—28no—
42#324—44no—
43#323—44no—
44C25C22⋊C411yescharacteristic
45C23×C4—22no—
46C23×C4—22no—
47C23×C4—22no—
48C23×C4—22no—
49C23×C4—22no—
50C23×C4—24no—
51C23×C4—42no—
52C23×C4C22⋊C411yescharacteristic
53C23×C4C22⋊C412yes—
54C23×C4C22×C411yescharacteristic
55C23×C4C2×D412yes—
56C22×C8—44no—
57#125—416no—
58#117—416no—
59C2×C4⋊C4—28no—
60C2×C4⋊C4—28no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—24no—
65C2×C4⋊C4—28no—
66C2×C4⋊C4—28no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4—42no—
69C2×C4⋊C4—42no—
70C2×C4⋊C4—44no—
71C2×C4⋊C4—44no—
72C2×C4⋊C4—84no—
73C2×C4⋊C4—84no—
74C2×C4⋊C4C22⋊C414yes—
75C2×C4⋊C4C22⋊C414yes—
76C2×C22⋊C4—24no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—28no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4—84no—
81C2×C42—24no—
82C2×C42—22no—
83C2×C42—42no—
84C2×C42C2×D414yes—
85C22⋊C8—44no—
86C22⋊C8—44no—
87#95—28no—
88#95—28no—
89C24—21no—
90C24—21no—
91C24—22no—
92C24—22no—
93C24—42no—
94C24—42no—
95C24#10112yes—
96C24C2×C22⋊C411yescharacteristic
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—28no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—24no—
110C22×C4—24no—
111C22×C4—48no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—48no—
115C22×C4—44no—
116C22×C4—44no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—44no—
120C22×C4—44no—
121C22×C4—44no—
122C22×C4—44no—
123C22×C4—44no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4#9914yes—
127C22×C4#10112yes—
128C22×C4C2×C22⋊C412yes—
129C22×C4#12014yes—
130C22×C4#13114yes—
131C2×C8—84no—
132C2×C8—84no—
133C2×C8—84no—
134C4⋊C4—28no—
135C4⋊C4—48no—
136C4⋊C4—44no—
137C4⋊C4—44no—
138C4⋊C4—416no—
139C4⋊C4—44no—
140C4⋊C4—88no—
141C4⋊C4—88no—
142C22⋊C4—48no—
143C22⋊C4—88no—
144C42—28no—
145C42—44no—
146C23—22no—
147C23—22no—
148C23—22no—
149C23—24no—
150C23—44no—
151C23—44no—
152C23—42no—
153C23—42no—
154C23—42no—
155C23—41no—
156C23—41no—
157C23—82no—
158C23—82no—
159C23—82no—
160C23—82no—
161C23#37911yescharacteristic
162C23#40812yes—
163C23C2×#9912yes—
164C23C2×#10111yescharacteristic
165C23C2×#10111yescharacteristic
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—24no—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—416no—
177C2×C4—44no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—44no—
184C2×C4—44no—
185C2×C4—44no—
186C2×C4—44no—
187C2×C4—44no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4—84no—
193C2×C4—84no—
194C2×C4—88no—
195C2×C4#40814yes—
196C8—164no—
197C22—22no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—44no—
204C22—42no—
205C22—84no—
206C22—84no—
207C22—82no—
208C22—81no—
209C22—81no—
210C22#147212yes—
211C22#147212yes—
212C22#178011yescharacteristic
213C22#178111yescharacteristic
214C22#178411yescharacteristic
215C22#179512yes—
216C22#179612yes—
217C4—24no—
218C4—44no—
219C4—44no—
220C4—42no—
221C4—84no—
222C4—84no—
223C2—22no—
224C2—22no—
225C2—82no—
226C2#830811yescharacteristic
227C2#832611yescharacteristic
228C2#832711yescharacteristic
229C2#851712yes—
230C2#852512yes—
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7593C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
#83002
#75932