Finite Group Databasebeta

← #163239

Subgroups — #163239

Order 512.

778 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 210 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×#2864C211yesmaximal, characteristic
2#20778C212yesmaximal
3#7717C218yesmaximal, complemented
4#7627C214yesmaximal
5#2864—22no—
6#2864—22no—
7#2864C414yes—
8#2864C2214yes—
9C2×#389C412yes—
10C2×#389C2212yes—
11C2×#387—22no—
12C2×#387—22no—
13C2×#382C411yescharacteristic
14C2×#382C411yescharacteristic
15C2×C43C2211yescharacteristic
16#1839C2214yes—
17#1831C2214yes—
18#1817—22no—
19#1797—22no—
20#1767—22no—
21#1749C2214yes—
22#1745—22no—
23#1271C22116yes—
24C22×C4⋊C4—22no—
25C22×C4⋊C4—22no—
26C22×C4⋊C4C2×C412yes—
27C22×C4⋊C4C2×C412yes—
28C22×C42C2×C411yescharacteristic
29C22×C42C2×C411yescharacteristic
30C22×C42D412yes—
31C22×C42D412yes—
32C22×C42C2311yescharacteristic
33C2×C4⋊C8—24no—
34C2×C22⋊C8—24no—
35C2×#97—24no—
36#389—22no—
37#389—22no—
38#389—22no—
39#389—22no—
40#389C2×C414yes—
41#389C2×C414yes—
42#389C2×C414yes—
43#389C2314yes—
44#387—24no—
45#387—24no—
46#387—24no—
47#387—24no—
48#387—24no—
49#387—24no—
50#387—24no—
51#387—24no—
52#382—22no—
53#382—22no—
54#382C2×C414yes—
55#382C2×C414yes—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—22no—
61C2×#95—22no—
62C43C2×C412yes—
63C43D412yes—
64C43D412yes—
65C43C2312yes—
66#336—24no—
67#336—24no—
68#336—24no—
69#336C2318yes—
70C23×C4—22no—
71C23×C4—22no—
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C412yes—
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22×C411yescharacteristic
76C23×C4C2×D412yes—
77C22×C8—28no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—44no—
85C2×C4⋊C4—48no—
86C2×C4⋊C4—48no—
87C2×C4⋊C4—44no—
88C2×C4⋊C4C22×C414yes—
89C2×C4⋊C4C22×C414yes—
90C2×C42—24no—
91C2×C42—42no—
92C2×C42—44no—
93C2×C42—42no—
94C2×C42—44no—
95C2×C42C22⋊C412yes—
96C2×C42C22⋊C412yes—
97C2×C42C22⋊C412yes—
98C2×C42C22⋊C412yes—
99C2×C42C22⋊C412yes—
100C2×C42C22⋊C411yescharacteristic
101C2×C42C22⋊C411yescharacteristic
102C2×C42C22×C412yes—
103C2×C42C22×C412yes—
104C2×C42C22×C411yescharacteristic
105C2×C42C2×D412yes—
106C2×C42C2×D412yes—
107C2×C42C2×D412yes—
108C2×C42C2×D412yes—
109C2×C42C2×D412yes—
110C2×C42C2411yescharacteristic
111C4⋊C8—48no—
112C22⋊C8—48no—
113#97—48no—
114#95—24no—
115#95—24no—
116#95—24no—
117#95—24no—
118#95—24no—
119#95—24no—
120#95—24no—
121#95—24no—
122C24C2×C22⋊C411yescharacteristic
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—24no—
128C22×C4—24no—
129C22×C4—24no—
130C22×C4—24no—
131C22×C4—24no—
132C22×C4—24no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—44no—
141C22×C4—44no—
142C22×C4—44no—
143C22×C4#10114yes—
144C22×C4C2×C22⋊C412yes—
145C22×C4C2×C22⋊C412yes—
146C22×C4C2×C22⋊C412yes—
147C22×C4C2×C22⋊C412yes—
148C22×C4C2×C22⋊C411yescharacteristic
149C22×C4C2×C22⋊C412yes—
150C22×C4C23×C411yescharacteristic
151C22×C4C22×D412yes—
152C2×C8—48no—
153C2×C8—48no—
154C2×C8—48no—
155C4⋊C4—28no—
156C4⋊C4—44no—
157C4⋊C4—44no—
158C4⋊C4—48no—
159C4⋊C4—48no—
160C4⋊C4—48no—
161C4⋊C4—48no—
162C4⋊C4—44no—
163C42—24no—
164C42—24no—
165C42—24no—
166C42—24no—
167C42—24no—
168C42—24no—
169C42—41no—
170C42—41no—
171C42—42no—
172C42—42no—
173C42—42no—
174C42—42no—
175C42—41no—
176C42—42no—
177C42—42no—
178C42—41no—
179C23—21no—
180C23—21no—
181C23—22no—
182C23C2×#10112yes—
183C23#42114yes—
184C23C22×C22⋊C411yescharacteristic
185C2×C4—28no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—44no—
207C2×C4—48no—
208C2×C4—48no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—88no—
217C2×C4—88no—
218C2×C4#41012yes—
219C2×C4C2×#10114yes—
220C2×C4#56812yes—
221C2×C4#57514yes—
222C8—88no—
223C22—24no—
224C22—22no—
225C22—24no—
226C22—22no—
227C22#153714yes—
228C22#287911yescharacteristic
229C22C2×#41011yescharacteristic
230C22C22×#10111yescharacteristic
231C22#289612yes—
232C22C2×#42112yes—
233C4—24no—
234C4—24no—
235C4—22no—
236C4—22no—
237C4—42no—
238C4—42no—
239C4—42no—
240C4—42no—
241C4—84no—
242C4—84no—
243C2—24no—
244C2#1739411yescharacteristic
245C2#1749712yes—
246C2#1757912yes—
247C2C2×#153711yescharacteristic
248C2#1775511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7717C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28642
#207782
#77172
#76272