Finite Group Databasebeta

← #478781

Subgroups — #478781

Order 512.

1002 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 216 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#15135C212yesmaximal
2#11292C214yesmaximal, complemented
3#11207C214yesmaximal, complemented
4#11199C214yesmaximal
5#11147C211yesmaximal, characteristic
6C22×C22⋊C8C2212yes—
7C2×#399—22no—
8C2×#389C2214yes—
9C2×#384C2212yes—
10C2×#380C2214yes—
11C2×C4×C22⋊C4C2211yescharacteristic
12C2×#340C2214yes—
13C2×#340C2214yes—
14C2×#340C2214yes—
15C2×#340C2214yes—
16C2×#336C2214yes—
17#1445—22no—
18#1444—22no—
19#1439—22no—
20C4⋊#95C2212yes—
21C24×C4C2311yescharacteristic
22C23×C8C2314yes—
23C22×C4⋊C4—22no—
24C22×C4⋊C4—24no—
25C22×C4⋊C4—22no—
26C22×C4⋊C4C2314yes—
27C22×C4⋊C4C2314yes—
28C22×C22⋊C4—24no—
29C22×C22⋊C4D411yescharacteristic
30C22×C22⋊C4D411yescharacteristic
31C22×C42C2312yes—
32C2×C22⋊C8—44no—
33C2×C22⋊C8—42no—
34#399—48no—
35#389—28no—
36#389—44no—
37#384—42no—
38#384—42no—
39#380—28no—
40#380—44no—
41C4×C22⋊C4—42no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—24no—
45C2×#95—24no—
46C2×#95D412yes—
47#340—28no—
48#340—44no—
49#340—44no—
50#340—44no—
51#336—44no—
52C25C2×D411yescharacteristic
53C23×C4—24no—
54C23×C4—24no—
55C23×C4—22no—
56C23×C4—24no—
57C23×C4—24no—
58C23×C4—24no—
59C23×C4C2×D411yescharacteristic
60C23×C4C2×D412yes—
61C23×C4C2×D412yes—
62C23×C4C2411yescharacteristic
63C22×C8—24no—
64C22×C8—24no—
65C22×C8—24no—
66C22×C8—24no—
67C22×C8—24no—
68C22×C8—24no—
69C22×C8—24no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—48no—
82C2×C4⋊C4—44no—
83C2×C4⋊C4—44no—
84C2×C4⋊C4—44no—
85C2×C4⋊C4—48no—
86C2×C4⋊C4—44no—
87C2×C4⋊C4—84no—
88C2×C4⋊C4—84no—
89C2×C4⋊C4#4118yes—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—48no—
95C2×C22⋊C4—48no—
96C2×C22⋊C4—84no—
97C2×C22⋊C4D814yes—
98C2×C42—24no—
99C2×C42—24no—
100C2×C42—22no—
101C2×C42—22no—
102C22⋊C8—48no—
103#95—42no—
104#95—48no—
105#95—48no—
106#95—44no—
107#95—44no—
108C24—21no—
109C24—22no—
110C24—24no—
111C24—22no—
112C24—21no—
113C24—24no—
114C24C2×D812yes—
115C24C22×D411yescharacteristic
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—24no—
119C22×C4—24no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—24no—
123C22×C4—24no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—28no—
127C22×C4—24no—
128C22×C4—44no—
129C22×C4—44no—
130C22×C4—44no—
131C22×C4—44no—
132C22×C4—44no—
133C22×C4—44no—
134C22×C4—44no—
135C22×C4—44no—
136C22×C4—44no—
137C22×C4—44no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—44no—
141C22×C4—44no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4C2×D814yes—
150C22×C4#13714yes—
151C22×C4C2×#4114yes—
152C22×C4H(5,𝔽2)14yes—
153C22×C4H(5,𝔽2)12yes—
154C22×C42−1+414yes—
155C2×C8—44no—
156C2×C8—44no—
157C2×C8—44no—
158C2×C8—44no—
159C2×C8—44no—
160C2×C8—44no—
161C2×C8—44no—
162C4⋊C4—28no—
163C4⋊C4—28no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—816no—
171C4⋊C4—88no—
172C4⋊C4—88no—
173C22⋊C4—44no—
174C22⋊C4—44no—
175C22⋊C4—816no—
176C42—44no—
177C42—42no—
178C42—42no—
179C23—22no—
180C23—24no—
181C23—28no—
182C23—41no—
183C23—44no—
184C23—41no—
185C23—44no—
186C23—42no—
187C23—42no—
188C23—42no—
189C23—44no—
190C23—44no—
191C23—44no—
192C23#53411yescharacteristic
193C23#53612yes—
194C23#54614yes—
195C23C22×D811yescharacteristic
196C23C2×#13712yes—
197C23#57511yescharacteristic
198C23#57714yes—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—28no—
202C2×C4—28no—
203C2×C4—44no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—41no—
210C2×C4—48no—
211C2×C4—48no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—48no—
215C2×C4—84no—
216C2×C4—84no—
217C2×C4—84no—
218C2×C4—84no—
219C2×C4—84no—
220C2×C4—84no—
221C2×C4—84no—
222C2×C4—84no—
223C2×C4—84no—
224C2×C4—84no—
225C2×C4—84no—
226C2×C4—84no—
227C2×C4—84no—
228C2×C4#53914yes—
229C2×C4#54014yes—
230C2×C4#54214yes—
231C2×C4#55214yes—
232C2×C4#55912yes—
233C2×C4#56012yes—
234C2×C4#56214yes—
235C2×C4#56314yes—
236C8—84no—
237C22—42no—
238C22—44no—
239C22—48no—
240C22—81no—
241C22—81no—
242C22—82no—
243C22—84no—
244C22#268714yes—
245C22#270912yes—
246C22#281711yescharacteristic
247C22#318611yescharacteristic
248C22#319511yescharacteristic
249C22#320012yes—
250C22#322612yes—
251C22#323712yes—
252C22#324414yes—
253C22#329414yes—
254C22#331714yes—
255C22#332014yes—
256C22#333114yes—
257C4—44no—
258C4—44no—
259C4—81no—
260C4—88no—
261C4—164no—
262C4#277314yes—
263C4#283714yes—
264C2—82no—
265C2#1716811yescharacteristic
266C2#3135014yes—
267C2#3137114yes—
268C2#3202612yes—
269C2#3203712yes—
270C2#3278611yescharacteristic
271C2#3279411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11292C2semidirect
3. #11207C2semidirect

Maximal subgroups

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

subgroupindex
#151352
#112922
#112072
#111992
#111472