Finite Group Databasebeta

← #354030

Subgroups — #354030

Order 512.

3678 conjugacy classes of subgroups; only the 738 normal subgroups are shown.

normal subgroups only

This group has 3678 conjugacy classes of subgroups — too many to draw. Only its 738 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×#2280C213yesmaximal
2#33996C211yesmaximal, characteristic, complemented
3C2×#1740C213yesmaximal, complemented
4#10936C214yesmaximal, complemented
5#10935C214yesmaximal
6C25×C4C2211yescharacteristic
7C22×#117C2213yes—
8C22×#117C2213yes—
9#2280—212no—
10#2280C4124yescomplemented
11C2×#381C2216yes—
12C2×#380—26no—
13C2×#379C2216yes—
14#2274—26no—
15#2273C2214yes—
16C4×#117—26no—
17C2×C4×C22⋊C4—26no—
18#1740—212no—
19C2×#342C22112yes—
20C26C2311yescharacteristic
21C24×C4—23no—
22C24×C4—23no—
23C24×C4—26no—
24C24×C4—212no—
25C24×C4C2×C4112yes—
26C24×C4C2311yescharacteristic
27C24×C4C2311yescharacteristic
28C2×#117—26no—
29C2×#117—26no—
30C2×#117—212no—
31C2×#117—26no—
32C2×#117—212no—
33C2×#117—212no—
34C2×#117C2×C4112yes—
35C2×#117C2×C4112yes—
36C2×#117D4112yes—
37C22×C4⋊C4—23no—
38C22×C4⋊C4D416yes—
39C22×C22⋊C4—212no—
40C22×C22⋊C4C2316yes—
41C22×C22⋊C4C2316yes—
42C22×C42—23no—
43C22×C42D416yes—
44#381—212no—
45#381C2×C4124yes—
46#380—248no—
47#379—212no—
48#379C2×C4124yes—
49C4×C4⋊C4—224no—
50C4×C22⋊C4—224no—
51C4×C22⋊C4—248no—
52#376—24no—
53C2×#95—212no—
54C43—24no—
55#342—224no—
56C25—23no—
57C25—23no—
58C25—26no—
59C25—48no—
60C25C22×C416yes—
61C25C2411yescharacteristic
62C23×C4—23no—
63C23×C4—23no—
64C23×C4—26no—
65C23×C4—212no—
66C23×C4—212no—
67C23×C4—224no—
68C23×C4—224no—
69C23×C4—224no—
70C23×C4—248no—
71C23×C4—26no—
72C23×C4—43no—
73C23×C4—41no—
74C23×C4—412no—
75C23×C4—412no—
76C23×C4—46no—
77C23×C4—46no—
78C23×C4—412no—
79C23×C4—48no—
80C23×C4—44no—
81C23×C4C42116yescomplemented
82C23×C4C22×C416yes—
83C23×C4C22×C416yes—
84C23×C4C2×D4112yes—
85C23×C4C2×D413yes—
86C23×C4C2×D413yes—
87C23×C4C2×D416yes—
88C23×C4#41112yes—
89#117—224no—
90#117—224no—
91#117—224no—
92#117—224no—
93#117—248no—
94#117—412no—
95#117—412no—
96C2×C4⋊C4—212no—
97C2×C4⋊C4—212no—
98C2×C4⋊C4—212no—
99C2×C4⋊C4—412no—
100C2×C4⋊C4C22⋊C4124yescomplemented
101C2×C22⋊C4—212no—
102C2×C22⋊C4—212no—
103C2×C22⋊C4—212no—
104C2×C22⋊C4—212no—
105C2×C22⋊C4—26no—
106C2×C22⋊C4—248no—
107C2×C22⋊C4—448no—
108C2×C22⋊C4C22×C4112yes—
109C2×C22⋊C4C22×C4112yes—
110C2×C22⋊C4C2×D4112yes—
111C2×C42—26no—
112C2×C42—26no—
113C2×C42—212no—
114C2×C42—212no—
115C2×C42—216no—
116C2×C42—412no—
117C2×C42C22⋊C4124yescomplemented
118#95—216no—
119#95—248no—
120C24—224no—
121C24—26no—
122C24—212no—
123C24—26no—
124C24—212no—
125C24—424no—
126C24—44no—
127C24—412no—
128C24—424no—
129C24—448no—
130C24—41no—
131C24—46no—
132C24—43no—
133C24—43no—
134C24—42no—
135C24—41no—
136C24C2×C4214yes—
137C24C23×C413yes—
138C24C22×D413yes—
139C24C22×D413yes—
140C24C2×#4116yes—
141C22×C4—26no—
142C22×C4—26no—
143C22×C4—212no—
144C22×C4—212no—
145C22×C4—212no—
146C22×C4—212no—
147C22×C4—248no—
148C22×C4—248no—
149C22×C4—212no—
150C22×C4—41no—
151C22×C4—46no—
152C22×C4—43no—
153C22×C4—43no—
154C22×C4—42no—
155C22×C4—41no—
156C22×C4—412no—
157C22×C4—412no—
158C22×C4—424no—
159C22×C4—448no—
160C22×C4—448no—
161C22×C4—424no—
162C22×C4—424no—
163C22×C4—424no—
164C22×C4—424no—
165C22×C4—412no—
166C22×C4C2×C4214yes—
167C22×C4C2×C4214yes—
168C22×C4C2×C22⋊C4124yes—
169C22×C4C2×C22⋊C416yes—
170C22×C4C2×C22⋊C416yes—
171C22×C4#117112yes—
172C22×C4C4×D4112yes—
173C22×C4C4×D4112yes—
174C22×C4C4×D4124yes—
175C22×C4#12013yes—
176C22×C4#12011yescharacteristic
177C22×C4#121112yes—
178C22×C4#121112yes—
179C22×C4#12316yes—
180C22×C4#12316yes—
181C22×C4#124112yes—
182C22×C4#12618yes—
183C22×C4#12714yes—
184C4⋊C4—248no—
185C4⋊C4—424no—
186C22⋊C4—224no—
187C22⋊C4—224no—
188C22⋊C4—224no—
189C22⋊C4—496no—
190C22⋊C4—46no—
191C22⋊C4—46no—
192C42—224no—
193C42—224no—
194C42—424no—
195C42—816no—
196C23—212no—
197C23—224no—
198C23—248no—
199C23—212no—
200C23—48no—
201C23—46no—
202C23—46no—
203C23—412no—
204C23—412no—
205C23—448no—
206C23—424no—
207C23—424no—
208C23—424no—
209C23—448no—
210C23—424no—
211C23—416no—
212C23—48no—
213C23—48no—
214C23—46no—
215C23—412no—
216C23—46no—
217C23C22×C4211yescharacteristic
218C23C22×C22⋊C413yes—
219C23C2×#11713yes—
220C23C2×C4×D416yes—
221C23C2×C4×D416yes—
222C23C2×#12011yescharacteristic
223C23C2×#12116yes—
224C23C2×#12313yes—
225C23C2×#12413yes—
226C23C2×#12612yes—
227C23C2×#12711yescharacteristic
228C2×C4—212no—
229C2×C4—224no—
230C2×C4—424no—
231C2×C4—46no—
232C2×C4—412no—
233C2×C4—46no—
234C2×C4—448no—
235C2×C4—416no—
236C2×C4—424no—
237C2×C4—424no—
238C2×C4—412no—
239C2×C4—812no—
240C2×C4—824no—
241C2×C4C4×C22⋊C4112yes—
242C2×C4C4×C22⋊C4112yes—
243C2×C4#38616yes—
244C2×C4#38616yes—
245C2×C4#386112yes—
246C2×C4#388124yes—
247C2×C4#388124yes—
248C2×C4#390112yes—
249C2×C4#390112yes—
250C22—26no—
251C22—212no—
252C22—26no—
253C22—212no—
254C22—224no—
255C22—43no—
256C22—46no—
257C22—43no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—424no—
262C22—412no—
263C22—448no—
264C22—44no—
265C22—424no—
266C22C2×C4×C22⋊C413yes—
267C22C2×#38613yes—
268C22C2×#38816yes—
269C22C2×#39013yes—
270C22#302514yes—
271C4—44no—
272C4—424no—
273C4—812no—
274C4#1437112yes—
275C4#143714yes—
276C2—23no—
277C2—23no—
278C2—26no—
279C2—48no—
280C2C2×#143711yescharacteristic
281C2#2095916yes—
subgroupG/H
centerC22×C4#120
commutatorC23C22×C42
Frattini, cosocleC25C24
socleC23C2×#120
prime radicalC26C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33996C2semidirect
3. C2×#1740C2semidirect
4. #10936C2semidirect
10. #2280C4semidirect
81. C23×C4C42semidirect
100. C2×C4⋊C4C22⋊C4semidirect
117. C2×C42C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22802
#339962
C2×#17402
#109362
#109352