Finite Group Databasebeta

← #115698

Subgroups — #115698

Order 512.

1197 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1197 conjugacy classes of subgroups — too many to draw. Only its 142 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×#2853C211yesmaximal, characteristic
2#20682C212yesmaximal, complemented
3#17392C212yesmaximal
4#8114C212yesmaximal, complemented
5C22×#123C2211yescharacteristic
6C23×C42C2211yescharacteristic
7C2×#407C2214yes—
8#2853—24no—
9#2853—24no—
10C2×#388—22no—
11C2×#387—22no—
12C2×#386C412yes—
13C2×#385C412yes—
14C2×#381C2211yescharacteristic
15#1817—28no—
16#1728—24no—
17#1468—28no—
18#1463—22no—
19#1462—22no—
20#1306—24no—
21C23×D4C2×C411yescharacteristic
22C24×C4C2×C412yes—
23C24×C4C2311yescharacteristic
24C22×M16—24no—
25C2×#123—22no—
26C2×#123—22no—
27C2×#123—24no—
28C2×#123—24no—
29C22×C4⋊C4—22no—
30C22×C4⋊C4D412yes—
31C22×C22⋊C4—22no—
32C22×C22⋊C4C2×C411yescharacteristic
33C22×C22⋊C4C2×C412yes—
34C22×C22⋊C4D412yes—
35C22×C42—24no—
36C22×C42—22no—
37C22×C42—22no—
38C22×C42—24no—
39C22×C42D414yes—
40#407—24no—
41#407—24no—
42#407—24no—
43#407—24no—
44C2×C22⋊C8—24no—
45C2×C22⋊C8D418yes—
46#388—216no—
47#387—216no—
48#386—24no—
49#386—24no—
50#385—24no—
51#385—24no—
52#381—22no—
53#381D414yes—
54C2×#95—22no—
55C2×#95—22no—
56C2×#95—22no—
57#336—216no—
58#324—28no—
59#323—28no—
60C25C22⋊C411yescharacteristic
61C25C22×C411yescharacteristic
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—24no—
65C22×D4—44no—
66C23×C4—22no—
67C23×C4—24no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—24no—
71C23×C4—48no—
72C23×C4—44no—
73C23×C4C22⋊C412yes—
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C414yes—
76C23×C4C2×D412yes—
77C23×C4C2×D412yes—
78C23×C4C2×D414yes—
79C23×C4C2×D412yes—
80C23×C4#4114yes—
81C2×M16—416no—
82C2×M16—44no—
83C2×M16—44no—
84C22×C8—28no—
85#123—42no—
86#123—44no—
87#123—44no—
88#123—42no—
89#123—42no—
90#123—42no—
91C2×C4⋊C4—28no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—48no—
97C2×C22⋊C4—28no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—24no—
104C2×C22⋊C4—24no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—48no—
107C2×C22⋊C4—44no—
108C2×C22⋊C4C22⋊C414yes—
109C2×C42—24no—
110C2×C42—28no—
111C2×C42—28no—
112C2×C42—24no—
113C2×C42—42no—
114C2×C42—44no—
115C2×C42—44no—
116C2×C42—42no—
117C2×C42—42no—
118C2×C42—42no—
119C22⋊C8—28no—
120C22⋊C8—28no—
121C22⋊C8—44no—
122C22⋊C8—44no—
123#95—28no—
124#95—28no—
125#95—28no—
126C24—24no—
127C24—21no—
128C24—22no—
129C24—44no—
130C24—44no—
131C24—42no—
132C24C2×C22⋊C411yescharacteristic
133C24C2×C22⋊C411yescharacteristic
134C24C2×C22⋊C411yescharacteristic
135C24C4×D414yes—
136C2×D4—44no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—44no—
140C2×D4—88no—
141C22×C4—28no—
142C22×C4—28no—
143C22×C4—216no—
144C22×C4—24no—
145C22×C4—28no—
146C22×C4—24no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—48no—
153C22×C4—48no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—48no—
157C22×C4—48no—
158C22×C4—416no—
159C22×C4—44no—
160C22×C4—48no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—42no—
169C22×C4#12012yes—
170C22×C4#12014yes—
171C22×C4#12012yes—
172C22×C4#12114yes—
173C22×C4#12114yes—
174C22×C4#12114yes—
175C22×C4#12312yes—
176C22×C4#12312yes—
177C22×C4#12412yes—
178C22×C4#12712yes—
179M16—88no—
180M16—88no—
181C2×C8—416no—
182C2×C8—48no—
183C4⋊C4—416no—
184C4⋊C4—44no—
185C4⋊C4—44no—
186C22⋊C4—416no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—44no—
190C22⋊C4—44no—
191C22⋊C4—44no—
192C22⋊C4—44no—
193C42—48no—
194C42—48no—
195C23—24no—
196C23—22no—
197C23—28no—
198C23—28no—
199C23—22no—
200C23—42no—
201C23—42no—
202C23—44no—
203C23—416no—
204C23—44no—
205C23—41no—
206C23—42no—
207C23—44no—
208C23—88no—
209C23—82no—
210C23—82no—
211C23#37911yescharacteristic
212C23#38612yes—
213C23#38612yes—
214C23#39012yes—
215C23#41014yes—
216D4—84no—
217D4—84no—
218C2×C4—24no—
219C2×C4—28no—
220C2×C4—24no—
221C2×C4—28no—
222C2×C4—28no—
223C2×C4—216no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—44no—
231C2×C4—48no—
232C2×C4—48no—
233C2×C4—416no—
234C2×C4—48no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—44no—
243C2×C4—42no—
244C2×C4—88no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4#39214yes—
248C2×C4#39414yes—
249C8—88no—
250C22—28no—
251C22—22no—
252C22—22no—
253C22—24no—
254C22—22no—
255C22—22no—
256C22—24no—
257C22—44no—
258C22—416no—
259C22—44no—
260C22—84no—
261C22—81no—
262C22—82no—
263C22—164no—
264C22#143811yescharacteristic
265C22#178212yes—
266C22#189114yes—
267C22#189914yes—
268C4—24no—
269C4—24no—
270C4—48no—
271C4—84no—
272C4—81no—
273C4—82no—
274C2—22no—
275C2—24no—
276C2—24no—
277C2—42no—
278C2—161no—
279C2#1096611yescharacteristic
280C2#1099312yes—
subgroupG/H
center, socleC22#1438
commutatorC25C22×C4
Frattini, cosocleC24×C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20682C2semidirect
4. #8114C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28532
#206822
#173922
#81142