Finite Group Databasebeta

← #189884

Subgroups — #189884

Order 512.

1160 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 242 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#34186C211yesmaximal, characteristic
2#21790C211yesmaximal, characteristic
3#13959C211yesmaximal, characteristic
4#8715C218yesmaximal, complemented
5#8022C212yesmaximal, complemented
6#8018C212yesmaximal, complemented
7C22×C4×D4C2211yescharacteristic
8C2×#432C2212yes—
9#2357—24no—
10#2354C418yescomplemented
11#2312—22no—
12#2309—21no—
13#2285C2212yes—
14C2×#383C2211yescharacteristic
15#2280C2214yes—
16#2274—21no—
17C2×#376C2211yescharacteristic
18#1955—24no—
19#1779C2214yes—
20#1445—21no—
21#1438—21no—
22#1437—22no—
23C22⋊#95C2214yes—
24#1294C2218yes—
25#1294C2218yes—
26C23×D4D411yescharacteristic
27C24×C4C2312yes—
28C2×C4×D4—24no—
29C2×C4×D4—24no—
30C2×C4×D4C2×C414yes—
31C2×C4×D4D414yes—
32C2×#117—22no—
33C2×#117C2×C418yes—
34C22×C4⋊C4D411yescharacteristic
35C22×C22⋊C4—24no—
36C22×C22⋊C4D412yes—
37C22×C42—21no—
38C22×C42C2311yescharacteristic
39#432—24no—
40#432—24no—
41C2×C4⋊C8—22no—
42C2×C22⋊C8—28no—
43C2×C22⋊C8C2314yes—
44C2×#97—22no—
45#390—28no—
46#388—216no—
47#384C2×C414yes—
48#383—24no—
49#382—22no—
50#381—24no—
51#380—22no—
52#380C2×C418yes—
53#380C2314yes—
54#379—24no—
55C4×C4⋊C4—28no—
56C4×C4⋊C4—22no—
57C4×C4⋊C4C2×C414yes—
58C4×C22⋊C4—24no—
59C4×C22⋊C4—24no—
60#376—24no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—24no—
65C2×#95C2312yes—
66C2×#95C2312yes—
67C25C2×D412yes—
68C22×D4—22no—
69C22×D4—22no—
70C22×D4—44no—
71C22×D4C22⋊C414yes—
72C23×C4—24no—
73C23×C4—24no—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4—84no—
77C23×C4C22×C414yes—
78C23×C4C2×D411yescharacteristic
79C23×C4C2×D412yes—
80C23×C4C2×D414yes—
81C23×C4C2×D411yescharacteristic
82C23×C4C2411yescharacteristic
83C22×C8—24no—
84C4×D4—216no—
85C4×D4—28no—
86C4×D4—28no—
87#117—48no—
88#117—88no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4C22⋊C414yes—
94C2×C4⋊C4C22×C418yes—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—28no—
99C2×C22⋊C4—28no—
100C2×C22⋊C4—48no—
101C2×C22⋊C4—48no—
102C2×C22⋊C4—84no—
103C2×C22⋊C4C22⋊C418yes—
104C2×C42—24no—
105C2×C42—28no—
106C2×C42—44no—
107C2×C42—42no—
108C2×C42C22×C412yes—
109C2×C42C2×D412yes—
110C2×C42#4114yes—
111C2×C42#4114yes—
112C4⋊C8—44no—
113C22⋊C8—24no—
114C22⋊C8—24no—
115C22⋊C8—88no—
116#97—44no—
117#95—28no—
118#95—24no—
119#95—24no—
120#95—24no—
121#95—24no—
122#95—42no—
123#95—42no—
124#95—42no—
125#95—84no—
126C24—24no—
127C24—22no—
128C24—22no—
129C24—24no—
130C24—44no—
131C24—44no—
132C24C2×C22⋊C414yes—
133C24C22×D411yescharacteristic
134C2×D4—28no—
135C2×D4—44no—
136C2×D4—44no—
137C2×D4—48no—
138C2×D4—48no—
139C22×C4—24no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—28no—
143C22×C4—28no—
144C22×C4—24no—
145C22×C4—28no—
146C22×C4—42no—
147C22×C4—48no—
148C22×C4—42no—
149C22×C4—44no—
150C22×C4—48no—
151C22×C4—44no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—44no—
155C22×C4—42no—
156C22×C4—44no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—44no—
160C22×C4—88no—
161C22×C4—88no—
162C22×C4C2×C22⋊C414yes—
163C22×C4C2×C22⋊C412yes—
164C22×C4C2×C22⋊C412yes—
165C22×C4#12312yes—
166C22×C4#12312yes—
167C22×C4#12312yes—
168C22×C4#12312yes—
169C22×C4C23×C411yescharacteristic
170C22×C4C22×D411yescharacteristic
171C22×C4C2×#4112yes—
172C22×C4C2×#4112yes—
173C2×C8—48no—
174C2×C8—44no—
175C4⋊C4—28no—
176C4⋊C4—48no—
177C4⋊C4—48no—
178C4⋊C4—84no—
179C22⋊C4—216no—
180C22⋊C4—48no—
181C22⋊C4—88no—
182C22⋊C4—88no—
183C22⋊C4—816no—
184C42—48no—
185C42—48no—
186C42—88no—
187C42#11718yes—
188C42#12314yes—
189C42#12314yes—
190C23—22no—
191C23—22no—
192C23—28no—
193C23—24no—
194C23—24no—
195C23—28no—
196C23—24no—
197C23—48no—
198C23—48no—
199C23—44no—
200C23—44no—
201C23—82no—
202C23—82no—
203C23—84no—
204C23—84no—
205C23—82no—
206C23—82no—
207C23C22×C22⋊C411yescharacteristic
208C23C2×#12311yescharacteristic
209C23C2×#12311yescharacteristic
210C23#57514yes—
211D4—416no—
212C2×C4—24no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—28no—
220C2×C4—24no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—416no—
226C2×C4—44no—
227C2×C4—48no—
228C2×C4—48no—
229C2×C4—48no—
230C2×C4—48no—
231C2×C4—88no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—84no—
236C2×C4—82no—
237C2×C4—84no—
238C2×C4—82no—
239C2×C4—84no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4#38114yes—
243C2×C4#38114yes—
244C2×C4C2×#11712yes—
245C2×C4C2×#12312yes—
246C8—84no—
247C22—21no—
248C22—21no—
249C22—24no—
250C22—24no—
251C22—21no—
252C22—21no—
253C22—44no—
254C22—48no—
255C22—48no—
256C22—44no—
257C22—44no—
258C22—88no—
259C22—88no—
260C22C2×#38111yescharacteristic
261C22#288312yes—
262C22#289612yes—
263C22#290112yes—
264C22#309312yes—
265C22#309312yes—
266C4—22no—
267C4—22no—
268C4—24no—
269C4—24no—
270C4—44no—
271C4—84no—
272C4—88no—
273C4—162no—
274C2—22no—
275C2—22no—
276C2—84no—
277C2#1762914yes—
278C2#2069011yescharacteristic
279C2#2070111yescharacteristic
280C2#2070611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8715C2semidirect
5. #8022C2semidirect
6. #8018C2semidirect
10. #2354C4semidirect

Maximal subgroups

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

subgroupindex
#341862
#217902
#139592
#87152
#80222
#80182