Finite Group Databasebeta

← #189941

Subgroups — #189941

Order 512.

1140 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34012C211yesmaximal, characteristic
2#20714C212yesmaximal
3#8727C214yesmaximal, complemented
4#8726C214yesmaximal, complemented
5#8038C212yesmaximal, complemented
6#8038C212yesmaximal, complemented
7C23×C42C2211yescharacteristic
8C2×#431C2214yes—
9#2285C2212yes—
10#2285C2212yes—
11#2285C2211yescharacteristic
12#2285C2211yescharacteristic
13#2283—22no—
14#2283—22no—
15#2283C414yes—
16#2280—22no—
17#2280C412yes—
18C4×#117—22no—
19C4×#117C412yes—
20#1817—22no—
21#1797C2214yes—
22#1797C2214yes—
23#1779—22no—
24C4×C22⋊C8—22no—
25C4×C22⋊C8—22no—
26#1296C2214yescomplemented
27#1296C2218yescomplemented
28#1296C2214yescomplemented
29C24×C4C2×C411yescharacteristic
30C24×C4D412yes—
31C2×#117—22no—
32C2×#117—22no—
33C2×#117C2×C412yes—
34C2×#117C2×C412yes—
35C22×C42—28no—
36C22×C42C2×C411yescharacteristic
37C22×C42C2×C412yes—
38C22×C42D412yes—
39C22×C42D412yes—
40C22×C42D412yes—
41C22×C42C2312yes—
42C22×C42C2311yescharacteristic
43#431—216no—
44C2×C4⋊C8C2318yes—
45C2×C22⋊C8—24no—
46C2×C22⋊C8—24no—
47C2×C22⋊C8—24no—
48C2×C4×C8—24no—
49#384C2311yescharacteristic
50#384C2312yes—
51#384C2311yescharacteristic
52#383—24no—
53#382—22no—
54#382—22no—
55#382C2×C412yes—
56#382C2×C412yes—
57#381—24no—
58#381—22no—
59#380—22no—
60#380—22no—
61#380—22no—
62#380—22no—
63#380C2×C414yes—
64#380C2×C414yes—
65#379—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4C2×C414yes—
71C4×C4⋊C4C2×C414yes—
72C4×C22⋊C4—24no—
73C4×C22⋊C4—24no—
74C4×C22⋊C4—24no—
75#376—24no—
76#376—22no—
77C43—22no—
78#336—24no—
79#336—24no—
80C25C22⋊C411yescharacteristic
81C23×C4—24no—
82C23×C4—28no—
83C23×C4—28no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—44no—
87C23×C4C22⋊C412yes—
88C23×C4C22⋊C412yes—
89C23×C4C22⋊C412yes—
90C23×C4C22⋊C412yes—
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22×C411yescharacteristic
93C23×C4C22×C412yes—
94C23×C4C2×D412yes—
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C22×C8—28no—
98#117—48no—
99#117—48no—
100#117—44no—
101#117—44no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4C22×C412yes—
107C2×C4⋊C4C22×C414yes—
108C2×C4⋊C4C22×C412yes—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C42—24no—
114C2×C42—28no—
115C2×C42—28no—
116C2×C42—28no—
117C2×C42—28no—
118C2×C42—28no—
119C2×C42—24no—
120C2×C42—24no—
121C2×C42—22no—
122C2×C42—22no—
123C2×C42C22⋊C412yes—
124C2×C42C22⋊C412yes—
125C2×C42C22⋊C411yescharacteristic
126C2×C42C22⋊C411yescharacteristic
127C2×C42C22×C412yes—
128C2×C42C22×C411yescharacteristic
129C2×C42C2×D412yes—
130C2×C42C2×D412yes—
131C2×C42C2×D412yes—
132C2×C42#4118yes—
133C2×C42C2411yescharacteristic
134C4⋊C8—216no—
135C22⋊C8—48no—
136C22⋊C8—48no—
137C22⋊C8—48no—
138C4×C8—48no—
139#95—24no—
140#95—24no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—24no—
145C24—22no—
146C24—24no—
147C24—41no—
148C24—41no—
149C24—42no—
150C24C2×C22⋊C411yescharacteristic
151C24C2×C22⋊C412yes—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—432no—
183C22×C4—416no—
184C22×C4C2×C22⋊C412yes—
185C22×C4C2×C22⋊C412yes—
186C22×C4C2×C22⋊C412yes—
187C22×C4C2×C22⋊C412yes—
188C22×C4C2×C22⋊C411yescharacteristic
189C22×C4#11714yes—
190C22×C4#12318yes—
191C22×C4#13518yes—
192C22×C4C23×C411yescharacteristic
193C22×C4C22×D412yes—
194C2×C8—48no—
195C2×C8—48no—
196C2×C8—48no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C22⋊C4—44no—
205C22⋊C4—44no—
206C22⋊C4—48no—
207C22⋊C4—48no—
208C42—24no—
209C42—24no—
210C42—24no—
211C42—22no—
212C42—24no—
213C42—24no—
214C42—22no—
215C42—48no—
216C42—44no—
217C42—44no—
218C42—48no—
219C42#11714yes—
220C42#12318yes—
221C42C2×#4114yes—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—48no—
233C23—44no—
234C23—44no—
235C23—48no—
236C23—44no—
237C23#38112yes—
238C23#41614yes—
239C23C22×C22⋊C411yescharacteristic
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—216no—
247C2×C4—216no—
248C2×C4—28no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—48no—
256C2×C4—48no—
257C2×C4—48no—
258C2×C4—48no—
259C2×C4—44no—
260C2×C4—48no—
261C2×C4—48no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4#38114yes—
270C2×C4#38112yes—
271C2×C4#41312yes—
272C2×C4#41312yes—
273C2×C4#41614yes—
274C2×C4C2×#11712yes—
275C2×C4C2×#12314yes—
276C2×C4C2×#13514yes—
277C2×C4#57514yes—
278C8—88no—
279C22—24no—
280C22—24no—
281C22—22no—
282C22—24no—
283C22—24no—
284C22—416no—
285C22—41no—
286C22—42no—
287C22—42no—
288C22—41no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22C2×#38111yescharacteristic
293C22C2×#41311yescharacteristic
294C22C2×#41311yescharacteristic
295C22C2×#41612yes—
296C22#289612yes—
297C4—22no—
298C4—24no—
299C4—24no—
300C4—42no—
301C4—42no—
302C4—82no—
303C4—84no—
304C4—82no—
305C4#182214yes—
306C4#309118yes—
307C2—44no—
308C2—42no—
309C2#1770011yescharacteristic
310C2#1771612yes—
311C2#1774011yescharacteristic
312C2C2×#182211yescharacteristic
313C2#2069912yes—
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8727C2semidirect
4. #8726C2semidirect
5. #8038C2semidirect
6. #8038C2semidirect
26. #1296C22semidirect
27. #1296C22semidirect
28. #1296C22semidirect

Maximal subgroups

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

subgroupindex
#340122
#207142
#87272
#87262
#80382