Finite Group Databasebeta

← #189918

Subgroups — #189918

Order 512.

1108 conjugacy classes of subgroups.

The subgroup lattice has 313 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34012C211yesmaximal, characteristic
2#20715C212yesmaximal
3#8720C218yesmaximal, complemented
4#8031C212yesmaximal, complemented
5#8031C212yesmaximal, complemented
6C23×C42C2211yescharacteristic
7C2×#432C2214yes—
8#2285C412yes—
9#2285C412yes—
10#2285C2211yescharacteristic
11#2285C2211yescharacteristic
12#2283—22no—
13#2283—22no—
14#2283C2214yes—
15#2280—22no—
16#2280C412yes—
17C4×#117—22no—
18C4×#117C412yes—
19#1817—22no—
20#1797—22no—
21#1797—22no—
22#1779—22no—
23#1749C2214yes—
24#1749C2214yes—
25#1295C2218yescomplemented
26#1295C2218yescomplemented
27C24×C4C2×C411yescharacteristic
28C24×C4D412yes—
29C2×#117—22no—
30C2×#117—22no—
31C2×#117C2×C412yes—
32C2×#117C2×C412yes—
33C22×C42—28no—
34C22×C42C2×C411yescharacteristic
35C22×C42C2×C412yes—
36C22×C42D412yes—
37C22×C42D412yes—
38C22×C42D412yes—
39C22×C42C2312yes—
40C22×C42C2311yescharacteristic
41#432—216no—
42C2×C4⋊C8—24no—
43C2×C22⋊C8—24no—
44C2×C22⋊C8—24no—
45C2×C22⋊C8—24no—
46C2×#97C2318yes—
47#384C2×C411yescharacteristic
48#384C2×C412yes—
49#384C2×C411yescharacteristic
50#383—24no—
51#382—22no—
52#382—22no—
53#382C2312yes—
54#382C2312yes—
55#381—22no—
56#381—24no—
57#380—22no—
58#380—22no—
59#380—22no—
60#380—22no—
61#380C2×C414yes—
62#380C2×C414yes—
63#379—22no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4C2×C414yes—
69C4×C4⋊C4C2×C414yes—
70C4×C22⋊C4—24no—
71C4×C22⋊C4—24no—
72C4×C22⋊C4—24no—
73#376—22no—
74#376—24no—
75C43—22no—
76#336—24no—
77#336—24no—
78C25C22⋊C411yescharacteristic
79C23×C4—24no—
80C23×C4—28no—
81C23×C4—42no—
82C23×C4—42no—
83C23×C4—44no—
84C23×C4—44no—
85C23×C4C22⋊C412yes—
86C23×C4C22⋊C412yes—
87C23×C4C22⋊C412yes—
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C412yes—
90C23×C4C22×C411yescharacteristic
91C23×C4C22×C412yes—
92C23×C4C2×D412yes—
93C23×C4C2×D412yes—
94C23×C4C2×D412yes—
95C22×C8—28no—
96#117—48no—
97#117—48no—
98#117—44no—
99#117—44no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4C22×C412yes—
105C2×C4⋊C4C22×C412yes—
106C2×C4⋊C4C22×C414yes—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—22no—
111C2×C42—24no—
112C2×C42—28no—
113C2×C42—28no—
114C2×C42—28no—
115C2×C42—28no—
116C2×C42—28no—
117C2×C42—24no—
118C2×C42—24no—
119C2×C42—22no—
120C2×C42—22no—
121C2×C42C22⋊C411yescharacteristic
122C2×C42C22⋊C411yescharacteristic
123C2×C42C22⋊C412yes—
124C2×C42C22⋊C412yes—
125C2×C42C22×C412yes—
126C2×C42C22×C411yescharacteristic
127C2×C42C2×D412yes—
128C2×C42C2×D412yes—
129C2×C42C2×D412yes—
130C2×C42#4118yes—
131C2×C42C2411yescharacteristic
132C4⋊C8—48no—
133C22⋊C8—48no—
134C22⋊C8—48no—
135C22⋊C8—48no—
136#97—216no—
137#95—24no—
138#95—24no—
139#95—24no—
140#95—24no—
141#95—24no—
142#95—24no—
143C24—22no—
144C24—24no—
145C24—41no—
146C24—41no—
147C24—42no—
148C24C2×C22⋊C411yescharacteristic
149C24C2×C22⋊C412yes—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—432no—
182C22×C4—416no—
183C22×C4C2×C22⋊C412yes—
184C22×C4C2×C22⋊C412yes—
185C22×C4C2×C22⋊C412yes—
186C22×C4C2×C22⋊C412yes—
187C22×C4C2×C22⋊C411yescharacteristic
188C22×C4#11714yes—
189C22×C4#12318yes—
190C22×C4C23×C411yescharacteristic
191C22×C4C22×D412yes—
192C2×C8—48no—
193C2×C8—48no—
194C2×C8—48no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C22⋊C4—44no—
203C22⋊C4—44no—
204C22⋊C4—48no—
205C22⋊C4—48no—
206C42—24no—
207C42—24no—
208C42—24no—
209C42—22no—
210C42—24no—
211C42—24no—
212C42—22no—
213C42—48no—
214C42—44no—
215C42—44no—
216C42—48no—
217C42#11714yes—
218C42#12318yes—
219C42C2×#4114yes—
220C23—21no—
221C23—21no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—48no—
231C23—44no—
232C23—44no—
233C23—48no—
234C23—44no—
235C23#38112yes—
236C23#42114yes—
237C23C22×C22⋊C411yescharacteristic
238C2×C4—216no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—28no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—48no—
255C2×C4—48no—
256C2×C4—48no—
257C2×C4—48no—
258C2×C4—48no—
259C2×C4—48no—
260C2×C4—44no—
261C2×C4—48no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—44no—
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#42114yes—
274C2×C4C2×#11712yes—
275C2×C4C2×#12314yes—
276C8—88no—
277C22—24no—
278C22—22no—
279C22—24no—
280C22—24no—
281C22—24no—
282C22—416no—
283C22—41no—
284C22—42no—
285C22—41no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22C2×#38111yescharacteristic
291C22C2×#41311yescharacteristic
292C22C2×#41311yescharacteristic
293C22C2×#42112yes—
294C22C2×#42112yes—
295C4—22no—
296C4—24no—
297C4—24no—
298C4—24no—
299C4—42no—
300C4—42no—
301C4—82no—
302C4—82no—
303C4—84no—
304C4#182214yes—
305C2—44no—
306C2—42no—
307C2#1766212yes—
308C2#1767311yescharacteristic
309C2#1767811yescharacteristic
310C2C2×#182211yescharacteristic
311C2#2070512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8720C2semidirect
4. #8031C2semidirect
5. #8031C2semidirect
25. #1295C22semidirect
26. #1295C22semidirect

Maximal subgroups

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

subgroupindex
#340122
#207152
#87202
#80312