Finite Group Databasebeta

← #189894

Subgroups — #189894

Order 512.

1244 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34012C211yesmaximal, characteristic
2C4×#407C212yesmaximal
3#8720C218yesmaximal, complemented
4#8028C212yesmaximal, complemented
5#8028C212yesmaximal, complemented
6C23×C42C2211yescharacteristic
7C2×#407—22no—
8C2×C4×M16C2214yes—
9#2285C412yes—
10#2285C412yes—
11#2285C2211yescharacteristic
12#2285C2211yescharacteristic
13#2283—22no—
14#2283—22no—
15#2283C2214yes—
16#2280—22no—
17#2280C412yes—
18C4×#117—22no—
19C4×#117C412yes—
20#1749C2214yes—
21#1749C2214yes—
22C4×C22⋊C8—22no—
23C4×C22⋊C8—22no—
24#1728—22no—
25#1295C2218yescomplemented
26#1295C2218yescomplemented
27C24×C4C2×C411yescharacteristic
28C24×C4D412yes—
29C22×M16—24no—
30C2×#117—22no—
31C2×#117—22no—
32C2×#117C2×C412yes—
33C2×#117C2×C412yes—
34C22×C42—28no—
35C22×C42C2×C411yescharacteristic
36C22×C42C2×C412yes—
37C22×C42D412yes—
38C22×C42D412yes—
39C22×C42D412yes—
40C22×C42C2312yes—
41C22×C42C2311yescharacteristic
42#407—216no—
43C2×C22⋊C8—24no—
44C2×C22⋊C8—24no—
45C4×M16—216no—
46C2×#97C2318yes—
47C2×C4×C8—24no—
48#384C2×C411yescharacteristic
49#384C2×C412yes—
50#384C2×C411yescharacteristic
51#383—24no—
52#382—22no—
53#382—22no—
54#382C2312yes—
55#382C2312yes—
56#381—22no—
57#381—24no—
58#380—22no—
59#380—22no—
60#380—22no—
61#380—22no—
62#380C2×C414yes—
63#380C2×C414yes—
64#379—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4C2×C414yes—
70C4×C4⋊C4C2×C414yes—
71C4×C22⋊C4—24no—
72C4×C22⋊C4—24no—
73C4×C22⋊C4—24no—
74#376—22no—
75#376—24no—
76C43—22no—
77#336—24no—
78#336—24no—
79C25C22⋊C411yescharacteristic
80C23×C4—28no—
81C23×C4—24no—
82C23×C4—28no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4—44no—
86C23×C4C22⋊C412yes—
87C23×C4C22⋊C412yes—
88C23×C4C22⋊C412yes—
89C23×C4C22⋊C412yes—
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22×C411yescharacteristic
92C23×C4C22×C412yes—
93C23×C4C2×D412yes—
94C23×C4C2×D412yes—
95C23×C4C2×D412yes—
96C2×M16—216no—
97C2×M16—48no—
98C2×M16—48no—
99C22×C8—28no—
100#117—48no—
101#117—48no—
102#117—44no—
103#117—44no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4C22×C412yes—
109C2×C4⋊C4C22×C414yes—
110C2×C4⋊C4C22×C412yes—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C42—24no—
116C2×C42—28no—
117C2×C42—28no—
118C2×C42—28no—
119C2×C42—28no—
120C2×C42—28no—
121C2×C42—24no—
122C2×C42—24no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42C22⋊C412yes—
126C2×C42C22⋊C412yes—
127C2×C42C22⋊C411yescharacteristic
128C2×C42C22⋊C411yescharacteristic
129C2×C42C22×C412yes—
130C2×C42C22×C411yescharacteristic
131C2×C42C2×D412yes—
132C2×C42C2×D412yes—
133C2×C42C2×D412yes—
134C2×C42#4118yes—
135C2×C42C2411yescharacteristic
136C22⋊C8—216no—
137C22⋊C8—216no—
138#97—216no—
139C4×C8—48no—
140#95—24no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—24no—
145#95—24no—
146C24—22no—
147C24—24no—
148C24—41no—
149C24—41no—
150C24—42no—
151C24C2×C22⋊C411yescharacteristic
152C24C2×C22⋊C412yes—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—22no—
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—24no—
173C22×C4—22no—
174C22×C4—24no—
175C22×C4—22no—
176C22×C4—42no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—432no—
184C22×C4—416no—
185C22×C4#10418yes—
186C22×C4C2×C22⋊C412yes—
187C22×C4C2×C22⋊C412yes—
188C22×C4C2×C22⋊C412yes—
189C22×C4C2×C22⋊C412yes—
190C22×C4C2×C22⋊C411yescharacteristic
191C22×C4#11714yes—
192C22×C4#12318yes—
193C22×C4C23×C411yescharacteristic
194C22×C4C22×D412yes—
195M16—432no—
196C2×C8—216no—
197C2×C8—48no—
198C2×C8—48no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C22⋊C4—44no—
207C22⋊C4—44no—
208C22⋊C4—48no—
209C22⋊C4—48no—
210C42—24no—
211C42—24no—
212C42—24no—
213C42—22no—
214C42—24no—
215C42—22no—
216C42—24no—
217C42—48no—
218C42—44no—
219C42—44no—
220C42—48no—
221C42#11714yes—
222C42#12318yes—
223C42C2×#4114yes—
224C23—21no—
225C23—21no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—48no—
235C23—44no—
236C23—44no—
237C23—48no—
238C23—44no—
239C23#38112yes—
240C23C2×#10414yes—
241C23C22×C22⋊C411yescharacteristic
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—28no—
249C2×C4—216no—
250C2×C4—216no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—48no—
258C2×C4—48no—
259C2×C4—48no—
260C2×C4—48no—
261C2×C4—44no—
262C2×C4—48no—
263C2×C4—48no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4#38112yes—
272C2×C4#38114yes—
273C2×C4#41312yes—
274C2×C4#41312yes—
275C2×C4C2×#10414yes—
276C2×C4C2×#10414yes—
277C2×C4#42114yes—
278C2×C4C2×#11712yes—
279C2×C4C2×#12314yes—
280C8—416no—
281C22—22no—
282C22—24no—
283C22—24no—
284C22—24no—
285C22—24no—
286C22—416no—
287C22—41no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—42no—
294C22C2×#38111yescharacteristic
295C22C2×#41311yescharacteristic
296C22C2×#41311yescharacteristic
297C22C22×#10412yes—
298C22C2×#42112yes—
299C4—22no—
300C4—24no—
301C4—24no—
302C4—42no—
303C4—42no—
304C4—82no—
305C4—82no—
306C4—84no—
307C4#182214yes—
308C4#182618yes—
309C2—42no—
310C2—44no—
311C2#1763811yescharacteristic
312C2#1764912yes—
313C2#1766511yescharacteristic
314C2C2×#182211yescharacteristic
315C2C2×#182612yes—
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. #8720C2semidirect
4. #8028C2semidirect
5. #8028C2semidirect
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
C4×#4072
#87202
#80282