Finite Group Databasebeta

← #189972

Subgroups — #189972

Order 512.

1354 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic
2#20626C212yesmaximal
3#8728C214yesmaximal, complemented
4#8725C212yesmaximal, complemented
5#8723C212yesmaximal, complemented
6#8041C214yesmaximal, complemented
7C22×#117C412yes—
8C23×C42C2211yescharacteristic
9C2×#423C2214yes—
10C2×#407—22no—
11#2285—22no—
12#2285—22no—
13#2285C412yes—
14#2285C412yes—
15#2285C2212yes—
16#2285C2212yes—
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#382C412yes—
20C2×#380—22no—
21C2×C4×C4⋊C4—22no—
22#1797—22no—
23#1797—22no—
24#1797C2214yes—
25#1797C2214yes—
26#1728—22no—
27#1296C2218yescomplemented
28#1296C2218yescomplemented
29C24×C4C2×C411yescharacteristic
30C24×C4D412yes—
31C22×M16—24no—
32C2×#117—22no—
33C2×#117—22no—
34C2×#117—48no—
35C2×#117C2×C414yes—
36C2×#117C2×C414yes—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4C2×C412yes—
40C22×C4⋊C4C2×C412yes—
41C22×C22⋊C4—22no—
42C22×C42—28no—
43C22×C42—22no—
44C22×C42C2×C411yescharacteristic
45C22×C42C2×C411yescharacteristic
46C22×C42C2×C411yescharacteristic
47C22×C42D412yes—
48C22×C42D412yes—
49C22×C42D412yes—
50C22×C42C2311yescharacteristic
51C22×C42C2311yescharacteristic
52C22×C42C2311yescharacteristic
53#423—216no—
54C2×C4⋊C8—24no—
55C2×C4⋊C8C2318yes—
56#407—48no—
57C2×C22⋊C8—24no—
58C2×C22⋊C8—24no—
59#384—22no—
60#384—22no—
61#384C2×C412yes—
62#384C2×C412yes—
63#384C2312yes—
64#384C2312yes—
65#382—24no—
66#382C2×C414yes—
67#382C2×C414yes—
68#380—24no—
69#380—24no—
70#380—24no—
71#380—24no—
72C4×C4⋊C4—24no—
73C4×C4⋊C4—24no—
74C4×C4⋊C4—24no—
75C4×C4⋊C4—24no—
76C2×#95—24no—
77#336—24no—
78#336—24no—
79C25C22⋊C411yescharacteristic
80C23×C4—24no—
81C23×C4—24no—
82C23×C4—24no—
83C23×C4—28no—
84C23×C4—24no—
85C23×C4—24no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—44no—
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22⋊C412yes—
91C23×C4C22⋊C412yes—
92C23×C4C22⋊C412yes—
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22×C411yescharacteristic
96C23×C4C22×C411yescharacteristic
97C23×C4C22×C411yescharacteristic
98C23×C4C2×D412yes—
99C23×C4C2×D412yes—
100C23×C4C2×D412yes—
101C2×M16—48no—
102C2×M16—48no—
103C2×M16—48no—
104C22×C8—28no—
105#117—48no—
106#117—48no—
107#117—48no—
108#117—48no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—44no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—48no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4C22×C414yes—
121C2×C4⋊C4C22×C414yes—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—48no—
125C2×C42—24no—
126C2×C42—28no—
127C2×C42—24no—
128C2×C42—24no—
129C2×C42—28no—
130C2×C42—28no—
131C2×C42—24no—
132C2×C42—24no—
133C2×C42—24no—
134C2×C42—24no—
135C2×C42—48no—
136C2×C42C22⋊C412yes—
137C2×C42C22⋊C412yes—
138C2×C42C22⋊C412yes—
139C2×C42C22×C411yescharacteristic
140C2×C42C22×C411yescharacteristic
141C2×C42C22×C411yescharacteristic
142C2×C42C2×D412yes—
143C2×C42C2×D412yes—
144C2×C42C2×D412yes—
145C2×C42#4118yes—
146C2×C42C2411yescharacteristic
147C4⋊C8—216no—
148C4⋊C8—48no—
149C22⋊C8—48no—
150C22⋊C8—48no—
151#95—28no—
152#95—28no—
153C24—22no—
154C24—22no—
155C24—22no—
156C24—24no—
157C24—42no—
158C24C2×C22⋊C411yescharacteristic
159C24C2×C22⋊C411yescharacteristic
160C24C2×C22⋊C411yescharacteristic
161C22×C4—232no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—416no—
198C22×C4—416no—
199C22×C4—48no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—48no—
203C22×C4#10014yes—
204C22×C4#10114yes—
205C22×C4C2×C22⋊C412yes—
206C22×C4C2×C22⋊C412yes—
207C22×C4C2×C22⋊C411yescharacteristic
208C22×C4C2×C22⋊C411yescharacteristic
209C22×C4C2×C22⋊C412yes—
210C22×C4C2×C22⋊C411yescharacteristic
211C22×C4#11714yes—
212C22×C4#12318yes—
213C22×C4AGL(1,ℤ/8)14yes—
214C22×C4#13714yes—
215C22×C4C23×C411yescharacteristic
216C22×C4C22×D412yes—
217M16—816no—
218C2×C8—48no—
219C2×C8—48no—
220C2×C8—48no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—48no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—48no—
229C4⋊C4—48no—
230C22⋊C4—48no—
231C22⋊C4—48no—
232C42—22no—
233C42—22no—
234C42—24no—
235C42—24no—
236C42—24no—
237C42—24no—
238C42—24no—
239C42—48no—
240C42—48no—
241C42#11714yes—
242C42#12318yes—
243C42C2×#4114yes—
244C23—21no—
245C23—22no—
246C23—21no—
247C23—28no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—21no—
256C23—48no—
257C23—48no—
258C23—44no—
259C23—44no—
260C23#38112yes—
261C23C2×#10012yes—
262C23C2×#10112yes—
263C23#41714yes—
264C23C22×C22⋊C411yescharacteristic
265C2×C4—28no—
266C2×C4—28no—
267C2×C4—28no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—28no—
279C2×C4—28no—
280C2×C4—28no—
281C2×C4—28no—
282C2×C4—28no—
283C2×C4—28no—
284C2×C4—28no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—48no—
289C2×C4—48no—
290C2×C4—48no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—48no—
296C2×C4—48no—
297C2×C4—88no—
298C2×C4—88no—
299C2×C4#38114yes—
300C2×C4#38112yes—
301C2×C4C2×#10012yes—
302C2×C4C2×#10012yes—
303C2×C4C2×#10112yes—
304C2×C4C2×#10112yes—
305C2×C4#41714yes—
306C2×C4C2×#11712yes—
307C2×C4C2×#12314yes—
308C2×C4C2×AGL(1,ℤ/8)12yes—
309C2×C4C2×#13712yes—
310C2×C4#57514yes—
311C8—88no—
312C22—22no—
313C22—24no—
314C22—22no—
315C22—24no—
316C22—24no—
317C22—22no—
318C22—22no—
319C22—24no—
320C22—416no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—42no—
326C22—42no—
327C22#156014yes—
328C22C2×#38111yescharacteristic
329C22C22×#10011yescharacteristic
330C22C22×#10111yescharacteristic
331C22C2×#41712yes—
332C22#289612yes—
333C4—22no—
334C4—22no—
335C4—22no—
336C4—24no—
337C4—24no—
338C4—42no—
339C4—84no—
340C4—84no—
341C4#182114yes—
342C4#309414yes—
343C4#309514yes—
344C2—24no—
345C2—42no—
346C2—42no—
347C2C2×#156011yescharacteristic
348C2#1773811yescharacteristic
349C2#1775112yes—
350C2C2×#182111yescharacteristic
351C2#2070212yes—
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. #8728C2semidirect
4. #8725C2semidirect
5. #8723C2semidirect
6. #8041C2semidirect
27. #1296C22semidirect
28. #1296C22semidirect

Maximal subgroups

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

subgroupindex
C2×#22852
#206262
#87282
#87252
#87232
#80412