Finite Group Databasebeta

← #160943

Subgroups — #160943

Order 512.

1306 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2296C211yesmaximal, characteristic
2#20714C212yesmaximal
3#7646C214yesmaximal, complemented
4#7641C214yesmaximal, complemented
5#7564C214yesmaximal, complemented
6C22×#117C411yescharacteristic
7C22×#117C411yescharacteristic
8C23×C42C2211yescharacteristic
9C2×#431—22no—
10#2296—22no—
11#2296—22no—
12#2296C412yes—
13#2296C412yes—
14#2296C2212yes—
15#2296C2212yes—
16C2×#389C412yes—
17C2×#389C2212yes—
18C2×#385—22no—
19C2×C4×C4⋊C4—22no—
20#1817C2214yes—
21#1797—22no—
22#1797C2214yes—
23#1779—22no—
24C4×C22⋊C8—22no—
25C4×C22⋊C8C2214yes—
26#1271C22116yescomplemented
27C24×C4C2×C411yescharacteristic
28C24×C4D412yes—
29C2×#117—21no—
30C2×#117—21no—
31C2×#117—28no—
32C2×#117—21no—
33C2×#117—21no—
34C2×#117—44no—
35C2×#117C2×C412yes—
36C2×#117C2×C412yes—
37C2×#117C2×C412yes—
38C2×#117C2×C412yes—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4C2×C412yes—
42C22×C4⋊C4C2×C412yes—
43C22×C22⋊C4—21no—
44C22×C22⋊C4—21no—
45C22×C42—21no—
46C22×C42—21no—
47C22×C42—44no—
48C22×C42C2×C411yescharacteristic
49C22×C42C2×C411yescharacteristic
50C22×C42C2×C411yescharacteristic
51C22×C42D412yes—
52C22×C42D412yes—
53C22×C42D412yes—
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57#431—48no—
58C2×C4⋊C8—24no—
59C2×C22⋊C8—24no—
60C2×C22⋊C8—24no—
61C2×C22⋊C8—24no—
62C2×C4×C8—24no—
63#389—22no—
64#389—22no—
65#389—22no—
66#389—22no—
67#389C2×C414yes—
68#389C2×C414yes—
69#389C2×C414yes—
70#389C2314yes—
71#385—24no—
72#385—24no—
73#385—24no—
74#385—24no—
75C4×C4⋊C4—24no—
76C4×C4⋊C4—24no—
77C4×C4⋊C4—24no—
78C4×C4⋊C4—24no—
79C2×#95—22no—
80C2×#95—22no—
81#336—24no—
82#336C2318yes—
83C25C22⋊C411yescharacteristic
84C23×C4—24no—
85C23×C4—24no—
86C23×C4—24no—
87C23×C4—22no—
88C23×C4—28no—
89C23×C4—22no—
90C23×C4—44no—
91C23×C4—44no—
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C412yes—
94C23×C4C22⋊C412yes—
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C412yes—
98C23×C4C22×C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C22×C411yescharacteristic
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103C23×C4C2×D412yes—
104C22×C8—28no—
105#117—28no—
106#117—28no—
107#117—24no—
108#117—28no—
109#117—44no—
110#117—44no—
111#117—44no—
112#117—44no—
113#117M1618yes—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—48no—
122C2×C4⋊C4—48no—
123C2×C4⋊C4—44no—
124C2×C4⋊C4C22×C414yes—
125C2×C4⋊C4C22×C414yes—
126C2×C22⋊C4—28no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—44no—
132C2×C42—28no—
133C2×C42—28no—
134C2×C42—22no—
135C2×C42—22no—
136C2×C42—22no—
137C2×C42—22no—
138C2×C42—42no—
139C2×C42—42no—
140C2×C42—44no—
141C2×C42—44no—
142C2×C42—42no—
143C2×C42—42no—
144C2×C42—42no—
145C2×C42—42no—
146C2×C42—44no—
147C2×C42C22⋊C412yes—
148C2×C42C22⋊C412yes—
149C2×C42C22⋊C412yes—
150C2×C42C22×C411yescharacteristic
151C2×C42C22×C411yescharacteristic
152C2×C42C22×C411yescharacteristic
153C2×C42C2×D412yes—
154C2×C42C2×D412yes—
155C2×C42C2×D412yes—
156C2×C42C2411yescharacteristic
157C4⋊C8—48no—
158C22⋊C8—48no—
159C22⋊C8—48no—
160C22⋊C8—48no—
161C4×C8—48no—
162#95—24no—
163#95—24no—
164#95—24no—
165#95—24no—
166C24—22no—
167C24—22no—
168C24—22no—
169C24—24no—
170C24—42no—
171C24C2×C22⋊C411yescharacteristic
172C24C2×C22⋊C411yescharacteristic
173C24C2×C22⋊C411yescharacteristic
174C22×C4—232no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—28no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—432no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4#9914yes—
217C22×C4#10114yes—
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C411yescharacteristic
220C22×C4C2×C22⋊C411yescharacteristic
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4C2×M1614yes—
225C22×C4#13518yes—
226C22×C4C23×C411yescharacteristic
227C22×C4C22×D412yes—
228C2×C8—48no—
229C2×C8—48no—
230C2×C8—48no—
231C4⋊C4—28no—
232C4⋊C4—24no—
233C4⋊C4—44no—
234C4⋊C4—48no—
235C4⋊C4—48no—
236C4⋊C4—44no—
237C4⋊C4—48no—
238C4⋊C4—48no—
239C4⋊C4C2×M1618yes—
240C22⋊C4—28no—
241C22⋊C4—28no—
242C22⋊C4—44no—
243C22⋊C4—44no—
244C42—28no—
245C42—28no—
246C42—24no—
247C42—24no—
248C42—24no—
249C42—24no—
250C42—42no—
251C42—41no—
252C42—41no—
253C42—41no—
254C42—42no—
255C42—41no—
256C42—44no—
257C42—44no—
258C23—21no—
259C23—21no—
260C23—22no—
261C23—28no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—48no—
271C23—48no—
272C23—44no—
273C23—44no—
274C23#40712yes—
275C23C2×#9912yes—
276C23C2×#10112yes—
277C23#41614yes—
278C23C22×C22⋊C411yescharacteristic
279C2×C4—28no—
280C2×C4—28no—
281C2×C4—28no—
282C2×C4—216no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—28no—
296C2×C4—28no—
297C2×C4—28no—
298C2×C4—28no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—48no—
303C2×C4—48no—
304C2×C4—48no—
305C2×C4—48no—
306C2×C4—48no—
307C2×C4—416no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—88no—
314C2×C4#40712yes—
315C2×C4C2×#9912yes—
316C2×C4C2×#9912yes—
317C2×C4C2×#10112yes—
318C2×C4C2×#10112yes—
319C2×C4#41614yes—
320C2×C4#42114yes—
321C2×C4C22×M1612yes—
322C2×C4C2×#13514yes—
323C8—88no—
324C22—22no—
325C22—24no—
326C22—24no—
327C22—22no—
328C22—24no—
329C22—24no—
330C22—22no—
331C22—22no—
332C22—416no—
333C22—41no—
334C22—41no—
335C22—42no—
336C22—41no—
337C22—41no—
338C22—42no—
339C22#154414yes—
340C22C2×#40711yescharacteristic
341C22C22×#9911yescharacteristic
342C22C22×#10111yescharacteristic
343C22C2×#41612yes—
344C22C2×#42112yes—
345C4—24no—
346C4—24no—
347C4—24no—
348C4—22no—
349C4—22no—
350C4—22no—
351C4—48no—
352C4—42no—
353C4—84no—
354C4#146314yes—
355C2—24no—
356C2—42no—
357C2—42no—
358C2C2×#146311yescharacteristic
359C2#1744312yes—
360C2#1757212yes—
361C2#1766411yescharacteristic
362C2C2×#154411yescharacteristic
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 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7646C2semidirect
4. #7641C2semidirect
5. #7564C2semidirect
26. #1271C22semidirect

Maximal subgroups

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

subgroupindex
C2×#22962
#207142
#76462
#76412
#75642