Finite Group Databasebeta

← #480288

Subgroups — #480288

Order 512.

1186 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C2×#1463C212yesmaximal
3#11514C214yesmaximal, complemented
4#11512C212yesmaximal, complemented
5#11511C212yesmaximal, complemented
6#11448C214yesmaximal, complemented
7C22×#117C411yescharacteristic
8C22×#117C411yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10C2×#407—22no—
11#2317—22no—
12#2317—22no—
13#2317C412yes—
14#2317C412yes—
15#2317C2212yes—
16#2317C2212yes—
17C2×#389—22no—
18C2×#385—22no—
19C2×#384C411yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#382C411yescharacteristic
22C2×#382C2211yescharacteristic
23#1463—22no—
24#1463—22no—
25#1463C2214yes—
26#1463C2214yes—
27C2×#324—22no—
28C2×#324C2214yes—
29#1325C2218yescomplemented
30#1324C2218yescomplemented
31C24×C4C2×C411yescharacteristic
32C24×C4D412yes—
33C22×M16—42no—
34C2×#117—21no—
35C2×#117—21no—
36C2×#117—28no—
37C2×#117—21no—
38C2×#117—21no—
39C2×#117—42no—
40C2×#117—42no—
41C2×#117C2×C412yes—
42C2×#117C2×C412yes—
43C2×#117C2×C412yes—
44C2×#117C2×C412yes—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—42no—
49C22×C4⋊C4—42no—
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4D412yes—
56C22×C4⋊C4D412yes—
57C22×C4⋊C4D412yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C42—21no—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66#407—24no—
67#407—24no—
68#407—42no—
69#407—42no—
70C2×C22⋊C8—24no—
71C2×C22⋊C8—42no—
72#389—24no—
73#389—24no—
74#389—24no—
75#389—24no—
76#385—24no—
77#385—24no—
78#385—24no—
79#385—24no—
80#384—22no—
81#384—22no—
82#384C2×C412yes—
83#384C2×C412yes—
84#384C2×C412yes—
85#384C2312yes—
86#382—22no—
87#382—22no—
88#382C2×C412yes—
89#382C2×C412yes—
90#382C2×C412yes—
91#382C2312yes—
92C2×#95—22no—
93C2×#95—22no—
94#324—24no—
95#324—24no—
96#324—24no—
97#324C2318yes—
98C25C22⋊C411yescharacteristic
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—24no—
102C23×C4—24no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—84no—
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C411yescharacteristic
115C23×C4C22⋊C412yes—
116C23×C4C22⋊C412yes—
117C23×C4C22⋊C412yes—
118C23×C4C22×C411yescharacteristic
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C2×M16—44no—
125C2×M16—44no—
126C2×M16—88no—
127C22×C8—44no—
128#117—28no—
129#117—28no—
130#117—24no—
131#117—24no—
132#117—24no—
133#117—44no—
134#117—44no—
135#117—44no—
136#117—44no—
137#117M1618yescomplemented
138C2×C4⋊C4—28no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—84no—
166C2×C4⋊C4—84no—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C411yescharacteristic
173C2×C4⋊C4C22×C411yescharacteristic
174C2×C4⋊C4C22×C411yescharacteristic
175C2×C4⋊C4C2×D412yes—
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2411yescharacteristic
179C2×C22⋊C4—28no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—42no—
193C2×C42—42no—
194C2×C42C22×C412yes—
195C2×C42C22×C412yes—
196C22⋊C8—28no—
197C22⋊C8—44no—
198C22⋊C8—44no—
199C22⋊C8—44no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—24no—
204C24—22no—
205C24—22no—
206C24—42no—
207C24—42no—
208C24#9912yes—
209C24#9912yes—
210C24C2×C22⋊C411yescharacteristic
211C24C2×C22⋊C411yescharacteristic
212C24C2×C22⋊C411yescharacteristic
213C22×C4—216no—
214C22×C4—216no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—48no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—84no—
269C22×C4—84no—
270C22×C4—84no—
271C22×C4#10114yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×M1614yes—
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D412yes—
281M16—88no—
282M16—88no—
283C2×C8—48no—
284C2×C8—88no—
285C4⋊C4—28no—
286C4⋊C4—28no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—84no—
302C4⋊C4—84no—
303C4⋊C4—84no—
304C4⋊C4—84no—
305C22⋊C4—28no—
306C22⋊C4—28no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C42—24no—
310C42—24no—
311C42—22no—
312C42—22no—
313C42—44no—
314C42—44no—
315C42C2×M1614yescomplemented
316C42C2×M1614yescomplemented
317C23—21no—
318C23—21no—
319C23—22no—
320C23—22no—
321C23—21no—
322C23—21no—
323C23—24no—
324C23—24no—
325C23—24no—
326C23—24no—
327C23—42no—
328C23—41no—
329C23—41no—
330C23—42no—
331C23—41no—
332C23—41no—
333C23—48no—
334C23—48no—
335C23#40712yes—
336C23C2×#9912yes—
337C23C2×#9911yescharacteristic
338C23C2×#9911yescharacteristic
339C23C2×#9912yes—
340C23C2×#10112yes—
341C23C22×C22⋊C411yescharacteristic
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—28no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—28no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—28no—
357C2×C4—28no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—48no—
369C2×C4—48no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4#40712yes—
395C2×C4C2×#10112yes—
396C2×C4C2×#10112yes—
397C2×C4C22×M1612yes—
398C8—88no—
399C22—24no—
400C22—24no—
401C22—21no—
402C22—22no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—22no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—42no—
413C22—42no—
414C22—48no—
415C22—48no—
416C22—42no—
417C22—42no—
418C22#146314yes—
419C22#212712yes—
420C22#213312yes—
421C22C2×#40711yescharacteristic
422C22C22×#9911yescharacteristic
423C22C22×#10111yescharacteristic
424C4—22no—
425C4—22no—
426C4—22no—
427C4—22no—
428C4—44no—
429C4—44no—
430C4—42no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—82no—
435C4—82no—
436C2—21no—
437C2—21no—
438C2—22no—
439C2—22no—
440C2—24no—
441C2—42no—
442C2C2×#146311yescharacteristic
443C2C2×#212711yescharacteristic
444C2C2×#213311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11514C2semidirect
4. #11512C2semidirect
5. #11511C2semidirect
6. #11448C2semidirect
29. #1325C22semidirect
30. #1324C22semidirect
137. #117M16semidirect
315. C42C2×M16semidirect
316. C42C2×M16semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#14632
#115142
#115122
#115112
#114482